Optimal. Leaf size=36 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a x^2+b x^5}}\right )}{3 \sqrt{b}} \]
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Rubi [A] time = 0.0475052, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2029, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a x^2+b x^5}}\right )}{3 \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 2029
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\sqrt{a x^2+b x^5}} \, dx &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^{5/2}}{\sqrt{a x^2+b x^5}}\right )\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a x^2+b x^5}}\right )}{3 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0120326, size = 59, normalized size = 1.64 \[ \frac{2 x \sqrt{a+b x^3} \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a+b x^3}}\right )}{3 \sqrt{b} \sqrt{x^2 \left (a+b x^3\right )}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.118, size = 480, normalized size = 13.3 \begin{align*} -4\,{\frac{{x}^{3/2} \left ( b{x}^{3}+a \right ) \left ( -1+i\sqrt{3} \right ) \left ( -bx+\sqrt [3]{-{b}^{2}a} \right ) ^{2}}{\sqrt{b{x}^{5}+a{x}^{2}}{b}^{2}\sqrt{ \left ( b{x}^{3}+a \right ) x} \left ( i\sqrt{3}-3 \right ) }\sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xb}{ \left ( -1+i\sqrt{3} \right ) \left ( -bx+\sqrt [3]{-{b}^{2}a} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-{b}^{2}a}+2\,bx+\sqrt [3]{-{b}^{2}a}}{ \left ( 1+i\sqrt{3} \right ) \left ( -bx+\sqrt [3]{-{b}^{2}a} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-{b}^{2}a}-2\,bx-\sqrt [3]{-{b}^{2}a}}{ \left ( -1+i\sqrt{3} \right ) \left ( -bx+\sqrt [3]{-{b}^{2}a} \right ) }}} \left ({\it EllipticF} \left ( \sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xb}{ \left ( -1+i\sqrt{3} \right ) \left ( -bx+\sqrt [3]{-{b}^{2}a} \right ) }}},\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) \left ( -1+i\sqrt{3} \right ) }{ \left ( i\sqrt{3}-3 \right ) \left ( 1+i\sqrt{3} \right ) }}} \right ) -{\it EllipticPi} \left ( \sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xb}{ \left ( -1+i\sqrt{3} \right ) \left ( -bx+\sqrt [3]{-{b}^{2}a} \right ) }}},{\frac{-1+i\sqrt{3}}{i\sqrt{3}-3}},\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) \left ( -1+i\sqrt{3} \right ) }{ \left ( i\sqrt{3}-3 \right ) \left ( 1+i\sqrt{3} \right ) }}} \right ) \right ){\frac{1}{\sqrt{{\frac{x \left ( -bx+\sqrt [3]{-{b}^{2}a} \right ) \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}+2\,bx+\sqrt [3]{-{b}^{2}a} \right ) \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}-2\,bx-\sqrt [3]{-{b}^{2}a} \right ) }{{b}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33068, size = 247, normalized size = 6.86 \begin{align*} \left [\frac{\log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - 4 \, \sqrt{b x^{5} + a x^{2}}{\left (2 \, b x^{3} + a\right )} \sqrt{b} \sqrt{x} - a^{2}\right )}{6 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{2 \, \sqrt{b x^{5} + a x^{2}} \sqrt{-b} \sqrt{x}}{2 \, b x^{3} + a}\right )}{3 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}}}{\sqrt{x^{2} \left (a + b x^{3}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28855, size = 55, normalized size = 1.53 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{b + \frac{a}{x^{3}}}}{\sqrt{-b}}\right )}{3 \, \sqrt{-b}} + \frac{2 \, \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right )}{3 \, \sqrt{-b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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