3.33 \(\int \frac{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\left (c+d x^2\right )^{7/2}} \, dx\)

Optimal. Leaf size=376 \[ -\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (a d (4 d e-c f)+b c (d e-4 c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 d^2 \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{e+f x^2} (d e (4 a d+b c)-c f (a d+4 b c))}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}+\frac{\sqrt{e+f x^2} \left (a d \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+3 c d e f+2 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} d^{5/2} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{x \left (e+f x^2\right )^{3/2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]

[Out]

((d*(b*c + 4*a*d)*e - c*(4*b*c + a*d)*f)*x*Sqrt[e + f*x^2])/(15*c^2*d^2*(c + d*x
^2)^(3/2)) - ((b*c - a*d)*x*(e + f*x^2)^(3/2))/(5*c*d*(c + d*x^2)^(5/2)) + ((b*c
*(2*d^2*e^2 + 3*c*d*e*f - 8*c^2*f^2) + a*d*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2))*
Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(
5/2)*d^(5/2)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])
- (e^(3/2)*Sqrt[f]*(b*c*(d*e - 4*c*f) + a*d*(4*d*e - c*f))*Sqrt[c + d*x^2]*Ellip
ticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*d^2*(d*e - c*f)*Sqrt
[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 1.13699, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{e^{3/2} \sqrt{f} \sqrt{c+d x^2} (a d (4 d e-c f)+b c (d e-4 c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{15 c^3 d^2 \sqrt{e+f x^2} (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{x \sqrt{e+f x^2} (d e (4 a d+b c)-c f (a d+4 b c))}{15 c^2 d^2 \left (c+d x^2\right )^{3/2}}+\frac{\sqrt{e+f x^2} \left (a d \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )+b c \left (-8 c^2 f^2+3 c d e f+2 d^2 e^2\right )\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{15 c^{5/2} d^{5/2} \sqrt{c+d x^2} (d e-c f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{x \left (e+f x^2\right )^{3/2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(7/2),x]

[Out]

((d*(b*c + 4*a*d)*e - c*(4*b*c + a*d)*f)*x*Sqrt[e + f*x^2])/(15*c^2*d^2*(c + d*x
^2)^(3/2)) - ((b*c - a*d)*x*(e + f*x^2)^(3/2))/(5*c*d*(c + d*x^2)^(5/2)) + ((b*c
*(2*d^2*e^2 + 3*c*d*e*f - 8*c^2*f^2) + a*d*(8*d^2*e^2 - 3*c*d*e*f - 2*c^2*f^2))*
Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(15*c^(
5/2)*d^(5/2)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])
- (e^(3/2)*Sqrt[f]*(b*c*(d*e - 4*c*f) + a*d*(4*d*e - c*f))*Sqrt[c + d*x^2]*Ellip
ticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(15*c^3*d^2*(d*e - c*f)*Sqrt
[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi in Sympy [A]  time = 108.873, size = 333, normalized size = 0.89 \[ \frac{x \left (e + f x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{5 c d \left (c + d x^{2}\right )^{\frac{5}{2}}} - \frac{x \sqrt{e + f x^{2}} \left (c f \left (a d + 4 b c\right ) - d e \left (4 a d + b c\right )\right )}{15 c^{2} d^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}} - \frac{f \sqrt{e + f x^{2}} \left (c f \left (a d + 4 b c\right ) - d e \left (4 a d + b c\right )\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{15 c^{\frac{3}{2}} d^{\frac{5}{2}} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (c f - d e\right )} + \frac{\sqrt{e + f x^{2}} \left (c f \left (2 c f \left (a d + 4 b c\right ) + d e \left (4 a d + b c\right )\right ) - d e \left (c f \left (a d + 4 b c\right ) + 2 d e \left (4 a d + b c\right )\right )\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{15 c^{\frac{5}{2}} d^{\frac{5}{2}} \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)*(f*x**2+e)**(3/2)/(d*x**2+c)**(7/2),x)

[Out]

x*(e + f*x**2)**(3/2)*(a*d - b*c)/(5*c*d*(c + d*x**2)**(5/2)) - x*sqrt(e + f*x**
2)*(c*f*(a*d + 4*b*c) - d*e*(4*a*d + b*c))/(15*c**2*d**2*(c + d*x**2)**(3/2)) -
f*sqrt(e + f*x**2)*(c*f*(a*d + 4*b*c) - d*e*(4*a*d + b*c))*elliptic_f(atan(sqrt(
d)*x/sqrt(c)), -c*f/(d*e) + 1)/(15*c**(3/2)*d**(5/2)*sqrt(c*(e + f*x**2)/(e*(c +
d*x**2)))*sqrt(c + d*x**2)*(c*f - d*e)) + sqrt(e + f*x**2)*(c*f*(2*c*f*(a*d + 4
*b*c) + d*e*(4*a*d + b*c)) - d*e*(c*f*(a*d + 4*b*c) + 2*d*e*(4*a*d + b*c)))*elli
ptic_e(atan(sqrt(d)*x/sqrt(c)), -c*f/(d*e) + 1)/(15*c**(5/2)*d**(5/2)*sqrt(c*(e
+ f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)*(c*f - d*e))

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Mathematica [C]  time = 2.35337, size = 382, normalized size = 1.02 \[ \frac{\sqrt{\frac{d}{c}} \left (-x \sqrt{\frac{d}{c}} \left (e+f x^2\right ) \left (\left (c+d x^2\right )^2 \left (a d \left (2 c^2 f^2+3 c d e f-8 d^2 e^2\right )+b c \left (8 c^2 f^2-3 c d e f-2 d^2 e^2\right )\right )+3 c^2 (b c-a d) (d e-c f)^2-c \left (c+d x^2\right ) (d e-c f) (2 a d (c f+2 d e)+b c (d e-7 c f))\right )-i e \sqrt{\frac{d x^2}{c}+1} \left (c+d x^2\right )^2 \sqrt{\frac{f x^2}{e}+1} \left (\left (a d \left (2 c^2 f^2+3 c d e f-8 d^2 e^2\right )+b c \left (8 c^2 f^2-3 c d e f-2 d^2 e^2\right )\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+(d e-c f) (a d (c f+8 d e)+2 b c (2 c f+d e)) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )\right )}{15 c^2 d^3 \left (c+d x^2\right )^{5/2} \sqrt{e+f x^2} (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)*(e + f*x^2)^(3/2))/(c + d*x^2)^(7/2),x]

[Out]

(Sqrt[d/c]*(-(Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)*(d*e - c*f)^2 - c*(d*e
- c*f)*(b*c*(d*e - 7*c*f) + 2*a*d*(2*d*e + c*f))*(c + d*x^2) + (a*d*(-8*d^2*e^2
+ 3*c*d*e*f + 2*c^2*f^2) + b*c*(-2*d^2*e^2 - 3*c*d*e*f + 8*c^2*f^2))*(c + d*x^2)
^2)) - I*e*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*((a*d*(-8*d^2*e
^2 + 3*c*d*e*f + 2*c^2*f^2) + b*c*(-2*d^2*e^2 - 3*c*d*e*f + 8*c^2*f^2))*Elliptic
E[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (d*e - c*f)*(a*d*(8*d*e + c*f) + 2*b*c*
(d*e + 2*c*f))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])))/(15*c^2*d^3*(d*
e - c*f)*(c + d*x^2)^(5/2)*Sqrt[e + f*x^2])

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Maple [B]  time = 0.055, size = 2860, normalized size = 7.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(f*x^2+e)^(3/2)/(d*x^2+c)^(7/2),x)

[Out]

-1/15*(-4*x^3*b*c^5*f^3*(-d/c)^(1/2)+2*x^5*b*c*d^4*e^3*(-d/c)^(1/2)-x^3*a*c^4*d*
f^3*(-d/c)^(1/2)+20*x^3*a*c*d^4*e^3*(-d/c)^(1/2)+5*x^3*b*c^2*d^3*e^3*(-d/c)^(1/2
)+15*x*a*c^2*d^3*e^3*(-d/c)^(1/2)-4*x*b*c^5*e*f^2*(-d/c)^(1/2)-2*x^7*a*c^2*d^3*f
^3*(-d/c)^(1/2)-3*x^7*a*c*d^4*e*f^2*(-d/c)^(1/2)+3*x^7*b*c^2*d^3*e*f^2*(-d/c)^(1
/2)+2*x^7*b*c*d^4*e^2*f*(-d/c)^(1/2)-10*x^5*a*c^2*d^3*e*f^2*(-d/c)^(1/2)+17*x^5*
a*c*d^4*e^2*f*(-d/c)^(1/2)-10*x^5*b*c^3*d^2*e*f^2*(-d/c)^(1/2)+8*x^5*b*c^2*d^3*e
^2*f*(-d/c)^(1/2)-17*x^3*a*c^3*d^2*e*f^2*(-d/c)^(1/2)+7*x^3*a*c^2*d^3*e^2*f*(-d/
c)^(1/2)-8*x^3*b*c^4*d*e*f^2*(-d/c)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(
1/2))*x^4*b*c*d^4*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*x^3*b*c^3*d^2*e^
2*f*(-d/c)^(1/2)-x*a*c^4*d*e*f^2*(-d/c)^(1/2)-11*x*a*c^3*d^2*e^2*f*(-d/c)^(1/2)+
x*b*c^4*d*e^2*f*(-d/c)^(1/2)+8*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*d
^5*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*EllipticE(x*(-d/c)^(1/2),(c*f/d
/e)^(1/2))*x^4*a*d^5*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*EllipticF(x*(
-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/
2)-4*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^5*e*f^2*((d*x^2+c)/c)^(1/2)*(
(f*x^2+e)/e)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^3*d^2*e^3*((d
*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))
*a*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*EllipticE(x*(-d/c)^(1/2
),(c*f/d/e)^(1/2))*b*c^5*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*Ellipti
cE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^3*d^2*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/
e)^(1/2)+8*x^7*a*d^5*e^2*f*(-d/c)^(1/2)-8*x^7*b*c^3*d^2*f^3*(-d/c)^(1/2)-6*x^5*a
*c^3*d^2*f^3*(-d/c)^(1/2)-9*x^5*b*c^4*d*f^3*(-d/c)^(1/2)+2*EllipticE(x*(-d/c)^(1
/2),(c*f/d/e)^(1/2))*x^4*a*c^2*d^3*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)
+3*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c*d^4*e^2*f*((d*x^2+c)/c)^(1/
2)*((f*x^2+e)/e)^(1/2)+8*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*b*c^3*d^2
*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*EllipticE(x*(-d/c)^(1/2),(c*f/d
/e)^(1/2))*x^4*b*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*Ellipti
cF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^3*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x
^2+e)/e)^(1/2)+4*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c^3*d^2*e*f^2*(
(d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+6*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2
))*x^2*a*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+16*EllipticE(x*(-
d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^4*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^
(1/2)-6*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^3*d^2*e^2*f*((d*x^2+c)
/c)^(1/2)*((f*x^2+e)/e)^(1/2)-14*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a
*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*EllipticF(x*(-d/c)^(1/2
),(c*f/d/e)^(1/2))*x^2*b*c^4*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*E
llipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*b*c*d^4*e^3*((d*x^2+c)/c)^(1/2)*((f
*x^2+e)/e)^(1/2)+16*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c*d^4*e^3*((
d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+4*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2)
)*x^2*b*c^2*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-16*EllipticE(x*(-d/c
)^(1/2),(c*f/d/e)^(1/2))*x^2*a*c*d^4*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)
-4*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*b*c^2*d^3*e^3*((d*x^2+c)/c)^(1/
2)*((f*x^2+e)/e)^(1/2)-EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^4*d*e*f^2*(
(d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-7*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2
))*a*c^3*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*EllipticF(x*(-d/c)^
(1/2),(c*f/d/e)^(1/2))*b*c^4*d*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*E
llipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^4*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x
^2+e)/e)^(1/2)+3*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c^3*d^2*e^2*f*((d*x
^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b
*c^4*d*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+4*EllipticF(x*(-d/c)^(1/2),
(c*f/d/e)^(1/2))*x^2*b*c^3*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-Ell
ipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c^2*d^3*e*f^2*((d*x^2+c)/c)^(1/2)*(
(f*x^2+e)/e)^(1/2)-7*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*a*c*d^4*e^2*f
*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-4*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1
/2))*x^4*b*c^3*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*EllipticF(x*(
-d/c)^(1/2),(c*f/d/e)^(1/2))*x^4*b*c^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/
e)^(1/2)+8*x^5*a*d^5*e^3*(-d/c)^(1/2))/(f*x^2+e)^(1/2)/c^3/(c*f-d*e)/(-d/c)^(1/2
)/(d*x^2+c)^(5/2)/d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(7/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b f x^{4} +{\left (b e + a f\right )} x^{2} + a e\right )} \sqrt{f x^{2} + e}}{{\left (d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}\right )} \sqrt{d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(7/2),x, algorithm="fricas")

[Out]

integral((b*f*x^4 + (b*e + a*f)*x^2 + a*e)*sqrt(f*x^2 + e)/((d^3*x^6 + 3*c*d^2*x
^4 + 3*c^2*d*x^2 + c^3)*sqrt(d*x^2 + c)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)*(f*x**2+e)**(3/2)/(d*x**2+c)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(7/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)*(f*x^2 + e)^(3/2)/(d*x^2 + c)^(7/2), x)