Optimal. Leaf size=52 \[ \frac{2 x}{a \sqrt{a x^2+b x^3}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.0583374, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2023, 2008, 206} \[ \frac{2 x}{a \sqrt{a x^2+b x^3}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2023
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a x^2+b x^3\right )^{3/2}} \, dx &=\frac{2 x}{a \sqrt{a x^2+b x^3}}+\frac{\int \frac{1}{\sqrt{a x^2+b x^3}} \, dx}{a}\\ &=\frac{2 x}{a \sqrt{a x^2+b x^3}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x}{\sqrt{a x^2+b x^3}}\right )}{a}\\ &=\frac{2 x}{a \sqrt{a x^2+b x^3}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0085071, size = 35, normalized size = 0.67 \[ \frac{2 x \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x}{a}+1\right )}{a \sqrt{x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 53, normalized size = 1. \begin{align*} 2\,{\frac{{x}^{3} \left ( bx+a \right ) }{ \left ( b{x}^{3}+a{x}^{2} \right ) ^{3/2}{a}^{5/2}} \left ({a}^{3/2}-{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) a\sqrt{bx+a} \right ) } \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^{2}}{{\left (b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.801286, size = 333, normalized size = 6.4 \begin{align*} \left [\frac{{\left (b x^{2} + a x\right )} \sqrt{a} \log \left (\frac{b x^{2} + 2 \, a x - 2 \, \sqrt{b x^{3} + a x^{2}} \sqrt{a}}{x^{2}}\right ) + 2 \, \sqrt{b x^{3} + a x^{2}} a}{a^{2} b x^{2} + a^{3} x}, \frac{2 \,{\left ({\left (b x^{2} + a x\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}{a x}\right ) + \sqrt{b x^{3} + a x^{2}} a\right )}}{a^{2} b x^{2} + a^{3} x}\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^{2}}{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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