3.222 \(\int \frac{\sqrt{\frac{a}{x^3}}}{\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=159 \[ \frac{2 \sqrt{x^2+1} x^2 \sqrt{\frac{a}{x^3}}}{x+1}-2 \sqrt{x^2+1} x \sqrt{\frac{a}{x^3}}+\frac{(x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} x^{3/2} \sqrt{\frac{a}{x^3}} F\left (2 \tan ^{-1}\left (\sqrt{x}\right )|\frac{1}{2}\right )}{\sqrt{x^2+1}}-\frac{2 (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} x^{3/2} \sqrt{\frac{a}{x^3}} E\left (2 \tan ^{-1}\left (\sqrt{x}\right )|\frac{1}{2}\right )}{\sqrt{x^2+1}} \]

[Out]

-2*Sqrt[a/x^3]*x*Sqrt[1 + x^2] + (2*Sqrt[a/x^3]*x^2*Sqrt[1 + x^2])/(1 + x) - (2*
Sqrt[a/x^3]*x^(3/2)*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*EllipticE[2*ArcTan[Sqrt[x]
], 1/2])/Sqrt[1 + x^2] + (Sqrt[a/x^3]*x^(3/2)*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*
EllipticF[2*ArcTan[Sqrt[x]], 1/2])/Sqrt[1 + x^2]

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Rubi [A]  time = 0.120332, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{2 \sqrt{x^2+1} x^2 \sqrt{\frac{a}{x^3}}}{x+1}-2 \sqrt{x^2+1} x \sqrt{\frac{a}{x^3}}+\frac{(x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} x^{3/2} \sqrt{\frac{a}{x^3}} F\left (2 \tan ^{-1}\left (\sqrt{x}\right )|\frac{1}{2}\right )}{\sqrt{x^2+1}}-\frac{2 (x+1) \sqrt{\frac{x^2+1}{(x+1)^2}} x^{3/2} \sqrt{\frac{a}{x^3}} E\left (2 \tan ^{-1}\left (\sqrt{x}\right )|\frac{1}{2}\right )}{\sqrt{x^2+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a/x^3]/Sqrt[1 + x^2],x]

[Out]

-2*Sqrt[a/x^3]*x*Sqrt[1 + x^2] + (2*Sqrt[a/x^3]*x^2*Sqrt[1 + x^2])/(1 + x) - (2*
Sqrt[a/x^3]*x^(3/2)*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*EllipticE[2*ArcTan[Sqrt[x]
], 1/2])/Sqrt[1 + x^2] + (Sqrt[a/x^3]*x^(3/2)*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*
EllipticF[2*ArcTan[Sqrt[x]], 1/2])/Sqrt[1 + x^2]

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Rubi in Sympy [A]  time = 15.9624, size = 148, normalized size = 0.93 \[ - \frac{2 x^{\frac{3}{2}} \sqrt{\frac{a}{x^{3}}} \sqrt{\frac{x^{2} + 1}{\left (x + 1\right )^{2}}} \left (x + 1\right ) E\left (2 \operatorname{atan}{\left (\sqrt{x} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{2} + 1}} + \frac{x^{\frac{3}{2}} \sqrt{\frac{a}{x^{3}}} \sqrt{\frac{x^{2} + 1}{\left (x + 1\right )^{2}}} \left (x + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt{x} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{2} + 1}} + \frac{2 x^{2} \sqrt{\frac{a}{x^{3}}} \sqrt{x^{2} + 1}}{x + 1} - 2 x \sqrt{\frac{a}{x^{3}}} \sqrt{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a/x**3)**(1/2)/(x**2+1)**(1/2),x)

[Out]

-2*x**(3/2)*sqrt(a/x**3)*sqrt((x**2 + 1)/(x + 1)**2)*(x + 1)*elliptic_e(2*atan(s
qrt(x)), 1/2)/sqrt(x**2 + 1) + x**(3/2)*sqrt(a/x**3)*sqrt((x**2 + 1)/(x + 1)**2)
*(x + 1)*elliptic_f(2*atan(sqrt(x)), 1/2)/sqrt(x**2 + 1) + 2*x**2*sqrt(a/x**3)*s
qrt(x**2 + 1)/(x + 1) - 2*x*sqrt(a/x**3)*sqrt(x**2 + 1)

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Mathematica [C]  time = 0.0337988, size = 74, normalized size = 0.47 \[ 2 x \sqrt{\frac{a}{x^3}} \left (-\sqrt{x^2+1}+(-1)^{3/4} \sqrt{x} \left (F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{x}\right )\right |-1\right )-E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{x}\right )\right |-1\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a/x^3]/Sqrt[1 + x^2],x]

[Out]

2*Sqrt[a/x^3]*x*(-Sqrt[1 + x^2] + (-1)^(3/4)*Sqrt[x]*(-EllipticE[I*ArcSinh[(-1)^
(1/4)*Sqrt[x]], -1] + EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[x]], -1]))

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Maple [C]  time = 0.044, size = 116, normalized size = 0.7 \[{x\sqrt{{\frac{a}{{x}^{3}}}} \left ( 2\,\sqrt{-i \left ( x+i \right ) }\sqrt{-i \left ( -x+i \right ) }\sqrt{ix}{\it EllipticE} \left ( \sqrt{-i \left ( x+i \right ) },1/2\,\sqrt{2} \right ) \sqrt{2}-\sqrt{-i \left ( x+i \right ) }\sqrt{-i \left ( -x+i \right ) }\sqrt{ix}{\it EllipticF} \left ( \sqrt{-i \left ( x+i \right ) },{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}-2\,{x}^{2}-2 \right ){\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a/x^3)^(1/2)/(x^2+1)^(1/2),x)

[Out]

(a/x^3)^(1/2)*x*(2*(-I*(x+I))^(1/2)*(-I*(-x+I))^(1/2)*(I*x)^(1/2)*EllipticE((-I*
(x+I))^(1/2),1/2*2^(1/2))*2^(1/2)-(-I*(x+I))^(1/2)*(-I*(-x+I))^(1/2)*(I*x)^(1/2)
*EllipticF((-I*(x+I))^(1/2),1/2*2^(1/2))*2^(1/2)-2*x^2-2)/(x^2+1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{3}}}}{\sqrt{x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^3)/sqrt(x^2 + 1),x, algorithm="maxima")

[Out]

integrate(sqrt(a/x^3)/sqrt(x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{\frac{a}{x^{3}}}}{\sqrt{x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^3)/sqrt(x^2 + 1),x, algorithm="fricas")

[Out]

integral(sqrt(a/x^3)/sqrt(x^2 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{3}}}}{\sqrt{x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a/x**3)**(1/2)/(x**2+1)**(1/2),x)

[Out]

Integral(sqrt(a/x**3)/sqrt(x**2 + 1), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{3}}}}{\sqrt{x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^3)/sqrt(x^2 + 1),x, algorithm="giac")

[Out]

integrate(sqrt(a/x^3)/sqrt(x^2 + 1), x)