\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \arcsin (c x))}{f+g x} \, dx\) [43]

Optimal result

Integrand size = 31, antiderivative size = 1648 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{f+g x} \, dx=\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 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^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} \arcsin (c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} \arctan \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 {i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (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} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}} \]

[Out]

-1/3*d^2*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g+1/5*d^2*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))*(-c^2*
d*x^2+d)^(1/2)/g-a*d^2*(c^2*f^2-g^2)^(5/2)*arctan((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,I*(I*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,I*(I*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)+I*b*d^2*(c^2*f^2-g
^2)^(5/2)*arcsin(c*x)*ln(1-I*(I*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*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)-1/9*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*arcsin
(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/g^2/(-c^2*x^2+1)^(1/2)-1/2*c^2*d^2*f*(c^2*f^2-2*g^2)*x*(a+b*arcsin(c*x))*(-c^2
*d*x^2+d)^(1/2)/g^4-1/16*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)+a*d^2*(c^2*f^2-g^2)^2*(-c
^2*d*x^2+d)^(1/2)/g^5+b*d^2*(c^2*f^2-g^2)^2*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/g^5+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)+1/8*c^2*d^2*f*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g^2-1/4*c^4*d^2
*f*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/g^2-1/3*d^2*(c^2*f^2-2*g^2)*(-c^2*x^2+1)*(a+b*arcsin(c*x))*(-c^2
*d*x^2+d)^(1/2)/g^3-1/4*c*d^2*f*(c^2*f^2-2*g^2)*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/g^4/(-c^2*x^2+1)^(1
/2)+1/2*c*d^2*(c^2*f^2-g^2)^2*x*(a+b*arcsin(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*arcsin(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*arcsin(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*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)-I*b*d^2*(c^2*f^2-g^2)^(5/2)*arcsin(c*x)*ln(1-I*(I*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)

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 1648, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.903, Rules used = {4861, 4851, 4741, 4737, 30, 4767, 4783, 4795, 272, 45, 4779, 12, 4849, 697, 4841, 6874, 739, 210, 1668, 4883, 4881, 8, 4857, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{f+g x} \, dx=-\frac {b d^2 x^5 \sqrt {d-c^2 d x^2} c^5}{25 g \sqrt {1-c^2 x^2}}+\frac {b d^2 f x^4 \sqrt {d-c^2 d x^2} c^5}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) c^4}{4 g^2}-\frac {b d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt {d-c^2 d x^2} c^3}{9 g^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2} c^3}{45 g \sqrt {1-c^2 x^2}}+\frac {b d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt {d-c^2 d x^2} c^3}{4 g^4 \sqrt {1-c^2 x^2}}-\frac {b d^2 f x^2 \sqrt {d-c^2 d x^2} c^3}{16 g^2 \sqrt {1-c^2 x^2}}-\frac {d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) c^2}{2 g^4}+\frac {d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) c^2}{8 g^2}-\frac {d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 c}{4 b g^4 \sqrt {1-c^2 x^2}}+\frac {d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 c}{2 b g^5 \sqrt {1-c^2 x^2}}-\frac {d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 c}{16 b g^2 \sqrt {1-c^2 x^2}}-\frac {b d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt {d-c^2 d x^2} c}{g^5 \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {d-c^2 d x^2} c}{3 g^3 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2} c}{15 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} \arcsin (c x)}{g^5}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 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} (a+b \arcsin (c x))}{3 g^3}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 g}-\frac {a d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b d^2 \left (c^2 f^2-g^2\right )^{5/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (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} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\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 {d^2 \left (c^2 f^2-g^2\right )^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g^4 (f+g x) c}+\frac {d^2 \left (c^2 f^2-g^2\right )^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b g^6 (f+g x) \sqrt {1-c^2 x^2} c} \]

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[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 - 2*g^2)*x*Sqrt[d - c^2*d*x^2])/(3*g^3*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^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]*ArcSin[c*x])/g^5 + (c^2*d^2*f*x*
Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[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*A
rcSin[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(4*g^2) - (d^2*(1 - c^2*x^2)*Sq
rt[d - c^2*d*x^2]*(a + b*ArcSin[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
*ArcSin[c*x]))/(3*g^3) + (d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(5*g) - (c*d^2*f*Sqrt[d
- c^2*d*x^2]*(a + b*ArcSin[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*ArcSin[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*ArcSin[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*ArcSin[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*ArcSin[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]*ArcTan[(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]) + (I*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d
- c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2])
- (I*b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[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, (I*E^(
I*ArcSin[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, (I*E^(I*ArcSin[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 210

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

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 3404

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

Rule 4737

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

Rule 4741

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

Rule 4767

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

Rule 4779

Int[((a_.) + ArcSin[(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*ArcSin[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[Si
mplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p
- 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rule 4783

Int[((a_.) + ArcSin[(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*ArcSin[c*x])^n/(f*(m + 2))), x] + (Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/S
qrt[1 - c^2*x^2]], Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x] - Dist[b*c*(n/(f*(m + 2)))*Si
mp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b,
c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 4795

Int[((a_.) + ArcSin[(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*ArcSin[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*ArcSin[c*x])^n, x], x] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(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*ArcSin[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 4841

Int[(((a_.) + ArcSin[(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*ArcSin[c*x])^n, u, x] - Dist
[b*c*n, Int[SimplifyIntegrand[u*((a + b*ArcSin[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 4849

Int[((a_.) + ArcSin[(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*ArcSin[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*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[
{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 4851

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

Rule 4857

Int[(((a_.) + ArcSin[(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*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a,
b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 4861

Int[((a_.) + ArcSin[(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*ArcSin[c*x])^n, x], x] /; F
reeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ[p - 1/2] &&  !GtQ[d, 0]

Rule 4881

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

Rule 4883

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{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 {c^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{g^4}+\frac {c^2 \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{g^3}-\frac {c^4 f x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{g^2}+\frac {c^4 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{g}+\frac {\left (-c^2 f^2+g^2\right )^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x)) \, 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} (a+b \arcsin (c x)) \, dx}{g \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} (a+b \arcsin (c x)) \, 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} (a+b \arcsin (c x)) \, dx}{g^3 \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} (a+b \arcsin (c x))}{f+g x} \, dx}{g^4 \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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))^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 (a+b \arcsin (c x))}{\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 (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 \arcsin (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 {\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 ) (a+b \arcsin (c x))^2}{(f+g x)^2} \, dx}{2 b c g^4 \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} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 \arcsin (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 {1}{f+g x}-\frac {c^2 \left (g x+\frac {f^2}{f+g x}\right )}{g^2}\right ) (a+b \arcsin (c x))}{\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} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 ) \arcsin (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} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 ) \arcsin (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} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 \arcsin (c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (c^2 f^2-g^2\right ) \arcsin (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} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \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 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\arcsin (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} \arcsin (c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \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 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \sin (x)} \, dx,x,\arcsin (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^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \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^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} \arcsin (c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c g^4 (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arctan \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 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\arcsin (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^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \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^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} \arcsin (c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c g^4 (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arctan \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 i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c f-2 i e^{i x} g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\arcsin (c x)\right )}{g^5 \sqrt {1-c^2 x^2}}-\frac {\left (2 i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c f-2 i e^{i x} g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\arcsin (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^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \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^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} \arcsin (c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c g^4 (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arctan \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 {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {\left (i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arcsin (c x)\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x} g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\arcsin (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^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \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^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} \arcsin (c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c g^4 (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arctan \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 {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{g^6 \sqrt {1-c^2 x^2}}+\frac {\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \arcsin (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^2-2 g^2\right ) x \sqrt {d-c^2 d x^2}}{3 g^3 \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^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} \arcsin (c x)}{g^5}+\frac {c^2 d^2 f x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{4 g^2}-\frac {d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{3 g^3}+\frac {d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{5 g}-\frac {c d^2 f \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c g^4 (f+g x)}-\frac {a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arctan \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 {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}}-\frac {i b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (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 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6 \sqrt {1-c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.13 (sec) , antiderivative size = 787, normalized size of antiderivative = 0.48 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{f+g x} \, dx=-\frac {d^2 \sqrt {d-c^2 d x^2} \left (-900 b c^3 f \left (c^2 f^2-2 g^2\right ) x^2-225 b c^5 f g^2 x^4+144 b c^5 g^3 x^5+400 b c g \left (c^2 f^2-2 g^2\right ) x \left (-3+c^2 x^2\right )+1800 c^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+900 c^4 f g^2 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-720 c^4 g^3 x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+1200 g \left (c^2 f^2-2 g^2\right ) \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {900 c f \left (c^2 f^2-2 g^2\right ) (a+b \arcsin (c x))^2}{b}+\frac {1800 \left (-c^2 f^2+g^2\right )^2 \left (-1+c^2 x^2\right ) (a+b \arcsin (c x))^2}{b c (f+g x)}-80 g^3 \left (6 b c x+b c^3 x^3-6 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-3 c^2 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))\right )+225 c f g^2 \left (b c^2 x^2-2 c x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))+\frac {(a+b \arcsin (c x))^2}{b}\right )-\frac {1800 \left (-c^2 f^2+g^2\right )^2 \left (c^2 g x (a+b \arcsin (c x))^2+\frac {\left (c^2 f^2-g^2\right ) (a+b \arcsin (c x))^2}{f+g x}-2 b c \left (b c g x-g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))-i \sqrt {c^2 f^2-g^2} \left ((a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-i b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+i b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )\right )}{b c g^2}\right )}{3600 g^4 \sqrt {1-c^2 x^2}} \]

[In]

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

[Out]

-1/3600*(d^2*Sqrt[d - c^2*d*x^2]*(-900*b*c^3*f*(c^2*f^2 - 2*g^2)*x^2 - 225*b*c^5*f*g^2*x^4 + 144*b*c^5*g^3*x^5
+ 400*b*c*g*(c^2*f^2 - 2*g^2)*x*(-3 + c^2*x^2) + 1800*c^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[1 - c^2*x^2]*(a + b*ArcS
in[c*x]) + 900*c^4*f*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - 720*c^4*g^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*
ArcSin[c*x]) + 1200*g*(c^2*f^2 - 2*g^2)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (900*c*f*(c^2*f^2 - 2*g^2)*(
a + b*ArcSin[c*x])^2)/b + (1800*(-(c^2*f^2) + g^2)^2*(-1 + c^2*x^2)*(a + b*ArcSin[c*x])^2)/(b*c*(f + g*x)) - 8
0*g^3*(6*b*c*x + b*c^3*x^3 - 6*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - 3*c^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcS
in[c*x])) + 225*c*f*g^2*(b*c^2*x^2 - 2*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) + (a + b*ArcSin[c*x])^2/b) -
(1800*(-(c^2*f^2) + g^2)^2*(c^2*g*x*(a + b*ArcSin[c*x])^2 + ((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^2)/(f + g*x)
- 2*b*c*(b*c*g*x - g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - I*Sqrt[c^2*f^2 - g^2]*((a + b*ArcSin[c*x])*(Log[1
+ (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f
^2 - g^2])]) - I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + I*b*PolyLog[2, (I*E^(I*Ar
cSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/(b*c*g^2)))/(g^4*Sqrt[1 - c^2*x^2])

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 2580, normalized size of antiderivative = 1.57

method	result	size
default	\(\text {Expression too large to display}\)	\(2580\)
parts	\(\text {Expression too large to display}\)	\(2580\)

[In]

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

[Out]

a/g*(1/5*(-(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)+c^2*d*f/g*(-1/8*(-2*(x+f/g)*c^2*d+2*
c^2*d*f/g)/c^2/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)-3/16*(4*c^2*d^2*(c^2*f^2-g^2
)/g^2-4*c^4*d^2*f^2/g^2)/c^2/d*(-1/4*(-2*(x+f/g)*c^2*d+2*c^2*d*f/g)/c^2/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)^(1/2)-1/8*(4*c^2*d^2*(c^2*f^2-g^2)/g^2-4*c^4*d^2*f^2/g^2)/c^2/d/(c^2*d)^(1/2)*arctan((c
^2*d)^(1/2)*x/(-(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))))-d*(c^2*f^2-g^2)/g^2*(1/3*(-(
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)^(3/2)+c^2*d*f/g*(-1/4*(-2*(x+f/g)*c^2*d+2*c^2*d*f/g)/c
^2/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)^(1/2)-1/8*(4*c^2*d^2*(c^2*f^2-g^2)/g^2-4*c^4*d
^2*f^2/g^2)/c^2/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(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)))-d*(c^2*f^2-g^2)/g^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*f/g/(c
^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(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))+d*(c^2*f^
2-g^2)/g^2/(-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)))))+b*(-1/16*(-d*(c^2*x^2-1)
)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arcsin(c*x)^2*f*(8*c^4*f^4-20*c^2*f^2*g^2+15*g^4)*d^2*c/g^6+1/800*(-d*(
c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4-16*I*(-c^2*x^2+1)^(1/2)*x^5*c^5+13*c^2*x^2+20*I*(-c^2*x^2+1)^(1/2)*x^
3*c^3-5*I*(-c^2*x^2+1)^(1/2)*x*c-1)*(I+5*arcsin(c*x))*d^2/(c^2*x^2-1)/g-1/256*(-d*(c^2*x^2-1))^(1/2)*(-8*I*(-c
^2*x^2+1)^(1/2)*x^4*c^4+8*c^5*x^5+8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3-I*(-c^2*x^2+1)^(1/2)+4*c*x)*f*(4*a
rcsin(c*x)+I)*d^2*c/(c^2*x^2-1)/g^2+1/288*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*c^3*x^3*(-c^2*x^2+1)
^(1/2)+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(4*c^2*f^2-7*g^2)*(I+3*arcsin(c*x))*d^2/(c^2*x^2-1)/g^3-1/16*(-d*(c^2*x^2
-1))^(1/2)*(-2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(c^2*f^2-2*g^2)*(I+2*arcsi
n(c*x))*d^2*c/(c^2*x^2-1)/g^4+1/16*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(8*c^4*f^4-18*c
^2*f^2*g^2+11*g^4)*(arcsin(c*x)+I)*d^2/(c^2*x^2-1)/g^5+1/16*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c
^2*x^2-1)*(8*c^4*f^4-18*c^2*f^2*g^2+11*g^4)*(arcsin(c*x)-I)*d^2/(c^2*x^2-1)/g^5-1/16*(-d*(c^2*x^2-1))^(1/2)*(2
*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(c^2*f^2-2*g^2)*(-I+2*arcsin(c*x))*d^2*c
/(c^2*x^2-1)/g^4+1/288*(-d*(c^2*x^2-1))^(1/2)*(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)
*x*c-5*c^2*x^2+1)*(4*c^2*f^2-7*g^2)*(-I+3*arcsin(c*x))*d^2/(c^2*x^2-1)/g^3-1/256*(-d*(c^2*x^2-1))^(1/2)*(8*I*(
-c^2*x^2+1)^(1/2)*c^4*x^4+8*c^5*x^5-8*I*(-c^2*x^2+1)^(1/2)*x^2*c^2-12*c^3*x^3+I*(-c^2*x^2+1)^(1/2)+4*c*x)*f*(-
I+4*arcsin(c*x))*d^2*c/(c^2*x^2-1)/g^2+1/800*(-d*(c^2*x^2-1))^(1/2)*(16*I*c^5*x^5*(-c^2*x^2+1)^(1/2)+16*c^6*x^
6-20*I*(-c^2*x^2+1)^(1/2)*x^3*c^3-28*c^4*x^4+5*I*(-c^2*x^2+1)^(1/2)*x*c+13*c^2*x^2-1)*(-I+5*arcsin(c*x))*d^2/(
c^2*x^2-1)/g+I*(-c^2*f^2+g^2)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(I*arcsin(c*x)*ln((I*c*f+(I*c*x+
(-c^2*x^2+1)^(1/2))*g-(-c^2*f^2+g^2)^(1/2))/(I*c*f-(-c^2*f^2+g^2)^(1/2)))-I*arcsin(c*x)*ln((I*c*f+(I*c*x+(-c^2
*x^2+1)^(1/2))*g+(-c^2*f^2+g^2)^(1/2))/(I*c*f+(-c^2*f^2+g^2)^(1/2)))+dilog((I*c*f+(I*c*x+(-c^2*x^2+1)^(1/2))*g
-(-c^2*f^2+g^2)^(1/2))/(I*c*f-(-c^2*f^2+g^2)^(1/2)))-dilog((I*c*f+(I*c*x+(-c^2*x^2+1)^(1/2))*g+(-c^2*f^2+g^2)^
(1/2))/(I*c*f+(-c^2*f^2+g^2)^(1/2))))*(c^4*f^4-2*c^2*f^2*g^2+g^4)*d^2/(c^2*x^2-1)/g^6)

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{g x + f} \,d x } \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(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)*arcsin(c*x))*sqr
t(-c^2*d*x^2 + d)/(g*x + f), x)

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{f+g x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{f + g x}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for m
ore details)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{f+g\,x} \,d x \]

[In]

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

[Out]

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