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

Optimal. Leaf size=539 \[ \frac{b^3 c^{3/2} \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a \sqrt{d} e \sqrt{c+d x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{b^2 \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{\sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d \sqrt{f} \sqrt{c+d x^2} \left (2 b c^2 f-a d (c f+d e)\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d^2 x}{c \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (d e-c f)}-\frac{d^2 \sqrt{e} \sqrt{c+d x^2} (2 a d f-3 b c f+b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

[Out]

-((d^2*x)/(c*(b*c - a*d)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) - (b^2*Sq
rt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(
(b*c - a*d)^2*Sqrt[e]*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e +
f*x^2]) - (d*Sqrt[f]*(2*b*c^2*f - a*d*(d*e + c*f))*Sqrt[c + d*x^2]*EllipticE[Ar
cTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[e]*(d*e - c*f
)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d^2*Sqrt[e]*(b*d*e
- 3*b*c*f + 2*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 -
(d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[f]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e
+ f*x^2))]*Sqrt[e + f*x^2]) + (b^3*c^(3/2)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)
/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*Sqrt[d]*(b*c - a*d)^2*
e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])

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Rubi [A]  time = 1.40994, antiderivative size = 539, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219 \[ \frac{b^3 c^{3/2} \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a \sqrt{d} e \sqrt{c+d x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{b^2 \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{\sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d \sqrt{f} \sqrt{c+d x^2} \left (2 b c^2 f-a d (c f+d e)\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d^2 x}{c \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (d e-c f)}-\frac{d^2 \sqrt{e} \sqrt{c+d x^2} (2 a d f-3 b c f+b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

-((d^2*x)/(c*(b*c - a*d)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) - (b^2*Sq
rt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(
(b*c - a*d)^2*Sqrt[e]*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e +
f*x^2]) - (d*Sqrt[f]*(2*b*c^2*f - a*d*(d*e + c*f))*Sqrt[c + d*x^2]*EllipticE[Ar
cTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[e]*(d*e - c*f
)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d^2*Sqrt[e]*(b*d*e
- 3*b*c*f + 2*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 -
(d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[f]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e
+ f*x^2))]*Sqrt[e + f*x^2]) + (b^3*c^(3/2)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)
/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(a*Sqrt[d]*(b*c - a*d)^2*
e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*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)**(3/2)/(f*x**2+e)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 7.18586, size = 1284, normalized size = 2.38 \[ \sqrt{d x^2+c} \sqrt{f x^2+e} \left (-\frac{x d^3}{c (b c-a d) (c f-d e)^2 \left (d x^2+c\right )}-\frac{f^3 x}{e (b e-a f) (d e-c f)^2 \left (f x^2+e\right )}\right )-\frac{\sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )} \left (\frac{i b^2 e f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right ) c^3}{a \sqrt{\frac{d}{c}} \sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )}}+\frac{i b d e f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right ) c^2}{\sqrt{\frac{d}{c}} \sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )}}+\frac{i b d e f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right ) c^2}{\sqrt{\frac{d}{c}} \sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )}}-\frac{2 i b^2 d e^2 f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right ) c^2}{a \sqrt{\frac{d}{c}} \sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )}}-\frac{i a d^2 e f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right ) c}{\sqrt{\frac{d}{c}} \sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )}}-\frac{2 i a d^2 e f^2 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right ) c}{\sqrt{\frac{d}{c}} \sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )}}+\frac{i b d^2 e^2 f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right ) c}{\sqrt{\frac{d}{c}} \sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )}}+\frac{i b^2 d^2 e^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right ) c}{a \sqrt{\frac{d}{c}} \sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )}}+\frac{i b d^3 e^3 \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{\sqrt{\frac{d}{c}} \sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )}}-\frac{i a d^3 e^2 f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{\sqrt{\frac{d}{c}} \sqrt{\left (d x^2+c\right ) \left (f x^2+e\right )}}\right )}{c (b c-a d) e (b e-a f) (c f-d e)^2 \sqrt{d x^2+c} \sqrt{f x^2+e}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x^2]*Sqrt[e + f*x^2]*(-((d^3*x)/(c*(b*c - a*d)*(-(d*e) + c*f)^2*(c +
d*x^2))) - (f^3*x)/(e*(b*e - a*f)*(d*e - c*f)^2*(e + f*x^2))) - (Sqrt[(c + d*x^2
)*(e + f*x^2)]*((I*b*d^3*e^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)]) - (I*a*d^3*e^2*f*Sqrt[1 + (d*x
^2)/c]*Sqrt[1 + (f*x^2)/e]*(EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*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)]) + (I*b*c^2*d*e*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)]) - (I*a*c*d^2*e*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)] - Ell
ipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f
*x^2)]) + (I*b*c*d^2*e^2*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*A
rcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)]) + (
I*b*c^2*d*e*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*c*d
^2*e*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*b^2*c*d^2*e^3*Sq
rt[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)]) - ((2*I)*b^2*c^2
*d*e^2*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSi
nh[Sqrt[d/c]*x], (c*f)/(d*e)])/(a*Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)]) + (I*
b^2*c^3*e*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*Sqrt[d/c]*Sqrt[(c + d*x^2)*(e + f*x^2)]))
)/(c*(b*c - a*d)*e*(b*e - a*f)*(-(d*e) + c*f)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]  time = 0.056, size = 956, normalized size = 1.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^3*a^2*c*d^2*f^3*(-d/c)^(1/2)+x^3*a^2*d^3*e*f^2*(-d/c)^(1/2)-x^3*a*b*c^2*d*f^3
*(-d/c)^(1/2)-x^3*a*b*d^3*e^2*f*(-d/c)^(1/2)-EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^
(1/2))*a^2*c*d^2*e*f^2*((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^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+E
llipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c*d^2*e^2*f*((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*b*d^3*e^3*((d*x^2+c
)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*c*d
^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-EllipticE(x*(-d/c)^(1/2),(c*f/d
/e)^(1/2))*a^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+EllipticE(x*(-d
/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c^2*d*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/
2)+EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f
*x^2+e)/e)^(1/2)+EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^
2*c^3*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*EllipticPi(x*(-d/c)^(1/2),
b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^2*c^2*d*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e
)/e)^(1/2)+EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^2*c*d^
2*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+x*a^2*c^2*d*f^3*(-d/c)^(1/2)+x*a^2
*d^3*e^2*f*(-d/c)^(1/2)-x*a*b*c^3*f^3*(-d/c)^(1/2)-x*a*b*d^3*e^3*(-d/c)^(1/2))*(
f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/e/c/(c*f-d*e)^2/a/(-d/c)^(1/2)/(a*f-b*e)/(a*d-b*c
)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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