3.90 \(\int (g+h x) (a+c x^2)^{3/2} (d+e x+f x^2) \, dx\)

Optimal. Leaf size=213 \[ \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) (-a e h-a f g+6 c d g)}{16 c^{3/2}}-\frac {\left (a+c x^2\right )^{5/2} \left (6 \left (2 a f h^2+c \left (5 f g^2-7 h (d h+e g)\right )\right )+5 c h x (5 f g-7 e h)\right )}{210 c^2 h}+\frac {x \left (a+c x^2\right )^{3/2} (6 c d g-a (e h+f g))}{24 c}+\frac {a x \sqrt {a+c x^2} (-a e h-a f g+6 c d g)}{16 c}+\frac {f \left (a+c x^2\right )^{5/2} (g+h x)^2}{7 c h} \]

[Out]

1/24*(6*c*d*g-a*(e*h+f*g))*x*(c*x^2+a)^(3/2)/c+1/7*f*(h*x+g)^2*(c*x^2+a)^(5/2)/c/h-1/210*(12*a*f*h^2+6*c*(5*f*
g^2-7*h*(d*h+e*g))+5*c*h*(-7*e*h+5*f*g)*x)*(c*x^2+a)^(5/2)/c^2/h+1/16*a^2*(-a*e*h-a*f*g+6*c*d*g)*arctanh(x*c^(
1/2)/(c*x^2+a)^(1/2))/c^(3/2)+1/16*a*(-a*e*h-a*f*g+6*c*d*g)*x*(c*x^2+a)^(1/2)/c

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Rubi [A]  time = 0.27, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1654, 780, 195, 217, 206} \[ \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) (-a e h-a f g+6 c d g)}{16 c^{3/2}}-\frac {\left (a+c x^2\right )^{5/2} \left (6 \left (2 a f h^2-7 c h (d h+e g)+5 c f g^2\right )+5 c h x (5 f g-7 e h)\right )}{210 c^2 h}+\frac {x \left (a+c x^2\right )^{3/2} (6 c d g-a (e h+f g))}{24 c}+\frac {a x \sqrt {a+c x^2} (-a e h-a f g+6 c d g)}{16 c}+\frac {f \left (a+c x^2\right )^{5/2} (g+h x)^2}{7 c h} \]

Antiderivative was successfully verified.

[In]

Int[(g + h*x)*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x]

[Out]

(a*(6*c*d*g - a*f*g - a*e*h)*x*Sqrt[a + c*x^2])/(16*c) + ((6*c*d*g - a*(f*g + e*h))*x*(a + c*x^2)^(3/2))/(24*c
) + (f*(g + h*x)^2*(a + c*x^2)^(5/2))/(7*c*h) - ((6*(5*c*f*g^2 + 2*a*f*h^2 - 7*c*h*(e*g + d*h)) + 5*c*h*(5*f*g
- 7*e*h)*x)*(a + c*x^2)^(5/2))/(210*c^2*h) + (a^2*(6*c*d*g - a*f*g - a*e*h)*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^
2]])/(16*c^(3/2))

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 206

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 1654

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]))

Rubi steps

\begin {align*} \int (g+h x) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}+\frac {\int (g+h x) \left ((7 c d-2 a f) h^2-c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{7 c h^2}\\ &=\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {(6 c d g-a f g-a e h) \int \left (a+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac {(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {(a (6 c d g-a f g-a e h)) \int \sqrt {a+c x^2} \, dx}{8 c}\\ &=\frac {a (6 c d g-a f g-a e h) x \sqrt {a+c x^2}}{16 c}+\frac {(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {\left (a^2 (6 c d g-a f g-a e h)\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{16 c}\\ &=\frac {a (6 c d g-a f g-a e h) x \sqrt {a+c x^2}}{16 c}+\frac {(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {\left (a^2 (6 c d g-a f g-a e h)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{16 c}\\ &=\frac {a (6 c d g-a f g-a e h) x \sqrt {a+c x^2}}{16 c}+\frac {(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {a^2 (6 c d g-a f g-a e h) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.63, size = 209, normalized size = 0.98 \[ \frac {\sqrt {a+c x^2} \left (-\frac {105 a^{5/2} \sqrt {\frac {c x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a e h+a f g-6 c d g)}{c^{3/2} \left (a+c x^2\right )}-\frac {96 a^3 f h}{c^2}+\frac {3 a^2 (112 d h+7 e (16 g+5 h x)+f x (35 g+16 h x))}{c}+2 a x (21 d (25 g+16 h x)+x (7 e (48 g+35 h x)+f x (245 g+192 h x)))+4 c x^3 (21 d (5 g+4 h x)+2 x (7 e (6 g+5 h x)+5 f x (7 g+6 h x)))\right )}{1680} \]

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x]

[Out]

(Sqrt[a + c*x^2]*((-96*a^3*f*h)/c^2 + (3*a^2*(112*d*h + 7*e*(16*g + 5*h*x) + f*x*(35*g + 16*h*x)))/c + 4*c*x^3
*(21*d*(5*g + 4*h*x) + 2*x*(7*e*(6*g + 5*h*x) + 5*f*x*(7*g + 6*h*x))) + 2*a*x*(21*d*(25*g + 16*h*x) + x*(7*e*(
48*g + 35*h*x) + f*x*(245*g + 192*h*x))) - (105*a^(5/2)*(-6*c*d*g + a*f*g + a*e*h)*Sqrt[1 + (c*x^2)/a]*ArcSinh
[(Sqrt[c]*x)/Sqrt[a]])/(c^(3/2)*(a + c*x^2))))/1680

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fricas [A]  time = 1.02, size = 477, normalized size = 2.24 \[ \left [\frac {105 \, {\left (a^{3} e h - {\left (6 \, a^{2} c d - a^{3} f\right )} g\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (240 \, c^{3} f h x^{6} + 280 \, {\left (c^{3} f g + c^{3} e h\right )} x^{5} + 336 \, a^{2} c e g + 48 \, {\left (7 \, c^{3} e g + {\left (7 \, c^{3} d + 8 \, a c^{2} f\right )} h\right )} x^{4} + 70 \, {\left (7 \, a c^{2} e h + {\left (6 \, c^{3} d + 7 \, a c^{2} f\right )} g\right )} x^{3} + 48 \, {\left (14 \, a c^{2} e g + {\left (14 \, a c^{2} d + a^{2} c f\right )} h\right )} x^{2} + 48 \, {\left (7 \, a^{2} c d - 2 \, a^{3} f\right )} h + 105 \, {\left (a^{2} c e h + {\left (10 \, a c^{2} d + a^{2} c f\right )} g\right )} x\right )} \sqrt {c x^{2} + a}}{3360 \, c^{2}}, \frac {105 \, {\left (a^{3} e h - {\left (6 \, a^{2} c d - a^{3} f\right )} g\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (240 \, c^{3} f h x^{6} + 280 \, {\left (c^{3} f g + c^{3} e h\right )} x^{5} + 336 \, a^{2} c e g + 48 \, {\left (7 \, c^{3} e g + {\left (7 \, c^{3} d + 8 \, a c^{2} f\right )} h\right )} x^{4} + 70 \, {\left (7 \, a c^{2} e h + {\left (6 \, c^{3} d + 7 \, a c^{2} f\right )} g\right )} x^{3} + 48 \, {\left (14 \, a c^{2} e g + {\left (14 \, a c^{2} d + a^{2} c f\right )} h\right )} x^{2} + 48 \, {\left (7 \, a^{2} c d - 2 \, a^{3} f\right )} h + 105 \, {\left (a^{2} c e h + {\left (10 \, a c^{2} d + a^{2} c f\right )} g\right )} x\right )} \sqrt {c x^{2} + a}}{1680 \, c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="fricas")

[Out]

[1/3360*(105*(a^3*e*h - (6*a^2*c*d - a^3*f)*g)*sqrt(c)*log(-2*c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(24
0*c^3*f*h*x^6 + 280*(c^3*f*g + c^3*e*h)*x^5 + 336*a^2*c*e*g + 48*(7*c^3*e*g + (7*c^3*d + 8*a*c^2*f)*h)*x^4 + 7
0*(7*a*c^2*e*h + (6*c^3*d + 7*a*c^2*f)*g)*x^3 + 48*(14*a*c^2*e*g + (14*a*c^2*d + a^2*c*f)*h)*x^2 + 48*(7*a^2*c
*d - 2*a^3*f)*h + 105*(a^2*c*e*h + (10*a*c^2*d + a^2*c*f)*g)*x)*sqrt(c*x^2 + a))/c^2, 1/1680*(105*(a^3*e*h - (
6*a^2*c*d - a^3*f)*g)*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (240*c^3*f*h*x^6 + 280*(c^3*f*g + c^3*e*h)
*x^5 + 336*a^2*c*e*g + 48*(7*c^3*e*g + (7*c^3*d + 8*a*c^2*f)*h)*x^4 + 70*(7*a*c^2*e*h + (6*c^3*d + 7*a*c^2*f)*
g)*x^3 + 48*(14*a*c^2*e*g + (14*a*c^2*d + a^2*c*f)*h)*x^2 + 48*(7*a^2*c*d - 2*a^3*f)*h + 105*(a^2*c*e*h + (10*
a*c^2*d + a^2*c*f)*g)*x)*sqrt(c*x^2 + a))/c^2]

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giac [A]  time = 0.25, size = 264, normalized size = 1.24 \[ \frac {1}{1680} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, c f h x + \frac {7 \, {\left (c^{6} f g + c^{6} h e\right )}}{c^{5}}\right )} x + \frac {6 \, {\left (7 \, c^{6} d h + 8 \, a c^{5} f h + 7 \, c^{6} g e\right )}}{c^{5}}\right )} x + \frac {35 \, {\left (6 \, c^{6} d g + 7 \, a c^{5} f g + 7 \, a c^{5} h e\right )}}{c^{5}}\right )} x + \frac {24 \, {\left (14 \, a c^{5} d h + a^{2} c^{4} f h + 14 \, a c^{5} g e\right )}}{c^{5}}\right )} x + \frac {105 \, {\left (10 \, a c^{5} d g + a^{2} c^{4} f g + a^{2} c^{4} h e\right )}}{c^{5}}\right )} x + \frac {48 \, {\left (7 \, a^{2} c^{4} d h - 2 \, a^{3} c^{3} f h + 7 \, a^{2} c^{4} g e\right )}}{c^{5}}\right )} - \frac {{\left (6 \, a^{2} c d g - a^{3} f g - a^{3} h e\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, c^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="giac")

[Out]

1/1680*sqrt(c*x^2 + a)*((2*((4*(5*(6*c*f*h*x + 7*(c^6*f*g + c^6*h*e)/c^5)*x + 6*(7*c^6*d*h + 8*a*c^5*f*h + 7*c
^6*g*e)/c^5)*x + 35*(6*c^6*d*g + 7*a*c^5*f*g + 7*a*c^5*h*e)/c^5)*x + 24*(14*a*c^5*d*h + a^2*c^4*f*h + 14*a*c^5
*g*e)/c^5)*x + 105*(10*a*c^5*d*g + a^2*c^4*f*g + a^2*c^4*h*e)/c^5)*x + 48*(7*a^2*c^4*d*h - 2*a^3*c^3*f*h + 7*a
^2*c^4*g*e)/c^5) - 1/16*(6*a^2*c*d*g - a^3*f*g - a^3*h*e)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)

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maple [A]  time = 0.00, size = 287, normalized size = 1.35 \[ -\frac {a^{3} e h \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}-\frac {a^{3} f g \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}+\frac {3 a^{2} d g \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 \sqrt {c}}-\frac {\sqrt {c \,x^{2}+a}\, a^{2} e h x}{16 c}-\frac {\sqrt {c \,x^{2}+a}\, a^{2} f g x}{16 c}+\frac {3 \sqrt {c \,x^{2}+a}\, a d g x}{8}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} a e h x}{24 c}-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} a f g x}{24 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} f h \,x^{2}}{7 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} d g x}{4}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} e h x}{6 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} f g x}{6 c}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {5}{2}} a f h}{35 c^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} d h}{5 c}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} e g}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x)

[Out]

1/7*h*f*x^2*(c*x^2+a)^(5/2)/c-2/35*h*f*a/c^2*(c*x^2+a)^(5/2)+1/6*x*(c*x^2+a)^(5/2)/c*e*h+1/6*x*(c*x^2+a)^(5/2)
/c*f*g-1/24*a/c*x*(c*x^2+a)^(3/2)*e*h-1/24*a/c*x*(c*x^2+a)^(3/2)*f*g-1/16*a^2/c*x*(c*x^2+a)^(1/2)*e*h-1/16*a^2
/c*x*(c*x^2+a)^(1/2)*f*g-1/16*a^3/c^(3/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))*e*h-1/16*a^3/c^(3/2)*ln(c^(1/2)*x+(c*x
^2+a)^(1/2))*f*g+1/5*(c*x^2+a)^(5/2)/c*d*h+1/5*(c*x^2+a)^(5/2)/c*e*g+1/4*d*g*x*(c*x^2+a)^(3/2)+3/8*d*g*a*x*(c*
x^2+a)^(1/2)+3/8*d*g*a^2/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2))

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maxima [A]  time = 0.45, size = 211, normalized size = 0.99 \[ \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} f h x^{2}}{7 \, c} + \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d g x + \frac {3}{8} \, \sqrt {c x^{2} + a} a d g x + \frac {3 \, a^{2} d g \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} e g}{5 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} d h}{5 \, c} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} a f h}{35 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (f g + e h\right )} x}{6 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (f g + e h\right )} a x}{24 \, c} - \frac {\sqrt {c x^{2} + a} {\left (f g + e h\right )} a^{2} x}{16 \, c} - \frac {{\left (f g + e h\right )} a^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(c*x^2+a)^(3/2)*(f*x^2+e*x+d),x, algorithm="maxima")

[Out]

1/7*(c*x^2 + a)^(5/2)*f*h*x^2/c + 1/4*(c*x^2 + a)^(3/2)*d*g*x + 3/8*sqrt(c*x^2 + a)*a*d*g*x + 3/8*a^2*d*g*arcs
inh(c*x/sqrt(a*c))/sqrt(c) + 1/5*(c*x^2 + a)^(5/2)*e*g/c + 1/5*(c*x^2 + a)^(5/2)*d*h/c - 2/35*(c*x^2 + a)^(5/2
)*a*f*h/c^2 + 1/6*(c*x^2 + a)^(5/2)*(f*g + e*h)*x/c - 1/24*(c*x^2 + a)^(3/2)*(f*g + e*h)*a*x/c - 1/16*sqrt(c*x
^2 + a)*(f*g + e*h)*a^2*x/c - 1/16*(f*g + e*h)*a^3*arcsinh(c*x/sqrt(a*c))/c^(3/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (g+h\,x\right )\,{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g + h*x)*(a + c*x^2)^(3/2)*(d + e*x + f*x^2),x)

[Out]

int((g + h*x)*(a + c*x^2)^(3/2)*(d + e*x + f*x^2), x)

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sympy [A]  time = 27.89, size = 768, normalized size = 3.61 \[ \frac {a^{\frac {5}{2}} e h x}{16 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} f g x}{16 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {3}{2}} d g x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {a^{\frac {3}{2}} d g x}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} e h x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} f g x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 \sqrt {a} c d g x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 \sqrt {a} c e h x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 \sqrt {a} c f g x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {a^{3} e h \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 c^{\frac {3}{2}}} - \frac {a^{3} f g \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 c^{\frac {3}{2}}} + \frac {3 a^{2} d g \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 \sqrt {c}} + a d h \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + a e g \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + a f h \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c d h \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c e g \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c f h \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + \frac {c^{2} d g x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} e h x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} f g x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(c*x**2+a)**(3/2)*(f*x**2+e*x+d),x)

[Out]

a**(5/2)*e*h*x/(16*c*sqrt(1 + c*x**2/a)) + a**(5/2)*f*g*x/(16*c*sqrt(1 + c*x**2/a)) + a**(3/2)*d*g*x*sqrt(1 +
c*x**2/a)/2 + a**(3/2)*d*g*x/(8*sqrt(1 + c*x**2/a)) + 17*a**(3/2)*e*h*x**3/(48*sqrt(1 + c*x**2/a)) + 17*a**(3/
2)*f*g*x**3/(48*sqrt(1 + c*x**2/a)) + 3*sqrt(a)*c*d*g*x**3/(8*sqrt(1 + c*x**2/a)) + 11*sqrt(a)*c*e*h*x**5/(24*
sqrt(1 + c*x**2/a)) + 11*sqrt(a)*c*f*g*x**5/(24*sqrt(1 + c*x**2/a)) - a**3*e*h*asinh(sqrt(c)*x/sqrt(a))/(16*c*
*(3/2)) - a**3*f*g*asinh(sqrt(c)*x/sqrt(a))/(16*c**(3/2)) + 3*a**2*d*g*asinh(sqrt(c)*x/sqrt(a))/(8*sqrt(c)) +
a*d*h*Piecewise((sqrt(a)*x**2/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + a*e*g*Piecewise((sqrt(a)*x**2
/2, Eq(c, 0)), ((a + c*x**2)**(3/2)/(3*c), True)) + a*f*h*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x*
*2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + c*d*h*Piecewise((-2
*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(
a)*x**4/4, True)) + c*e*g*Piecewise((-2*a**2*sqrt(a + c*x**2)/(15*c**2) + a*x**2*sqrt(a + c*x**2)/(15*c) + x**
4*sqrt(a + c*x**2)/5, Ne(c, 0)), (sqrt(a)*x**4/4, True)) + c*f*h*Piecewise((8*a**3*sqrt(a + c*x**2)/(105*c**3)
- 4*a**2*x**2*sqrt(a + c*x**2)/(105*c**2) + a*x**4*sqrt(a + c*x**2)/(35*c) + x**6*sqrt(a + c*x**2)/7, Ne(c, 0
)), (sqrt(a)*x**6/6, True)) + c**2*d*g*x**5/(4*sqrt(a)*sqrt(1 + c*x**2/a)) + c**2*e*h*x**7/(6*sqrt(a)*sqrt(1 +
c*x**2/a)) + c**2*f*g*x**7/(6*sqrt(a)*sqrt(1 + c*x**2/a))

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