Optimal. Leaf size=131 \[ -\frac{63 b^4 \sqrt{a+b x^2}}{256 x^2}-\frac{21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 \sqrt{a}}-\frac{9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac{\left (a+b x^2\right )^{9/2}}{10 x^{10}} \]
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Rubi [A] time = 0.0826313, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ -\frac{63 b^4 \sqrt{a+b x^2}}{256 x^2}-\frac{21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 \sqrt{a}}-\frac{9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac{\left (a+b x^2\right )^{9/2}}{10 x^{10}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{9/2}}{x^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{9/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac{1}{20} (9 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{7/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac{9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac{\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac{1}{160} \left (63 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac{9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac{\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac{1}{64} \left (21 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac{9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac{\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac{1}{256} \left (63 b^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{63 b^4 \sqrt{a+b x^2}}{256 x^2}-\frac{21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac{9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac{\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac{1}{512} \left (63 b^5\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{63 b^4 \sqrt{a+b x^2}}{256 x^2}-\frac{21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac{9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac{\left (a+b x^2\right )^{9/2}}{10 x^{10}}+\frac{1}{256} \left (63 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )\\ &=-\frac{63 b^4 \sqrt{a+b x^2}}{256 x^2}-\frac{21 b^3 \left (a+b x^2\right )^{3/2}}{128 x^4}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{160 x^6}-\frac{9 b \left (a+b x^2\right )^{7/2}}{80 x^8}-\frac{\left (a+b x^2\right )^{9/2}}{10 x^{10}}-\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{256 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0479415, size = 109, normalized size = 0.83 \[ -\frac{2858 a^2 b^3 x^6+2024 a^3 b^2 x^4+784 a^4 b x^2+128 a^5+2455 a b^4 x^8+315 b^5 x^{10} \sqrt{\frac{b x^2}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^2}{a}+1}\right )+965 b^5 x^{10}}{1280 x^{10} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 213, normalized size = 1.6 \begin{align*} -{\frac{1}{10\,a{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{b}{80\,{a}^{2}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{2}}{160\,{a}^{3}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{3}}{128\,{a}^{4}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{7\,{b}^{4}}{256\,{a}^{5}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{7\,{b}^{5}}{256\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{9\,{b}^{5}}{256\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{63\,{b}^{5}}{1280\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{21\,{b}^{5}}{256\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{63\,{b}^{5}}{256}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{63\,{b}^{5}}{256\,a}\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 = 1.72751, size = 497, normalized size = 3.79 \begin{align*} \left [\frac{315 \, \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 (965 \, a b^{4} x^{8} + 1490 \, a^{2} b^{3} x^{6} + 1368 \, a^{3} b^{2} x^{4} + 656 \, a^{4} b x^{2} + 128 \, a^{5}\right )} \sqrt{b x^{2} + a}}{2560 \, a x^{10}}, \frac{315 \, \sqrt{-a} b^{5} x^{10} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (965 \, a b^{4} x^{8} + 1490 \, a^{2} b^{3} x^{6} + 1368 \, a^{3} b^{2} x^{4} + 656 \, a^{4} b x^{2} + 128 \, a^{5}\right )} \sqrt{b x^{2} + a}}{1280 \, a x^{10}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.3829, size = 153, normalized size = 1.17 \begin{align*} - \frac{a^{4} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{10 x^{9}} - \frac{41 a^{3} b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{80 x^{7}} - \frac{171 a^{2} b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{160 x^{5}} - \frac{149 a b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{128 x^{3}} - \frac{193 b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{256 x} - \frac{63 b^{5} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{256 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.66556, size = 139, normalized size = 1.06 \begin{align*} \frac{1}{1280} \, b^{5}{\left (\frac{315 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{965 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 2370 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 2688 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 1470 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{b x^{2} + a} a^{4}}{b^{5} x^{10}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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