Optimal. Leaf size=113 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}-\frac{5 b^3 \sqrt{a+b x^2}}{128 a x^2}-\frac{5 b^2 \sqrt{a+b x^2}}{64 x^4}-\frac{5 b \left (a+b x^2\right )^{3/2}}{48 x^6}-\frac{\left (a+b x^2\right )^{5/2}}{8 x^8} \]
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Rubi [A] time = 0.0672193, antiderivative size = 113, 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} \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}-\frac{5 b^3 \sqrt{a+b x^2}}{128 a x^2}-\frac{5 b^2 \sqrt{a+b x^2}}{64 x^4}-\frac{5 b \left (a+b x^2\right )^{3/2}}{48 x^6}-\frac{\left (a+b x^2\right )^{5/2}}{8 x^8} \]
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 )^{5/2}}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{5/2}}{8 x^8}+\frac{1}{16} (5 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{5 b \left (a+b x^2\right )^{3/2}}{48 x^6}-\frac{\left (a+b x^2\right )^{5/2}}{8 x^8}+\frac{1}{32} \left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{5 b^2 \sqrt{a+b x^2}}{64 x^4}-\frac{5 b \left (a+b x^2\right )^{3/2}}{48 x^6}-\frac{\left (a+b x^2\right )^{5/2}}{8 x^8}+\frac{1}{128} \left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{5 b^2 \sqrt{a+b x^2}}{64 x^4}-\frac{5 b^3 \sqrt{a+b x^2}}{128 a x^2}-\frac{5 b \left (a+b x^2\right )^{3/2}}{48 x^6}-\frac{\left (a+b x^2\right )^{5/2}}{8 x^8}-\frac{\left (5 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{256 a}\\ &=-\frac{5 b^2 \sqrt{a+b x^2}}{64 x^4}-\frac{5 b^3 \sqrt{a+b x^2}}{128 a x^2}-\frac{5 b \left (a+b x^2\right )^{3/2}}{48 x^6}-\frac{\left (a+b x^2\right )^{5/2}}{8 x^8}-\frac{\left (5 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}\\ &=-\frac{5 b^2 \sqrt{a+b x^2}}{64 x^4}-\frac{5 b^3 \sqrt{a+b x^2}}{128 a x^2}-\frac{5 b \left (a+b x^2\right )^{3/2}}{48 x^6}-\frac{\left (a+b x^2\right )^{5/2}}{8 x^8}+\frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0101187, size = 39, normalized size = 0.35 \[ -\frac{b^4 \left (a+b x^2\right )^{7/2} \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{b x^2}{a}+1\right )}{7 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 159, normalized size = 1.4 \begin{align*} -{\frac{1}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{b}{48\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{2}}{192\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{4}}{128\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{4}}{384\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{5\,{b}^{4}}{128\,{a}^{2}}\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 = 1.6166, size = 433, normalized size = 3.83 \begin{align*} \left [\frac{15 \, \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 (15 \, a b^{3} x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a^{3} b x^{2} + 48 \, a^{4}\right )} \sqrt{b x^{2} + a}}{768 \, a^{2} x^{8}}, -\frac{15 \, \sqrt{-a} b^{4} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (15 \, a b^{3} x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a^{3} b x^{2} + 48 \, a^{4}\right )} \sqrt{b x^{2} + a}}{384 \, a^{2} x^{8}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.15153, size = 150, normalized size = 1.33 \begin{align*} - \frac{a^{3}}{8 \sqrt{b} x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{23 a^{2} \sqrt{b}}{48 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{127 a b^{\frac{3}{2}}}{192 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{133 b^{\frac{5}{2}}}{384 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 b^{\frac{7}{2}}}{128 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{5 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{128 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.53279, size = 127, normalized size = 1.12 \begin{align*} -\frac{1}{384} \, b^{4}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} + 73 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a - 55 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} + 15 \, \sqrt{b x^{2} + a} a^{3}}{a b^{4} x^{8}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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