3.1.46 \(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x))}{f+g x} \, dx\) [46]

Optimal. Leaf size=1536 \[ \frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}}{g^5}+\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2}}{g^5 \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}} \]

[Out]

1/4*b*c^3*d^2*f*(c^2*f^2+2*g^2)*x^2*(c^2*d*x^2+d)^(1/2)/g^4/(c^2*x^2+1)^(1/2)-1/4*c*d^2*f*(c^2*f^2+2*g^2)*(a+b
*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/g^4/(c^2*x^2+1)^(1/2)+b*d^2*(c^2*f^2+g^2)^2*arcsinh(c*x)*(c^2*d*x^2+d)^
(1/2)/g^5+1/3*d^2*(c^2*f^2+2*g^2)*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/g^3-1/2*c^2*d^2*f*(c^2*f^
2+2*g^2)*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/g^4-b*c*d^2*(c^2*f^2+g^2)^2*x*(c^2*d*x^2+d)^(1/2)/g^5/(c^2*x
^2+1)^(1/2)-1/3*b*c*d^2*(c^2*f^2+2*g^2)*x*(c^2*d*x^2+d)^(1/2)/g^3/(c^2*x^2+1)^(1/2)+1/16*b*c^3*d^2*f*x^2*(c^2*
d*x^2+d)^(1/2)/g^2/(c^2*x^2+1)^(1/2)-1/9*b*c^3*d^2*(c^2*f^2+2*g^2)*x^3*(c^2*d*x^2+d)^(1/2)/g^3/(c^2*x^2+1)^(1/
2)+1/16*b*c^5*d^2*f*x^4*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*x^2+1)^(1/2)+1/16*c*d^2*f*(a+b*arcsinh(c*x))^2*(c^2*d*x^2
+d)^(1/2)/b/g^2/(c^2*x^2+1)^(1/2)+b*d^2*(c^2*f^2+g^2)^(5/2)*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(
c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^6/(c^2*x^2+1)^(1/2)-b*d^2*(c^2*f^2+g^2)^(5/2)*arcsinh(c*x)*ln(1+(c*
x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^6/(c^2*x^2+1)^(1/2)-1/3*d^2*(c^2*x^2+1
)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/g+1/5*d^2*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/g+a*d^
2*(c^2*f^2+g^2)^2*(c^2*d*x^2+d)^(1/2)/g^5-1/2*c*d^2*(c^2*f^2+g^2)^2*x*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)
/b/g^5/(c^2*x^2+1)^(1/2)-1/2*d^2*(c^2*f^2+g^2)^3*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c/g^6/(g*x+f)/(c^2
*x^2+1)^(1/2)+1/2*d^2*(c^2*f^2+g^2)^2*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)/b/c/g^4/(g*x+
f)-1/4*c^4*d^2*f*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/g^2+2/15*b*c*d^2*x*(c^2*d*x^2+d)^(1/2)/g/(c^2*x^2+
1)^(1/2)-1/45*b*c^3*d^2*x^3*(c^2*d*x^2+d)^(1/2)/g/(c^2*x^2+1)^(1/2)-1/25*b*c^5*d^2*x^5*(c^2*d*x^2+d)^(1/2)/g/(
c^2*x^2+1)^(1/2)-a*d^2*(c^2*f^2+g^2)^(5/2)*arctanh((-c^2*f*x+g)/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2))*(c^2*d*
x^2+d)^(1/2)/g^6/(c^2*x^2+1)^(1/2)+b*d^2*(c^2*f^2+g^2)^(5/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^
2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^6/(c^2*x^2+1)^(1/2)-b*d^2*(c^2*f^2+g^2)^(5/2)*polylog(2,-(c*x+(c^2*x^2+1)
^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^6/(c^2*x^2+1)^(1/2)-1/8*c^2*d^2*f*x*(a+b*arcsinh(c*
x))*(c^2*d*x^2+d)^(1/2)/g^2

________________________________________________________________________________________

Rubi [A]
time = 1.78, antiderivative size = 1536, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 29, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.967, Rules used = {5845, 5840, 5785, 5783, 30, 5798, 5806, 5812, 272, 45, 5804, 12, 5839, 697, 5835, 6874, 267, 739, 212, 5856, 1668, 5855, 8, 5843, 3403, 2296, 2221, 2317, 2438} \begin {gather*} -\frac {b d^2 x^5 \sqrt {c^2 d x^2+d} c^5}{25 g \sqrt {c^2 x^2+1}}+\frac {b d^2 f x^4 \sqrt {c^2 d x^2+d} c^5}{16 g^2 \sqrt {c^2 x^2+1}}-\frac {d^2 f x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) c^4}{4 g^2}-\frac {b d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {c^2 d x^2+d} c^3}{9 g^3 \sqrt {c^2 x^2+1}}-\frac {b d^2 x^3 \sqrt {c^2 d x^2+d} c^3}{45 g \sqrt {c^2 x^2+1}}+\frac {b d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {c^2 d x^2+d} c^3}{4 g^4 \sqrt {c^2 x^2+1}}+\frac {b d^2 f x^2 \sqrt {c^2 d x^2+d} c^3}{16 g^2 \sqrt {c^2 x^2+1}}-\frac {d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) c^2}{2 g^4}-\frac {d^2 f x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) c^2}{8 g^2}-\frac {d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{4 b g^4 \sqrt {c^2 x^2+1}}-\frac {d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{2 b g^5 \sqrt {c^2 x^2+1}}+\frac {d^2 f \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{16 b g^2 \sqrt {c^2 x^2+1}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {c^2 d x^2+d} c}{g^5 \sqrt {c^2 x^2+1}}-\frac {b d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {c^2 d x^2+d} c}{3 g^3 \sqrt {c^2 x^2+1}}+\frac {2 b d^2 x \sqrt {c^2 d x^2+d} c}{15 g \sqrt {c^2 x^2+1}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{g^5}+\frac {d^2 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}-\frac {d^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {a d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {c^2 x^2+1}}\right )}{g^6 \sqrt {c^2 x^2+1}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^6 \sqrt {c^2 x^2+1}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}+1\right )}{g^6 \sqrt {c^2 x^2+1}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {c^2 d x^2+d} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {c^2 x^2+1}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {c^2 d x^2+d} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {c^2 x^2+1}}+\frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {c^2 d x^2+d}}{g^5}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) c}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^6 (f+g x) \sqrt {c^2 x^2+1} c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(f + g*x),x]

[Out]

(a*d^2*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2])/g^5 + (2*b*c*d^2*x*Sqrt[d + c^2*d*x^2])/(15*g*Sqrt[1 + c^2*x^2])
- (b*c*d^2*(c^2*f^2 + g^2)^2*x*Sqrt[d + c^2*d*x^2])/(g^5*Sqrt[1 + c^2*x^2]) - (b*c*d^2*(c^2*f^2 + 2*g^2)*x*Sq
rt[d + c^2*d*x^2])/(3*g^3*Sqrt[1 + c^2*x^2]) + (b*c^3*d^2*f*x^2*Sqrt[d + c^2*d*x^2])/(16*g^2*Sqrt[1 + c^2*x^2]
) + (b*c^3*d^2*f*(c^2*f^2 + 2*g^2)*x^2*Sqrt[d + c^2*d*x^2])/(4*g^4*Sqrt[1 + c^2*x^2]) - (b*c^3*d^2*x^3*Sqrt[d
+ c^2*d*x^2])/(45*g*Sqrt[1 + c^2*x^2]) - (b*c^3*d^2*(c^2*f^2 + 2*g^2)*x^3*Sqrt[d + c^2*d*x^2])/(9*g^3*Sqrt[1 +
c^2*x^2]) + (b*c^5*d^2*f*x^4*Sqrt[d + c^2*d*x^2])/(16*g^2*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^5*Sqrt[d + c^2*d*
x^2])/(25*g*Sqrt[1 + c^2*x^2]) + (b*d^2*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/g^5 - (c^2*d^2*f*x
*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*g^2) - (c^2*d^2*f*(c^2*f^2 + 2*g^2)*x*Sqrt[d + c^2*d*x^2]*(a + b
*ArcSinh[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(4*g^2) - (d^2*(1 + c^2*x^2
)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*g) + (d^2*(c^2*f^2 + 2*g^2)*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(
a + b*ArcSinh[c*x]))/(3*g^3) + (d^2*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(5*g) + (c*d^2*f
*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*g^2*Sqrt[1 + c^2*x^2]) - (c*d^2*f*(c^2*f^2 + 2*g^2)*Sqrt[d
+ c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*g^4*Sqrt[1 + c^2*x^2]) - (c*d^2*(c^2*f^2 + g^2)^2*x*Sqrt[d + c^2*d*x
^2]*(a + b*ArcSinh[c*x])^2)/(2*b*g^5*Sqrt[1 + c^2*x^2]) - (d^2*(c^2*f^2 + g^2)^3*Sqrt[d + c^2*d*x^2]*(a + b*Ar
cSinh[c*x])^2)/(2*b*c*g^6*(f + g*x)*Sqrt[1 + c^2*x^2]) + (d^2*(c^2*f^2 + g^2)^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2
*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)) - (a*d^2*(c^2*f^2 + g^2)^(5/2)*Sqrt[d + c^2*d*x^2]*ArcTa
nh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/(g^6*Sqrt[1 + c^2*x^2]) + (b*d^2*(c^2*f^2 + g^2)^(5
/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^6*Sqrt[1 + c^
2*x^2]) - (b*d^2*(c^2*f^2 + g^2)^(5/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt
[c^2*f^2 + g^2])])/(g^6*Sqrt[1 + c^2*x^2]) + (b*d^2*(c^2*f^2 + g^2)^(5/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^
ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^6*Sqrt[1 + c^2*x^2]) - (b*d^2*(c^2*f^2 + g^2)^(5/2)*Sqrt[d +
c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^6*Sqrt[1 + c^2*x^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 697

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rule 5785

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcSinh[c*x])^n/2), x] + (Dist[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[(a + b*ArcSinh[c*x])^
n/Sqrt[1 + c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[x*(a + b*ArcSinh[c*x
])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5798

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e
, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 5804

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[
SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p -
1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 5806

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(
f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/(f*(m + 2))), x] + (Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2]
/Sqrt[1 + c^2*x^2]], Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x], x] - Dist[b*c*(n/(f*(m + 2)))
*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a,
b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 5812

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Dist[f^2*((m - 1)/(c^2*
(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)
))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0
]

Rule 5835

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*(x_)^2)^(p_.))/((d_) + (e_.)*(x_))^2
, x_Symbol] :> With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcSinh[c*x])^n, u, x] - Di
st[b*c*n, Int[SimplifyIntegrand[u*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x], x]] /; FreeQ[{a, b
, c, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]

Rule 5839

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.) + (g_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :
> Simp[(f + g*x)^m*(d + e*x^2)*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] - Dist[1/(b*c*Sqrt[d]*
(n + 1)), Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; Fr
eeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 5840

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Int[ExpandIntegrand[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n, (f + g*x)^m*(d + e*x^2)^(p - 1/2), x], x] /; Fr
eeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IGtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 5843

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{
a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 5845

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f + g*x)^m*(1 + c^2*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] /;
FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && IntegerQ[p - 1/2] &&  !GtQ[d, 0]

Rule 5855

Int[ArcSinh[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e
*x^2)^p*ArcSinh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] &&
IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]

Rule 5856

Int[(ArcSinh[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x^2)^p, RFx*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x]
&& IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {\left (-c^4 f^3-2 c^2 f g^2\right ) \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g^4}+\frac {c^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g^3}-\frac {c^4 f x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g^2}+\frac {c^4 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g}+\frac {\left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g^4 (f+g x)}\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {\left (c^4 d^2 f \sqrt {d+c^2 d x^2}\right ) \int x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{g \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx}{g^4 \sqrt {1+c^2 x^2}}-\frac {\left (c^2 d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{g^4 \sqrt {1+c^2 x^2}}+\frac {\left (c^2 d^2 \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2}\right ) \int x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{g^3 \sqrt {1+c^2 x^2}}\\ &=-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {\left (c^4 d^2 f \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{4 g^2 \sqrt {1+c^2 x^2}}+\frac {\left (b c^5 d^2 f \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {\left (b c^5 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {-2+c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{g \sqrt {1+c^2 x^2}}-\frac {\left (d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (-g+2 c^2 f x+c^2 g x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c g^4 \sqrt {1+c^2 x^2}}-\frac {\left (c^2 d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 g^4 \sqrt {1+c^2 x^2}}+\frac {\left (b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{2 g^4 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 g^3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}+\frac {\left (c^2 d^2 f \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 g^2 \sqrt {1+c^2 x^2}}+\frac {\left (b c^3 d^2 f \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{8 g^2 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+3 c^4 x^4\right ) \, dx}{15 g \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (\frac {c^2 x}{g}+\frac {1+\frac {c^2 f^2}{g^2}}{f+g x}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{g^4 \sqrt {1+c^2 x^2}}\\ &=\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}+\frac {\left (d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {a \left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt {1+c^2 x^2}}+\frac {b \left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right ) \sinh ^{-1}(c x)}{g^2 (f+g x) \sqrt {1+c^2 x^2}}\right ) \, dx}{g^4 \sqrt {1+c^2 x^2}}\\ &=\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}+\frac {\left (a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{g^6 \sqrt {1+c^2 x^2}}+\frac {\left (b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right ) \sinh ^{-1}(c x)}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{g^6 \sqrt {1+c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}}{g^5}+\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}+\frac {\left (a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {c^2 g^2 \left (c^2 f^2+g^2\right )}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{c^2 g^8 \sqrt {1+c^2 x^2}}+\frac {\left (b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {c^2 g x \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}+\frac {\left (c^2 f^2+g^2\right ) \sinh ^{-1}(c x)}{(f+g x) \sqrt {1+c^2 x^2}}\right ) \, dx}{g^6 \sqrt {1+c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}}{g^5}+\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}+\frac {\left (b c^2 d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{g^5 \sqrt {1+c^2 x^2}}+\frac {\left (a d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{g^6 \sqrt {1+c^2 x^2}}+\frac {\left (b d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {\sinh ^{-1}(c x)}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{g^6 \sqrt {1+c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}}{g^5}+\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {\left (b c d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}\right ) \int 1 \, dx}{g^5 \sqrt {1+c^2 x^2}}-\frac {\left (a d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{c^2 f^2+g^2-x^2} \, dx,x,\frac {g-c^2 f x}{\sqrt {1+c^2 x^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {\left (b d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{g^6 \sqrt {1+c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}}{g^5}+\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2}}{g^5 \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {\left (2 b d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c e^x f-g+e^{2 x} g} \, dx,x,\sinh ^{-1}(c x)\right )}{g^6 \sqrt {1+c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}}{g^5}+\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2}}{g^5 \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {\left (2 b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g-2 \sqrt {c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )}{g^5 \sqrt {1+c^2 x^2}}-\frac {\left (2 b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g+2 \sqrt {c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )}{g^5 \sqrt {1+c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}}{g^5}+\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2}}{g^5 \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}-\frac {\left (b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f-2 \sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {\left (b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f+2 \sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{g^6 \sqrt {1+c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}}{g^5}+\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2}}{g^5 \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}-\frac {\left (b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f-2 \sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {\left (b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f+2 \sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{g^6 \sqrt {1+c^2 x^2}}\\ &=\frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}}{g^5}+\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2}}{g^5 \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^4 (f+g x)}-\frac {a d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 22.44, size = 7163, normalized size = 4.66 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(f + g*x),x]

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3927\) vs. \(2(1420)=2840\).
time = 2.64, size = 3928, normalized size = 2.56

method	result	size
default	\(\text {Expression too large to display}\)	\(3928\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/(g*x+f),x,method=_RETURNVERBOSE)

[Out]

-b*d^2*(d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)/g^6*ln(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*
f^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))*arcsinh(c*x)*c^4*f^4-b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)^(1/2)/
g^5*x*c^5*f^4+b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g^5*arcsinh(c*x)*c^4*f^4+b*d^2*(d*(c^2*x^2+1))^(1/2)*(c^
2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)/g^2*ln((-(c*x+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+
g^2)^(1/2)))*arcsinh(c*x)-b*d^2*(d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)/g^2*ln(((c*x+(c^2*
x^2+1)^(1/2))*g+c*f+(c^2*f^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))*arcsinh(c*x)-1/2*b*(d*(c^2*x^2+1))^(1/2)/(
c^2*x^2+1)^(1/2)*f^5*arcsinh(c*x)^2*c^5*d^2/g^6-5/4*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*f^3*arcsinh(c*x)
^2*c^3*d^2/g^4-15/16*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*f*arcsinh(c*x)^2*c*d^2/g^2+1/5*b*(d*(c^2*x^2+1)
)^(1/2)*d^2/(c^2*x^2+1)/g*arcsinh(c*x)*x^6*c^6+14/15*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g*arcsinh(c*x)*x^
4*c^4+a/g^5*d^2*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)*c^4*f^4+2*a/g^3*d^2*((x+f/g)^2
*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)*c^2*f^2-3*a/g^3*d^3/(d*(c^2*f^2+g^2)/g^2)^(1/2)*ln((2*d*
(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^
2*f^2+g^2)/g^2)^(1/2))/(x+f/g))*c^2*f^2-3*a/g^5*d^3/(d*(c^2*f^2+g^2)/g^2)^(1/2)*ln((2*d*(c^2*f^2+g^2)/g^2-2*c^
2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))
/(x+f/g))*c^4*f^4-7/8*a/g^2*c^2*d^2*f*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)*x-15/8*a
/g^2*c^2*d^3*f*ln((-c^2*d*f/g+(x+f/g)*c^2*d)/(c^2*d)^(1/2)+((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2
)/g^2)^(1/2))/(c^2*d)^(1/2)-1/2*a/g^4*d^2*c^4*f^3*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1
/2)*x-5/2*a/g^4*d^3*c^4*f^3*ln((-c^2*d*f/g+(x+f/g)*c^2*d)/(c^2*d)^(1/2)+((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d
*(c^2*f^2+g^2)/g^2)^(1/2))/(c^2*d)^(1/2)-a/g^6*d^3*c^6*f^5*ln((-c^2*d*f/g+(x+f/g)*c^2*d)/(c^2*d)^(1/2)+((x+f/g
)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(c^2*d)^(1/2)-a/g^7*d^3/(d*(c^2*f^2+g^2)/g^2)^(1/2)*
ln((2*d*(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/
g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(x+f/g))*c^6*f^6+8/3*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g^3*arcsinh(c*x)*x
^2*c^4*f^2-1/4*b*(d*(c^2*x^2+1))^(1/2)*f*c^6*d^2/(c^2*x^2+1)/g^2*arcsinh(c*x)*x^5-11/8*b*(d*(c^2*x^2+1))^(1/2)
*f*c^4*d^2/(c^2*x^2+1)/g^2*arcsinh(c*x)*x^3-9/8*b*(d*(c^2*x^2+1))^(1/2)*f*c^2*d^2/(c^2*x^2+1)/g^2*arcsinh(c*x)
*x-1/4*a/g^2*c^2*d*f*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(3/2)*x-a/g*d^3/(d*(c^2*f^2+g^2
)/g^2)^(1/2)*ln((2*d*(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^(1/2)*((x+f/g)^2*c^2*d-2*c^
2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(x+f/g))-1/9*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)^(1/2)/g^3*x^3
*c^5*f^2+1/16*b*(d*(c^2*x^2+1))^(1/2)*f*c^5*d^2/(c^2*x^2+1)^(1/2)/g^2*x^4+9/16*b*(d*(c^2*x^2+1))^(1/2)*f*c^3*d
^2/(c^2*x^2+1)^(1/2)/g^2*x^2-7/3*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)^(1/2)/g^3*x*c^3*f^2+1/4*b*(d*(c^2*x^2
+1))^(1/2)*f^3*c^5*d^2/(c^2*x^2+1)^(1/2)/g^4*x^2+1/5*a/g*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/
g^2)^(5/2)+1/3*a/g*d*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(3/2)-1/25*b*(d*(c^2*x^2+1))^(1
/2)*d^2/(c^2*x^2+1)^(1/2)/g*x^5*c^5-b*d^2*(d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)/g^2*dilo
g(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))+b*d^2*(d*(c^2*x^2+1))^(1/2)*(
c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)/g^2*dilog((-(c*x+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2
*f^2+g^2)^(1/2)))-11/45*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)^(1/2)/g*x^3*c^3-23/15*b*(d*(c^2*x^2+1))^(1/2)*
d^2/(c^2*x^2+1)^(1/2)/g*x*c+33/128*b*(d*(c^2*x^2+1))^(1/2)*f*c*d^2/(c^2*x^2+1)^(1/2)/g^2+1/8*b*(d*(c^2*x^2+1))
^(1/2)*f^3*c^3*d^2/(c^2*x^2+1)^(1/2)/g^4+2*b*d^2*(d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)/g
^4*ln((-(c*x+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))*arcsinh(c*x)*c^2*f^2-2*
b*d^2*(d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)/g^4*ln(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f
^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))*arcsinh(c*x)*c^2*f^2+34/15*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g
*arcsinh(c*x)*x^2*c^2+7/3*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g^3*arcsinh(c*x)*c^2*f^2+1/3*a/g^3*d*((x+f/g
)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(3/2)*c^2*f^2+b*d^2*(d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/
2)/(c^2*x^2+1)^(1/2)/g^6*ln((-(c*x+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))*a
rcsinh(c*x)*c^4*f^4+23/15*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g*arcsinh(c*x)+a/g*d^2*((x+f/g)^2*c^2*d-2*c^
2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)-1/2*b*(d*(c^2*x^2+1))^(1/2)*f^3*c^6*d^2/(c^2*x^2+1)/g^4*arcsinh(c*x
)*x^3-1/2*b*(d*(c^2*x^2+1))^(1/2)*f^3*c^4*d^2/(...

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/(g*x+f),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/(g*x+f),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^4 + 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 + 2*b*c^2*d^2*x^2 + b*d^2)*arcsinh(c*x))*sq
rt(c^2*d*x^2 + d)/(g*x + f), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{f + g x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))/(g*x+f),x)

[Out]

Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))/(f + g*x), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/(g*x+f),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{f+g\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2))/(f + g*x),x)

[Out]

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^(5/2))/(f + g*x), x)

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