Optimal. Leaf size=91 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}+\frac{a^2 x \sqrt{a+b x^2}}{16 b}+\frac{1}{8} a x^3 \sqrt{a+b x^2}+\frac{1}{6} x^3 \left (a+b x^2\right )^{3/2} \]
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Rubi [A] time = 0.0306598, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 217, 206} \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}+\frac{a^2 x \sqrt{a+b x^2}}{16 b}+\frac{1}{8} a x^3 \sqrt{a+b x^2}+\frac{1}{6} x^3 \left (a+b x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right )^{3/2} \, dx &=\frac{1}{6} x^3 \left (a+b x^2\right )^{3/2}+\frac{1}{2} a \int x^2 \sqrt{a+b x^2} \, dx\\ &=\frac{1}{8} a x^3 \sqrt{a+b x^2}+\frac{1}{6} x^3 \left (a+b x^2\right )^{3/2}+\frac{1}{8} a^2 \int \frac{x^2}{\sqrt{a+b x^2}} \, dx\\ &=\frac{a^2 x \sqrt{a+b x^2}}{16 b}+\frac{1}{8} a x^3 \sqrt{a+b x^2}+\frac{1}{6} x^3 \left (a+b x^2\right )^{3/2}-\frac{a^3 \int \frac{1}{\sqrt{a+b x^2}} \, dx}{16 b}\\ &=\frac{a^2 x \sqrt{a+b x^2}}{16 b}+\frac{1}{8} a x^3 \sqrt{a+b x^2}+\frac{1}{6} x^3 \left (a+b x^2\right )^{3/2}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{16 b}\\ &=\frac{a^2 x \sqrt{a+b x^2}}{16 b}+\frac{1}{8} a x^3 \sqrt{a+b x^2}+\frac{1}{6} x^3 \left (a+b x^2\right )^{3/2}-\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.115548, size = 83, normalized size = 0.91 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (3 a^2+14 a b x^2+8 b^2 x^4\right )-\frac{3 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{48 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 75, normalized size = 0.8 \begin{align*}{\frac{x}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{ax}{24\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}x}{16\,b}\sqrt{b{x}^{2}+a}}-{\frac{{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61282, size = 346, normalized size = 3.8 \begin{align*} \left [\frac{3 \, a^{3} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (8 \, b^{3} x^{5} + 14 \, a b^{2} x^{3} + 3 \, a^{2} b x\right )} \sqrt{b x^{2} + a}}{96 \, b^{2}}, \frac{3 \, a^{3} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (8 \, b^{3} x^{5} + 14 \, a b^{2} x^{3} + 3 \, a^{2} b x\right )} \sqrt{b x^{2} + a}}{48 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.97693, size = 119, normalized size = 1.31 \begin{align*} \frac{a^{\frac{5}{2}} x}{16 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 a^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{11 \sqrt{a} b x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + \frac{b^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.66679, size = 85, normalized size = 0.93 \begin{align*} \frac{1}{48} \,{\left (2 \,{\left (4 \, b x^{2} + 7 \, a\right )} x^{2} + \frac{3 \, a^{2}}{b}\right )} \sqrt{b x^{2} + a} x + \frac{a^{3} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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