Optimal. Leaf size=98 \[ \frac{3 b^2 x^{3/2}}{a^4}-\frac{9 b^3 \sqrt{x}}{a^5}+\frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{11/2}}-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (a x+b)} \]
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Rubi [A] time = 0.0406906, antiderivative size = 98, 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 = {263, 47, 50, 63, 205} \[ \frac{3 b^2 x^{3/2}}{a^4}-\frac{9 b^3 \sqrt{x}}{a^5}+\frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{11/2}}-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (a x+b)} \]
Antiderivative was successfully verified.
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Rule 263
Rule 47
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{\left (a+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^{9/2}}{(b+a x)^2} \, dx\\ &=-\frac{x^{9/2}}{a (b+a x)}+\frac{9 \int \frac{x^{7/2}}{b+a x} \, dx}{2 a}\\ &=\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (b+a x)}-\frac{(9 b) \int \frac{x^{5/2}}{b+a x} \, dx}{2 a^2}\\ &=-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (b+a x)}+\frac{\left (9 b^2\right ) \int \frac{x^{3/2}}{b+a x} \, dx}{2 a^3}\\ &=\frac{3 b^2 x^{3/2}}{a^4}-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (b+a x)}-\frac{\left (9 b^3\right ) \int \frac{\sqrt{x}}{b+a x} \, dx}{2 a^4}\\ &=-\frac{9 b^3 \sqrt{x}}{a^5}+\frac{3 b^2 x^{3/2}}{a^4}-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (b+a x)}+\frac{\left (9 b^4\right ) \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{2 a^5}\\ &=-\frac{9 b^3 \sqrt{x}}{a^5}+\frac{3 b^2 x^{3/2}}{a^4}-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (b+a x)}+\frac{\left (9 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{a^5}\\ &=-\frac{9 b^3 \sqrt{x}}{a^5}+\frac{3 b^2 x^{3/2}}{a^4}-\frac{9 b x^{5/2}}{5 a^3}+\frac{9 x^{7/2}}{7 a^2}-\frac{x^{9/2}}{a (b+a x)}+\frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0048274, size = 27, normalized size = 0.28 \[ \frac{2 x^{11/2} \, _2F_1\left (2,\frac{11}{2};\frac{13}{2};-\frac{a x}{b}\right )}{11 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 83, normalized size = 0.9 \begin{align*}{\frac{2}{7\,{a}^{2}}{x}^{{\frac{7}{2}}}}-{\frac{4\,b}{5\,{a}^{3}}{x}^{{\frac{5}{2}}}}+2\,{\frac{{b}^{2}{x}^{3/2}}{{a}^{4}}}-8\,{\frac{{b}^{3}\sqrt{x}}{{a}^{5}}}-{\frac{{b}^{4}}{{a}^{5} \left ( ax+b \right ) }\sqrt{x}}+9\,{\frac{{b}^{4}}{{a}^{5}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \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 = 1.84736, size = 477, normalized size = 4.87 \begin{align*} \left [\frac{315 \,{\left (a b^{3} x + b^{4}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (10 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a b^{3} x - 315 \, b^{4}\right )} \sqrt{x}}{70 \,{\left (a^{6} x + a^{5} b\right )}}, \frac{315 \,{\left (a b^{3} x + b^{4}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{x} \sqrt{\frac{b}{a}}}{b}\right ) +{\left (10 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a b^{3} x - 315 \, b^{4}\right )} \sqrt{x}}{35 \,{\left (a^{6} x + a^{5} b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11506, size = 119, normalized size = 1.21 \begin{align*} \frac{9 \, b^{4} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{5}} - \frac{b^{4} \sqrt{x}}{{\left (a x + b\right )} a^{5}} + \frac{2 \,{\left (5 \, a^{12} x^{\frac{7}{2}} - 14 \, a^{11} b x^{\frac{5}{2}} + 35 \, a^{10} b^{2} x^{\frac{3}{2}} - 140 \, a^{9} b^{3} \sqrt{x}\right )}}{35 \, a^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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