Optimal. Leaf size=137 \[ \frac{3 b^4 \sqrt{a+b x^2}}{256 a^2 x^2}-\frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{5/2}}-\frac{b^3 \sqrt{a+b x^2}}{128 a x^4}-\frac{b^2 \sqrt{a+b x^2}}{32 x^6}-\frac{b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac{\left (a+b x^2\right )^{5/2}}{10 x^{10}} \]
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Rubi [A] time = 0.0834827, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{3 b^4 \sqrt{a+b x^2}}{256 a^2 x^2}-\frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{5/2}}-\frac{b^3 \sqrt{a+b x^2}}{128 a x^4}-\frac{b^2 \sqrt{a+b x^2}}{32 x^6}-\frac{b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac{\left (a+b x^2\right )^{5/2}}{10 x^{10}} \]
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^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{5/2}}{10 x^{10}}+\frac{1}{4} b \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac{b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac{\left (a+b x^2\right )^{5/2}}{10 x^{10}}+\frac{1}{32} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{b^2 \sqrt{a+b x^2}}{32 x^6}-\frac{b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac{\left (a+b x^2\right )^{5/2}}{10 x^{10}}+\frac{1}{64} b^3 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{b^2 \sqrt{a+b x^2}}{32 x^6}-\frac{b^3 \sqrt{a+b x^2}}{128 a x^4}-\frac{b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac{\left (a+b x^2\right )^{5/2}}{10 x^{10}}-\frac{\left (3 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )}{256 a}\\ &=-\frac{b^2 \sqrt{a+b x^2}}{32 x^6}-\frac{b^3 \sqrt{a+b x^2}}{128 a x^4}+\frac{3 b^4 \sqrt{a+b x^2}}{256 a^2 x^2}-\frac{b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac{\left (a+b x^2\right )^{5/2}}{10 x^{10}}+\frac{\left (3 b^5\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{512 a^2}\\ &=-\frac{b^2 \sqrt{a+b x^2}}{32 x^6}-\frac{b^3 \sqrt{a+b x^2}}{128 a x^4}+\frac{3 b^4 \sqrt{a+b x^2}}{256 a^2 x^2}-\frac{b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac{\left (a+b x^2\right )^{5/2}}{10 x^{10}}+\frac{\left (3 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{256 a^2}\\ &=-\frac{b^2 \sqrt{a+b x^2}}{32 x^6}-\frac{b^3 \sqrt{a+b x^2}}{128 a x^4}+\frac{3 b^4 \sqrt{a+b x^2}}{256 a^2 x^2}-\frac{b \left (a+b x^2\right )^{3/2}}{16 x^8}-\frac{\left (a+b x^2\right )^{5/2}}{10 x^{10}}-\frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0102124, size = 39, normalized size = 0.28 \[ \frac{b^5 \left (a+b x^2\right )^{7/2} \, _2F_1\left (\frac{7}{2},6;\frac{9}{2};\frac{b x^2}{a}+1\right )}{7 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 179, normalized size = 1.3 \begin{align*} -{\frac{1}{10\,a{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,b}{80\,{a}^{2}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{2}}{160\,{a}^{3}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{3}}{640\,{a}^{4}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{b}^{4}}{1280\,{a}^{5}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{b}^{5}}{1280\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{5}}{256\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{5}}{256}\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}^{5}}{256\,{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 = 1.7218, size = 489, normalized size = 3.57 \begin{align*} \left [\frac{15 \, \sqrt{a} b^{5} x^{10} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (15 \, a b^{4} x^{8} - 10 \, a^{2} b^{3} x^{6} - 248 \, a^{3} b^{2} x^{4} - 336 \, a^{4} b x^{2} - 128 \, a^{5}\right )} \sqrt{b x^{2} + a}}{2560 \, a^{3} x^{10}}, \frac{15 \, \sqrt{-a} b^{5} x^{10} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (15 \, a b^{4} x^{8} - 10 \, a^{2} b^{3} x^{6} - 248 \, a^{3} b^{2} x^{4} - 336 \, a^{4} b x^{2} - 128 \, a^{5}\right )} \sqrt{b x^{2} + a}}{1280 \, a^{3} x^{10}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.7643, size = 175, normalized size = 1.28 \begin{align*} - \frac{a^{3}}{10 \sqrt{b} x^{11} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{29 a^{2} \sqrt{b}}{80 x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{73 a b^{\frac{3}{2}}}{160 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{129 b^{\frac{5}{2}}}{640 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{b^{\frac{7}{2}}}{256 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b^{\frac{9}{2}}}{256 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 b^{5} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{256 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.8112, size = 146, normalized size = 1.07 \begin{align*} \frac{1}{1280} \, b^{5}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 70 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a - 128 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} + 70 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} - 15 \, \sqrt{b x^{2} + a} a^{4}}{a^{2} b^{5} x^{10}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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