Optimal. Leaf size=126 \[ -\frac{105}{16} a^{3/2} b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{16 x^2}+\frac{35}{16} b^3 \left (a+b x^2\right )^{3/2}+\frac{105}{16} a b^3 \sqrt{a+b x^2}-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}-\frac{3 b \left (a+b x^2\right )^{7/2}}{8 x^4} \]
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Rubi [A] time = 0.0763166, antiderivative size = 126, 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, 50, 63, 208} \[ -\frac{105}{16} a^{3/2} b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{16 x^2}+\frac{35}{16} b^3 \left (a+b x^2\right )^{3/2}+\frac{105}{16} a b^3 \sqrt{a+b x^2}-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}-\frac{3 b \left (a+b x^2\right )^{7/2}}{8 x^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{9/2}}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{9/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}+\frac{1}{4} (3 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{7/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{3 b \left (a+b x^2\right )^{7/2}}{8 x^4}-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}+\frac{1}{16} \left (21 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{16 x^2}-\frac{3 b \left (a+b x^2\right )^{7/2}}{8 x^4}-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}+\frac{1}{32} \left (105 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac{35}{16} b^3 \left (a+b x^2\right )^{3/2}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{16 x^2}-\frac{3 b \left (a+b x^2\right )^{7/2}}{8 x^4}-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}+\frac{1}{32} \left (105 a b^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^2\right )\\ &=\frac{105}{16} a b^3 \sqrt{a+b x^2}+\frac{35}{16} b^3 \left (a+b x^2\right )^{3/2}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{16 x^2}-\frac{3 b \left (a+b x^2\right )^{7/2}}{8 x^4}-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}+\frac{1}{32} \left (105 a^2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=\frac{105}{16} a b^3 \sqrt{a+b x^2}+\frac{35}{16} b^3 \left (a+b x^2\right )^{3/2}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{16 x^2}-\frac{3 b \left (a+b x^2\right )^{7/2}}{8 x^4}-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}+\frac{1}{16} \left (105 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )\\ &=\frac{105}{16} a b^3 \sqrt{a+b x^2}+\frac{35}{16} b^3 \left (a+b x^2\right )^{3/2}-\frac{21 b^2 \left (a+b x^2\right )^{5/2}}{16 x^2}-\frac{3 b \left (a+b x^2\right )^{7/2}}{8 x^4}-\frac{\left (a+b x^2\right )^{9/2}}{6 x^6}-\frac{105}{16} a^{3/2} b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0119343, size = 39, normalized size = 0.31 \[ \frac{b^3 \left (a+b x^2\right )^{11/2} \, _2F_1\left (4,\frac{11}{2};\frac{13}{2};\frac{b x^2}{a}+1\right )}{11 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 168, normalized size = 1.3 \begin{align*} -{\frac{1}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{5\,b}{24\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{35\,{b}^{2}}{48\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{35\,{b}^{3}}{48\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{15\,{b}^{3}}{16\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{21\,{b}^{3}}{16\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{b}^{3}}{16} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{105\,{b}^{3}}{16}{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{105\,a{b}^{3}}{16}\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 = 1.62125, size = 451, normalized size = 3.58 \begin{align*} \left [\frac{315 \, a^{\frac{3}{2}} b^{3} x^{6} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (16 \, b^{4} x^{8} + 208 \, a b^{3} x^{6} - 165 \, a^{2} b^{2} x^{4} - 50 \, a^{3} b x^{2} - 8 \, a^{4}\right )} \sqrt{b x^{2} + a}}{96 \, x^{6}}, \frac{315 \, \sqrt{-a} a b^{3} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (16 \, b^{4} x^{8} + 208 \, a b^{3} x^{6} - 165 \, a^{2} b^{2} x^{4} - 50 \, a^{3} b x^{2} - 8 \, a^{4}\right )} \sqrt{b x^{2} + a}}{48 \, x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.82761, size = 175, normalized size = 1.39 \begin{align*} - \frac{105 a^{\frac{3}{2}} b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16} - \frac{a^{5}}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{29 a^{4} \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{215 a^{3} b^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{43 a^{2} b^{\frac{5}{2}}}{48 x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{14 a b^{\frac{7}{2}} x}{3 \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{b^{\frac{9}{2}} x^{3}}{3 \sqrt{\frac{a}{b x^{2}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.85332, size = 143, normalized size = 1.13 \begin{align*} \frac{1}{48} \,{\left (\frac{315 \, a^{2} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 16 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} + 192 \, \sqrt{b x^{2} + a} a - \frac{165 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 280 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3} + 123 \, \sqrt{b x^{2} + a} a^{4}}{b^{3} x^{6}}\right )} b^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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