Optimal. Leaf size=229 \[ \frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b x^3}}-\frac{2 \sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b x^3}}+\frac{2 x \left (a+b x^2\right )}{\sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}} \]
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Rubi [A] time = 0.340734, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b x^3}}-\frac{2 \sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{b^{3/4} \sqrt{a x+b x^3}}+\frac{2 x \left (a+b x^2\right )}{\sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}} \]
Antiderivative was successfully verified.
[In] Int[x/Sqrt[a*x + b*x^3],x]
[Out]
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Rubi in Sympy [A] time = 31.567, size = 214, normalized size = 0.93 \[ - \frac{2 \sqrt [4]{a} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt{x} \left (a + b x^{2}\right )} + \frac{\sqrt [4]{a} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{3}{4}} \sqrt{x} \left (a + b x^{2}\right )} + \frac{2 \sqrt{a x + b x^{3}}}{\sqrt{b} \left (\sqrt{a} + \sqrt{b} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**3+a*x)**(1/2),x)
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Mathematica [C] time = 0.0761848, size = 108, normalized size = 0.47 \[ \frac{2 i x^2 \sqrt{\frac{b x^2}{a}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{b} x}{\sqrt{a}}\right )^{3/2} \sqrt{x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[x/Sqrt[a*x + b*x^3],x]
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Maple [A] time = 0.023, size = 158, normalized size = 0.7 \[{\frac{1}{b}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^3+a*x)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{b x^{3} + a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(b*x^3 + a*x),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{b x^{3} + a x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(b*x^3 + a*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{x \left (a + b x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**3+a*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{b x^{3} + a x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(b*x^3 + a*x),x, algorithm="giac")
[Out]