Optimal. Leaf size=84 \[ \frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2} \]
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Rubi [A] time = 0.023137, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {195, 217, 206} \[ \frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^{5/2} \, dx &=\frac{1}{6} x \left (a+b x^2\right )^{5/2}+\frac{1}{6} (5 a) \int \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2}+\frac{1}{8} \left (5 a^2\right ) \int \sqrt{a+b x^2} \, dx\\ &=\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2}+\frac{1}{16} \left (5 a^3\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2}+\frac{1}{16} \left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.10306, size = 76, normalized size = 0.9 \[ \frac{1}{48} \sqrt{a+b x^2} \left (\frac{15 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{\frac{b x^2}{a}+1}}+33 a^2 x+26 a b x^3+8 b^2 x^5\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 66, normalized size = 0.8 \begin{align*}{\frac{x}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,ax}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}x}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \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 = 1.63455, size = 347, normalized size = 4.13 \begin{align*} \left [\frac{15 \, 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} + 26 \, a b^{2} x^{3} + 33 \, a^{2} b x\right )} \sqrt{b x^{2} + a}}{96 \, b}, -\frac{15 \, a^{3} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (8 \, b^{3} x^{5} + 26 \, a b^{2} x^{3} + 33 \, a^{2} b x\right )} \sqrt{b x^{2} + a}}{48 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.18772, size = 97, normalized size = 1.15 \begin{align*} \frac{11 a^{\frac{5}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{16} + \frac{13 a^{\frac{3}{2}} b x^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{24} + \frac{\sqrt{a} b^{2} x^{5} \sqrt{1 + \frac{b x^{2}}{a}}}{6} + \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14066, size = 85, normalized size = 1.01 \begin{align*} -\frac{5 \, a^{3} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, \sqrt{b}} + \frac{1}{48} \,{\left (2 \,{\left (4 \, b^{2} x^{2} + 13 \, a b\right )} x^{2} + 33 \, a^{2}\right )} \sqrt{b x^{2} + a} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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