Optimal. Leaf size=69 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}+\frac{5 a}{b^3 \sqrt{x}}+\frac{1}{b x^{3/2} (a x+b)}-\frac{5}{3 b^2 x^{3/2}} \]
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Rubi [A] time = 0.0242281, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 51, 63, 205} \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}+\frac{5 a}{b^3 \sqrt{x}}+\frac{1}{b x^{3/2} (a x+b)}-\frac{5}{3 b^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^{9/2}} \, dx &=\int \frac{1}{x^{5/2} (b+a x)^2} \, dx\\ &=\frac{1}{b x^{3/2} (b+a x)}+\frac{5 \int \frac{1}{x^{5/2} (b+a x)} \, dx}{2 b}\\ &=-\frac{5}{3 b^2 x^{3/2}}+\frac{1}{b x^{3/2} (b+a x)}-\frac{(5 a) \int \frac{1}{x^{3/2} (b+a x)} \, dx}{2 b^2}\\ &=-\frac{5}{3 b^2 x^{3/2}}+\frac{5 a}{b^3 \sqrt{x}}+\frac{1}{b x^{3/2} (b+a x)}+\frac{\left (5 a^2\right ) \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{2 b^3}\\ &=-\frac{5}{3 b^2 x^{3/2}}+\frac{5 a}{b^3 \sqrt{x}}+\frac{1}{b x^{3/2} (b+a x)}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=-\frac{5}{3 b^2 x^{3/2}}+\frac{5 a}{b^3 \sqrt{x}}+\frac{1}{b x^{3/2} (b+a x)}+\frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0045889, size = 27, normalized size = 0.39 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};-\frac{a x}{b}\right )}{3 b^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 60, normalized size = 0.9 \begin{align*}{\frac{{a}^{2}}{{b}^{3} \left ( ax+b \right ) }\sqrt{x}}+5\,{\frac{{a}^{2}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) }-{\frac{2}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}+4\,{\frac{a}{{b}^{3}\sqrt{x}}} \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 = 1.91305, size = 402, normalized size = 5.83 \begin{align*} \left [\frac{15 \,{\left (a^{2} x^{3} + a b x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - b}{a x + b}\right ) + 2 \,{\left (15 \, a^{2} x^{2} + 10 \, a b x - 2 \, b^{2}\right )} \sqrt{x}}{6 \,{\left (a b^{3} x^{3} + b^{4} x^{2}\right )}}, -\frac{15 \,{\left (a^{2} x^{3} + a b x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{\frac{a}{b}}}{a \sqrt{x}}\right ) -{\left (15 \, a^{2} x^{2} + 10 \, a b x - 2 \, b^{2}\right )} \sqrt{x}}{3 \,{\left (a b^{3} x^{3} + b^{4} x^{2}\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 = 1.13217, size = 78, normalized size = 1.13 \begin{align*} \frac{5 \, a^{2} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{a^{2} \sqrt{x}}{{\left (a x + b\right )} b^{3}} + \frac{2 \,{\left (6 \, a x - b\right )}}{3 \, b^{3} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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