Optimal. Leaf size=116 \[ \frac{3 b^3 \sqrt{a+b x^2}}{128 a^2 x^2}-\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{5/2}}-\frac{b^2 \sqrt{a+b x^2}}{64 a x^4}-\frac{b \sqrt{a+b x^2}}{16 x^6}-\frac{\left (a+b x^2\right )^{3/2}}{8 x^8} \]
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Rubi [A] time = 0.0693875, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ \frac{3 b^3 \sqrt{a+b x^2}}{128 a^2 x^2}-\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{5/2}}-\frac{b^2 \sqrt{a+b x^2}}{64 a x^4}-\frac{b \sqrt{a+b x^2}}{16 x^6}-\frac{\left (a+b x^2\right )^{3/2}}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2}}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{3/2}}{8 x^8}+\frac{1}{16} (3 b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{b \sqrt{a+b x^2}}{16 x^6}-\frac{\left (a+b x^2\right )^{3/2}}{8 x^8}+\frac{1}{32} b^2 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{b \sqrt{a+b x^2}}{16 x^6}-\frac{b^2 \sqrt{a+b x^2}}{64 a x^4}-\frac{\left (a+b x^2\right )^{3/2}}{8 x^8}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )}{128 a}\\ &=-\frac{b \sqrt{a+b x^2}}{16 x^6}-\frac{b^2 \sqrt{a+b x^2}}{64 a x^4}+\frac{3 b^3 \sqrt{a+b x^2}}{128 a^2 x^2}-\frac{\left (a+b x^2\right )^{3/2}}{8 x^8}+\frac{\left (3 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{256 a^2}\\ &=-\frac{b \sqrt{a+b x^2}}{16 x^6}-\frac{b^2 \sqrt{a+b x^2}}{64 a x^4}+\frac{3 b^3 \sqrt{a+b x^2}}{128 a^2 x^2}-\frac{\left (a+b x^2\right )^{3/2}}{8 x^8}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{128 a^2}\\ &=-\frac{b \sqrt{a+b x^2}}{16 x^6}-\frac{b^2 \sqrt{a+b x^2}}{64 a x^4}+\frac{3 b^3 \sqrt{a+b x^2}}{128 a^2 x^2}-\frac{\left (a+b x^2\right )^{3/2}}{8 x^8}-\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0093422, size = 39, normalized size = 0.34 \[ -\frac{b^4 \left (a+b x^2\right )^{5/2} \, _2F_1\left (\frac{5}{2},5;\frac{7}{2};\frac{b x^2}{a}+1\right )}{5 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 142, normalized size = 1.2 \begin{align*} -{\frac{1}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{b}{16\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{2}}{64\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{4}}{128\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{3\,{b}^{4}}{128\,{a}^{3}}\sqrt{b{x}^{2}+a}} \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 = 1.65344, size = 419, normalized size = 3.61 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{4} x^{8} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (3 \, a b^{3} x^{6} - 2 \, a^{2} b^{2} x^{4} - 24 \, a^{3} b x^{2} - 16 \, a^{4}\right )} \sqrt{b x^{2} + a}}{256 \, a^{3} x^{8}}, \frac{3 \, \sqrt{-a} b^{4} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (3 \, a b^{3} x^{6} - 2 \, a^{2} b^{2} x^{4} - 24 \, a^{3} b x^{2} - 16 \, a^{4}\right )} \sqrt{b x^{2} + a}}{128 \, a^{3} x^{8}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.1865, size = 148, normalized size = 1.28 \begin{align*} - \frac{a^{2}}{8 \sqrt{b} x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 a \sqrt{b}}{16 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{13 b^{\frac{3}{2}}}{64 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{b^{\frac{5}{2}}}{128 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b^{\frac{7}{2}}}{128 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{128 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.94811, size = 127, normalized size = 1.09 \begin{align*} \frac{1}{128} \, b^{4}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 11 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a - 11 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} + 3 \, \sqrt{b x^{2} + a} a^{3}}{a^{2} b^{4} x^{8}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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