3.88 \(\int \frac{1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=814 \[ \frac{e^{3/2} \sqrt{d x^2+c} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) b^4}{a c (b c-a d)^2 \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{f^{3/2} \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) b^2}{(b c-a d)^2 \sqrt{e} (b e-a f) (d e-c f) \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \sqrt{f} (2 b d e-b c f-a d f) \sqrt{d x^2+c} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) b^2}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{d \sqrt{f} \left (b c \left (5 d^2 e^2-7 c d f e-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d f e-3 c^2 f^2\right )\right ) \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 (b c-a d)^2 \sqrt{e} (d e-c f)^3 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d^2 \sqrt{e} \sqrt{f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt{d x^2+c} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt{d x^2+c} \sqrt{f x^2+e}}-\frac{d^2 x}{3 c (b c-a d) (d e-c f) \left (d x^2+c\right )^{3/2} \sqrt{f x^2+e}} \]

[Out]

-(d^2*x)/(3*c*(b*c - a*d)*(d*e - c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]) - (d^2*
(b*c*(5*d*e - 9*c*f) - 2*a*d*(d*e - 3*c*f))*x)/(3*c^2*(b*c - a*d)^2*(d*e - c*f)^
2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]) + (b^2*f^(3/2)*Sqrt[c + d*x^2]*EllipticE[ArcT
an[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/((b*c - a*d)^2*Sqrt[e]*(b*e - a*f)*(d
*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[f]*(b
*c*(5*d^2*e^2 - 7*c*d*e*f - 6*c^2*f^2) - a*d*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2)
)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^
2*(b*c - a*d)^2*Sqrt[e]*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt
[e + f*x^2]) - (b^2*Sqrt[e]*Sqrt[f]*(2*b*d*e - b*c*f - a*d*f)*Sqrt[c + d*x^2]*El
lipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*(b*e - a
*f)^2*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d^2*
Sqrt[e]*Sqrt[f]*(b*c*(7*d*e - 15*c*f) - a*d*(d*e - 9*c*f))*Sqrt[c + d*x^2]*Ellip
ticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(b*c - a*d)^2*(d*e -
c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^4*e^(3/2)*Sqr
t[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/
(c*f)])/(a*c*(b*c - a*d)^2*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*
x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 2.69766, antiderivative size = 814, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{e^{3/2} \sqrt{d x^2+c} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) b^4}{a c (b c-a d)^2 \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{f^{3/2} \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) b^2}{(b c-a d)^2 \sqrt{e} (b e-a f) (d e-c f) \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \sqrt{f} (2 b d e-b c f-a d f) \sqrt{d x^2+c} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) b^2}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{d \sqrt{f} \left (b c \left (5 d^2 e^2-7 c d f e-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d f e-3 c^2 f^2\right )\right ) \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 (b c-a d)^2 \sqrt{e} (d e-c f)^3 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d^2 \sqrt{e} \sqrt{f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt{d x^2+c} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt{d x^2+c} \sqrt{f x^2+e}}-\frac{d^2 x}{3 c (b c-a d) (d e-c f) \left (d x^2+c\right )^{3/2} \sqrt{f x^2+e}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^2)*(c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]

[Out]

-(d^2*x)/(3*c*(b*c - a*d)*(d*e - c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]) - (d^2*
(b*c*(5*d*e - 9*c*f) - 2*a*d*(d*e - 3*c*f))*x)/(3*c^2*(b*c - a*d)^2*(d*e - c*f)^
2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]) + (b^2*f^(3/2)*Sqrt[c + d*x^2]*EllipticE[ArcT
an[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/((b*c - a*d)^2*Sqrt[e]*(b*e - a*f)*(d
*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[f]*(b
*c*(5*d^2*e^2 - 7*c*d*e*f - 6*c^2*f^2) - a*d*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2)
)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^
2*(b*c - a*d)^2*Sqrt[e]*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt
[e + f*x^2]) - (b^2*Sqrt[e]*Sqrt[f]*(2*b*d*e - b*c*f - a*d*f)*Sqrt[c + d*x^2]*El
lipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*(b*e - a
*f)^2*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d^2*
Sqrt[e]*Sqrt[f]*(b*c*(7*d*e - 15*c*f) - a*d*(d*e - 9*c*f))*Sqrt[c + d*x^2]*Ellip
ticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(3*c^2*(b*c - a*d)^2*(d*e -
c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^4*e^(3/2)*Sqr
t[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/
(c*f)])/(a*c*(b*c - a*d)^2*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*
x^2))]*Sqrt[e + f*x^2])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)/(d*x**2+c)**(5/2)/(f*x**2+e)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 9.02774, size = 2744, normalized size = 3.37 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x^2)*(c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]

[Out]

Sqrt[c + d*x^2]*Sqrt[e + f*x^2]*(-(d^3*x)/(3*c*(b*c - a*d)*(-(d*e) + c*f)^2*(c +
d*x^2)^2) - (d^3*(-5*b*c*d*e + 2*a*d^2*e + 10*b*c^2*f - 7*a*c*d*f)*x)/(3*c^2*(b
*c - a*d)^2*(-(d*e) + c*f)^3*(c + d*x^2)) + (f^4*x)/(e*(b*e - a*f)*(d*e - c*f)^3
*(e + f*x^2))) + (Sqrt[(c + d*x^2)*(e + f*x^2)]*(((5*I)*b^2*c*d^4*e^4*Sqrt[1 + (
d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] -
EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e
+ f*x^2)]) - ((2*I)*a*b*d^5*e^4*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(Ellipti
cE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f
)/(d*e)]))/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)]) - ((10*I)*b^2*c^2*d^3*e^3*f
*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f
)/(d*e)] - EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c]*Sqrt[(c +
d*x^2)*(e + f*x^2)]) + ((2*I)*a*b*c*d^4*e^3*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x
^2)/e]*(EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - EllipticF[I*ArcSinh[Sqr
t[d/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)]) + ((2*I)*a^2
*d^5*e^3*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(EllipticE[I*ArcSinh[Sqrt[d/c
]*x], (c*f)/(d*e)] - EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c]
*Sqrt[(c + d*x^2)*(e + f*x^2)]) + ((10*I)*a*b*c^2*d^3*e^2*f^2*Sqrt[1 + (d*x^2)/c
]*Sqrt[1 + (f*x^2)/e]*(EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - Elliptic
F[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)
]) - ((7*I)*a^2*c*d^4*e^2*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(EllipticE
[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/
(d*e)]))/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)]) - ((3*I)*b^2*c^4*d*e*f^3*Sqrt
[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*
e)] - EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c]*Sqrt[(c + d*x^
2)*(e + f*x^2)]) + ((6*I)*a*b*c^3*d^2*e*f^3*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)
/e]*(EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - EllipticF[I*ArcSinh[Sqrt[d
/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)]) - ((3*I)*a^2*c^
2*d^3*e*f^3*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(EllipticE[I*ArcSinh[Sqrt[d/
c]*x], (c*f)/(d*e)] - EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c
]*Sqrt[(c + d*x^2)*(e + f*x^2)]) + ((4*I)*b^2*c^2*d^3*e^3*f*Sqrt[1 + (d*x^2)/c]*
Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*S
qrt[(c + d*x^2)*(e + f*x^2)]) - (I*a*b*c*d^4*e^3*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 +
(f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*Sqrt[(c +
d*x^2)*(e + f*x^2)]) - ((9*I)*b^2*c^3*d^2*e^2*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (
f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*Sqrt[(c + d
*x^2)*(e + f*x^2)]) + ((2*I)*a*b*c^2*d^3*e^2*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f
*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*Sqrt[(c + d*
x^2)*(e + f*x^2)]) + (I*a^2*c*d^4*e^2*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e
]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e
+ f*x^2)]) - ((3*I)*b^2*c^4*d*e*f^3*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Ell
ipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*
x^2)]) + ((15*I)*a*b*c^3*d^2*e*f^3*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Ellip
ticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^
2)]) - ((9*I)*a^2*c^2*d^3*e*f^3*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Elliptic
F[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)]
) + ((3*I)*b^3*c^2*d^3*e^4*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b
*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*Sqrt[d/c]*Sqrt[(c + d*x^2)*(
e + f*x^2)]) - ((9*I)*b^3*c^3*d^2*e^3*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*
EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*Sqrt[d/c]*Sqrt[
(c + d*x^2)*(e + f*x^2)]) + ((9*I)*b^3*c^4*d*e^2*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1
+ (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*Sq
rt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)]) - ((3*I)*b^3*c^5*e*f^3*Sqrt[1 + (d*x^2)/c
]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e
)])/(a*Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)])))/(3*c^2*(b*c - a*d)^2*e*(b*e -
a*f)*(-(d*e) + c*f)^3*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.088, size = 4115, normalized size = 5.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x)

[Out]

-1/3*(-3*x*a^2*b*c*d^5*e^4*(-d/c)^(1/2)+6*x*a*b^2*c^2*d^4*e^4*(-d/c)^(1/2)+3*Ell
ipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^3*c^6*e*f^3*((d*x^2+
c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)
/(-d/c)^(1/2))*b^3*c^3*d^3*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*Ellipti
cF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^3*c*d^5*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f
*x^2+e)/e)^(1/2)+5*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b^2*c*d^5*e^4
*((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))*a^2*b*c^4*d^2*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-10*EllipticE(x*
(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b*c^3*d^3*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+
e)/e)^(1/2)-2*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b*c^2*d^4*e^3*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))*
a*b^2*c^5*d*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+10*EllipticE(x*(-d/c)^
(1/2),(c*f/d/e)^(1/2))*a*b^2*c^3*d^3*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/
2)-9*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*b*c^4*d^2*e*f^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^2*b*c^3
*d^3*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticF(x*(-d/c)^(1/2),
(c*f/d/e)^(1/2))*a^2*b*c^2*d^4*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+9*E
llipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^2*c^4*d^2*e^2*f^2*((d*x^2+c)/c)^(1/
2)*((f*x^2+e)/e)^(1/2)-3*x^5*a^3*c^2*d^4*f^4*(-d/c)^(1/2)+2*x^5*a^3*d^6*e^2*f^2*
(-d/c)^(1/2)-6*x^3*a^3*c^3*d^3*f^4*(-d/c)^(1/2)+2*x^3*a^3*d^6*e^3*f*(-d/c)^(1/2)
-2*x^3*a^2*b*d^6*e^4*(-d/c)^(1/2)-3*x*a^3*c^4*d^2*f^4*(-d/c)^(1/2)-3*x*a*b^2*c^6
*f^4*(-d/c)^(1/2)+2*x*a^2*b*c^2*d^4*e^3*f*(-d/c)^(1/2)-11*x*a*b^2*c^3*d^3*e^3*f*
(-d/c)^(1/2)-3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*x^2*
b^3*c^2*d^4*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*EllipticE(x*(-d/c)^(1/
2),(c*f/d/e)^(1/2))*x^2*a^3*d^6*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*
EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*b*d^6*e^4*((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^2*a^3*d^6*e^3*
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))*x^2*a^2*b*d^6*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-9*EllipticPi(x*(
-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^3*c^5*d*e^2*f^2*((d*x^2+c)/c)^(
1/2)*((f*x^2+e)/e)^(1/2)+9*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)
^(1/2))*b^3*c^4*d^2*e^3*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))*a^3*c^3*d^3*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e
)^(1/2)+7*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^3*c^2*d^4*e^2*f^2*((d*x^2+
c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^3*
c*d^5*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*EllipticE(x*(-d/c)^(1/2),(
c*f/d/e)^(1/2))*a^2*b*c*d^5*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-5*Ellipt
icE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^2*c^2*d^4*e^4*((d*x^2+c)/c)^(1/2)*((f*x^
2+e)/e)^(1/2)+6*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^3*c^3*d^3*e*f^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^3*c^2*d^4*e^2*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))*a^3*c*d^5*e^3*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))*a^2*b*c*d^5*e^4*((d*x^2+c)/c)^(1/2)*
((f*x^2+e)/e)^(1/2)+5*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b^2*c^2*d^4*e^
4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+10*x^5*a^2*b*c^2*d^4*e*f^3*(-d/c)^(1/2
)+2*x^5*a^2*b*c*d^5*e^2*f^2*(-d/c)^(1/2)-10*x^5*a*b^2*c^2*d^4*e^2*f^2*(-d/c)^(1/
2)+5*x^5*a*b^2*c*d^5*e^3*f*(-d/c)^(1/2)+11*x^3*a^2*b*c^3*d^3*e*f^3*(-d/c)^(1/2)+
12*x^3*a^2*b*c^2*d^4*e^2*f^2*(-d/c)^(1/2)-x^3*a^2*b*c*d^5*e^3*f*(-d/c)^(1/2)-11*
x^3*a*b^2*c^3*d^3*e^2*f^2*(-d/c)^(1/2)-4*x^3*a*b^2*c^2*d^4*e^3*f*(-d/c)^(1/2)+11
*x*a^2*b*c^3*d^3*e^2*f^2*(-d/c)^(1/2)-6*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2)
)*x^2*a^2*b*c^3*d^3*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-10*EllipticE(x
*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*b*c^2*d^4*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f
*x^2+e)/e)^(1/2)-2*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*b*c*d^5*e^3
*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))*x^2*a*b^2*c^4*d^2*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+10*Ellipt
icE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b^2*c^2*d^4*e^3*f*((d*x^2+c)/c)^(1/2)*
((f*x^2+e)/e)^(1/2)-9*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*b*c^3*d^
3*e*f^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))*x^2*a^2*b*c^2*d^4*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*
EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*b*c*d^5*e^3*f*((d*x^2+c)/c)^(1
/2)*((f*x^2+e)/e)^(1/2)+9*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b^2*c^
3*d^3*e^2*f^2*((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))*a*b^2*c^3*d^3*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3
*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*x^2*b^3*c^5*d*e*f^
3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-9*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-
f/e)^(1/2)/(-d/c)^(1/2))*x^2*b^3*c^4*d^2*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/
e)^(1/2)+9*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*x^2*b^3*
c^3*d^3*e^3*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))*x^2*a^3*c^2*d^4*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+
7*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^3*c*d^5*e^2*f^2*((d*x^2+c)/c)^
(1/2)*((f*x^2+e)/e)^(1/2)-5*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b^2*
c*d^5*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+6*EllipticF(x*(-d/c)^(1/2),(c*
f/d/e)^(1/2))*x^2*a^3*c^2*d^4*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-7*x^
5*a^3*c*d^5*e*f^3*(-d/c)^(1/2)+6*x^5*a^2*b*c^3*d^3*f^4*(-d/c)^(1/2)-2*x^5*a^2*b*
d^6*e^3*f*(-d/c)^(1/2)-3*x^5*a*b^2*c^4*d^2*f^4*(-d/c)^(1/2)-8*x^3*a^3*c^2*d^4*e*
f^3*(-d/c)^(1/2)-4*x^3*a^3*c*d^5*e^2*f^2*(-d/c)^(1/2)+12*x^3*a^2*b*c^4*d^2*f^4*(
-d/c)^(1/2)-6*x^3*a*b^2*c^5*d*f^4*(-d/c)^(1/2)+5*x^3*a*b^2*c*d^5*e^4*(-d/c)^(1/2
)-8*x*a^3*c^2*d^4*e^2*f^2*(-d/c)^(1/2)+3*x*a^3*c*d^5*e^3*f*(-d/c)^(1/2)+6*x*a^2*
b*c^5*d*f^4*(-d/c)^(1/2)-14*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b^2*
c^2*d^4*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2))/(f*x^2+e)^(1/2)/(c*f-d*e)
^3/(a*d-b*c)^2/c^2/(-d/c)^(1/2)/(a*f-b*e)/e/a/(d*x^2+c)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)),x, algorithm="fricas")

[Out]

Timed out

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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(1/(b*x**2+a)/(d*x**2+c)**(5/2)/(f*x**2+e)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)), x)