Optimal. Leaf size=179 \[ \frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2048 a^{5/2}}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}} \]
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Rubi [A] time = 0.121738, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2048 a^{5/2}}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}} \]
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 )^{9/2}}{x^{15}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{9/2}}{x^8} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{1}{28} (9 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{7/2}}{x^7} \, dx,x,x^2\right )\\ &=-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{1}{16} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{1}{32} \left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{1}{256} \left (9 b^4\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{1}{512} \left (3 b^5\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}-\frac{\left (9 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )}{2048 a}\\ &=-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}+\frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{\left (9 b^7\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{4096 a^2}\\ &=-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}+\frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}+\frac{\left (9 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{2048 a^2}\\ &=-\frac{3 b^4 \sqrt{a+b x^2}}{256 x^6}-\frac{3 b^5 \sqrt{a+b x^2}}{1024 a x^4}+\frac{9 b^6 \sqrt{a+b x^2}}{2048 a^2 x^2}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{128 x^8}-\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{80 x^{10}}-\frac{3 b \left (a+b x^2\right )^{7/2}}{56 x^{12}}-\frac{\left (a+b x^2\right )^{9/2}}{14 x^{14}}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2048 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0123766, size = 39, normalized size = 0.22 \[ \frac{b^7 \left (a+b x^2\right )^{11/2} \, _2F_1\left (\frac{11}{2},8;\frac{13}{2};\frac{b x^2}{a}+1\right )}{11 a^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.29, size = 253, normalized size = 1.4 \begin{align*} -{\frac{1}{14\,a{x}^{14}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{b}{56\,{a}^{2}{x}^{12}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{2}}{560\,{a}^{3}{x}^{10}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{3}}{4480\,{a}^{4}{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{4}}{8960\,{a}^{5}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{5}}{7168\,{a}^{6}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{{b}^{6}}{2048\,{a}^{7}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{b}^{7}}{2048\,{a}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{9\,{b}^{7}}{14336\,{a}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{9\,{b}^{7}}{10240\,{a}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{b}^{7}}{2048\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{9\,{b}^{7}}{2048}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{9\,{b}^{7}}{2048\,{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 = 2.1011, size = 628, normalized size = 3.51 \begin{align*} \left [\frac{315 \, \sqrt{a} b^{7} x^{14} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (315 \, a b^{6} x^{12} - 210 \, a^{2} b^{5} x^{10} - 14168 \, a^{3} b^{4} x^{8} - 39056 \, a^{4} b^{3} x^{6} - 44928 \, a^{5} b^{2} x^{4} - 24320 \, a^{6} b x^{2} - 5120 \, a^{7}\right )} \sqrt{b x^{2} + a}}{143360 \, a^{3} x^{14}}, \frac{315 \, \sqrt{-a} b^{7} x^{14} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (315 \, a b^{6} x^{12} - 210 \, a^{2} b^{5} x^{10} - 14168 \, a^{3} b^{4} x^{8} - 39056 \, a^{4} b^{3} x^{6} - 44928 \, a^{5} b^{2} x^{4} - 24320 \, a^{6} b x^{2} - 5120 \, a^{7}\right )} \sqrt{b x^{2} + a}}{71680 \, a^{3} x^{14}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.8739, size = 231, normalized size = 1.29 \begin{align*} - \frac{a^{5}}{14 \sqrt{b} x^{15} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{23 a^{4} \sqrt{b}}{56 x^{13} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{541 a^{3} b^{\frac{3}{2}}}{560 x^{11} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5249 a^{2} b^{\frac{5}{2}}}{4480 x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{6653 a b^{\frac{7}{2}}}{8960 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{1027 b^{\frac{9}{2}}}{5120 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b^{\frac{11}{2}}}{2048 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{9 b^{\frac{13}{2}}}{2048 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{9 b^{7} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2048 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.35695, size = 184, normalized size = 1.03 \begin{align*} \frac{1}{71680} \, b^{7}{\left (\frac{315 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{315 \,{\left (b x^{2} + a\right )}^{\frac{13}{2}} - 2100 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} a - 8393 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a^{2} + 9216 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{3} - 5943 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{4} + 2100 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{5} - 315 \, \sqrt{b x^{2} + a} a^{6}}{a^{2} b^{7} x^{14}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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