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

Optimal. Leaf size=344 \[ \frac{b^2 e^{3/2} \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{e+f x^2} (b e-a f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f^{3/2} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{\sqrt{e} \sqrt{e+f x^2} (b e-a f) (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-a d f-b c f+2 b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{e+f x^2} (b e-a f)^2 (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

[Out]

(f^(3/2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)]
)/(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]) - (Sqrt[e]*Sqrt[f]*(2*b*d*e - b*c*f - a*d*f)*Sqrt[c + d*x^2]*EllipticF
[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*e - a*f)^2*(d*e - c*f)*Sqr
t[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*e^(3/2)*Sqrt[c + d*x^
2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a
*c*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 = 0.710956, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{b^2 e^{3/2} \sqrt{c+d x^2} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{e+f x^2} (b e-a f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f^{3/2} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{\sqrt{e} \sqrt{e+f x^2} (b e-a f) (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} (-a d f-b c f+2 b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{e+f x^2} (b e-a f)^2 (d e-c f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

(f^(3/2)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)]
)/(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]) - (Sqrt[e]*Sqrt[f]*(2*b*d*e - b*c*f - a*d*f)*Sqrt[c + d*x^2]*EllipticF
[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*e - a*f)^2*(d*e - c*f)*Sqr
t[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*e^(3/2)*Sqrt[c + d*x^
2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a
*c*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 [A]  time = 94.2393, size = 289, normalized size = 0.84 \[ - \frac{\sqrt{c} f \sqrt{e + f x^{2}} \left (a d f + b c f - 2 b d e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{\sqrt{d} e \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a f - b e\right )^{2} \left (c f - d e\right )} + \frac{f^{\frac{3}{2}} \sqrt{c + d x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{\sqrt{e} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}} \left (a f - b e\right ) \left (c f - d e\right )} + \frac{b^{2} e^{\frac{3}{2}} \sqrt{c + d x^{2}} \Pi \left (1 - \frac{b e}{a f}; \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{a c \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}} \left (a f - b e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c)*f*sqrt(e + f*x**2)*(a*d*f + b*c*f - 2*b*d*e)*elliptic_f(atan(sqrt(d)*x/
sqrt(c)), -c*f/(d*e) + 1)/(sqrt(d)*e*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(
c + d*x**2)*(a*f - b*e)**2*(c*f - d*e)) + f**(3/2)*sqrt(c + d*x**2)*elliptic_e(a
tan(sqrt(f)*x/sqrt(e)), 1 - d*e/(c*f))/(sqrt(e)*sqrt(e*(c + d*x**2)/(c*(e + f*x*
*2)))*sqrt(e + f*x**2)*(a*f - b*e)*(c*f - d*e)) + b**2*e**(3/2)*sqrt(c + d*x**2)
*elliptic_pi(1 - b*e/(a*f), atan(sqrt(f)*x/sqrt(e)), 1 - d*e/(c*f))/(a*c*sqrt(f)
*sqrt(e*(c + d*x**2)/(c*(e + f*x**2)))*sqrt(e + f*x**2)*(a*f - b*e)**2)

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Mathematica [C]  time = 0.919241, size = 221, normalized size = 0.64 \[ \frac{-i b e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i a d e f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-a f^2 x \sqrt{\frac{d}{c}} \left (c+d x^2\right )}{a e \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2} (a f-b e) (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

(-(a*Sqrt[d/c]*f^2*x*(c + d*x^2)) - I*a*d*e*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^
2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*b*e*(-(d*e) + c*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]*e*(-(b*e) + a*f)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt
[e + f*x^2])

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Maple [A]  time = 0.044, size = 303, normalized size = 0.9 \[{\frac{1}{ \left ( af-be \right ) ae \left ( cf-de \right ) \left ( df{x}^{4}+cf{x}^{2}+de{x}^{2}+ce \right ) } \left ({x}^{3}ad{f}^{2}\sqrt{-{\frac{d}{c}}}-\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) adef-{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{bc}{ad}},{1\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ) bcef\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{bc}{ad}},{1\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ) bd{e}^{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+xac{f}^{2}\sqrt{-{\frac{d}{c}}} \right ) \sqrt{f{x}^{2}+e}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^3*a*d*f^2*(-d/c)^(1/2)-((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*d*e*f-EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/
2)/(-d/c)^(1/2))*b*c*e*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*d*e^2*((d*x^2+c)/c)^(1/2)*((f*x^
2+e)/e)^(1/2)+x*a*c*f^2*(-d/c)^(1/2))*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/a/e/(a*f-b
*e)/(-d/c)^(1/2)/(c*f-d*e)/(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. \[ \int \frac{1}{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + c}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^2 + a)*sqrt(d*x^2 + c)*(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)*sqrt(d*x^2 + c)*(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. \[ \int \frac{1}{\left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (e + f x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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