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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]