Optimal. Leaf size=85 \[ \frac{7 b^2 \sqrt{x}}{a^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}-\frac{7 b x^{3/2}}{3 a^3}+\frac{7 x^{5/2}}{5 a^2}-\frac{x^{7/2}}{a (a x+b)} \]
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Rubi [A] time = 0.0334017, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{7 b^2 \sqrt{x}}{a^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}-\frac{7 b x^{3/2}}{3 a^3}+\frac{7 x^{5/2}}{5 a^2}-\frac{x^{7/2}}{a (a x+b)} \]
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^{3/2}}{\left (a+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^{7/2}}{(b+a x)^2} \, dx\\ &=-\frac{x^{7/2}}{a (b+a x)}+\frac{7 \int \frac{x^{5/2}}{b+a x} \, dx}{2 a}\\ &=\frac{7 x^{5/2}}{5 a^2}-\frac{x^{7/2}}{a (b+a x)}-\frac{(7 b) \int \frac{x^{3/2}}{b+a x} \, dx}{2 a^2}\\ &=-\frac{7 b x^{3/2}}{3 a^3}+\frac{7 x^{5/2}}{5 a^2}-\frac{x^{7/2}}{a (b+a x)}+\frac{\left (7 b^2\right ) \int \frac{\sqrt{x}}{b+a x} \, dx}{2 a^3}\\ &=\frac{7 b^2 \sqrt{x}}{a^4}-\frac{7 b x^{3/2}}{3 a^3}+\frac{7 x^{5/2}}{5 a^2}-\frac{x^{7/2}}{a (b+a x)}-\frac{\left (7 b^3\right ) \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{2 a^4}\\ &=\frac{7 b^2 \sqrt{x}}{a^4}-\frac{7 b x^{3/2}}{3 a^3}+\frac{7 x^{5/2}}{5 a^2}-\frac{x^{7/2}}{a (b+a x)}-\frac{\left (7 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{a^4}\\ &=\frac{7 b^2 \sqrt{x}}{a^4}-\frac{7 b x^{3/2}}{3 a^3}+\frac{7 x^{5/2}}{5 a^2}-\frac{x^{7/2}}{a (b+a x)}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0044028, size = 27, normalized size = 0.32 \[ \frac{2 x^{9/2} \, _2F_1\left (2,\frac{9}{2};\frac{11}{2};-\frac{a x}{b}\right )}{9 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 71, normalized size = 0.8 \begin{align*}{\frac{2}{5\,{a}^{2}}{x}^{{\frac{5}{2}}}}-{\frac{4\,b}{3\,{a}^{3}}{x}^{{\frac{3}{2}}}}+6\,{\frac{{b}^{2}\sqrt{x}}{{a}^{4}}}+{\frac{{b}^{3}}{{a}^{4} \left ( ax+b \right ) }\sqrt{x}}-7\,{\frac{{b}^{3}}{{a}^{4}\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 = 1.77733, size = 427, normalized size = 5.02 \begin{align*} \left [\frac{105 \,{\left (a b^{2} x + b^{3}\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 (6 \, a^{3} x^{3} - 14 \, a^{2} b x^{2} + 70 \, a b^{2} x + 105 \, b^{3}\right )} \sqrt{x}}{30 \,{\left (a^{5} x + a^{4} b\right )}}, -\frac{105 \,{\left (a b^{2} x + b^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{x} \sqrt{\frac{b}{a}}}{b}\right ) -{\left (6 \, a^{3} x^{3} - 14 \, a^{2} b x^{2} + 70 \, a b^{2} x + 105 \, b^{3}\right )} \sqrt{x}}{15 \,{\left (a^{5} x + a^{4} b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 64.3751, size = 542, normalized size = 6.38 \begin{align*} \begin{cases} \tilde{\infty } x^{\frac{9}{2}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{5}{2}}}{5 a^{2}} & \text{for}\: b = 0 \\\frac{2 x^{\frac{9}{2}}}{9 b^{2}} & \text{for}\: a = 0 \\\frac{12 i a^{4} \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{1}{a}}}{30 i a^{6} \sqrt{b} x \sqrt{\frac{1}{a}} + 30 i a^{5} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{28 i a^{3} b^{\frac{3}{2}} x^{\frac{5}{2}} \sqrt{\frac{1}{a}}}{30 i a^{6} \sqrt{b} x \sqrt{\frac{1}{a}} + 30 i a^{5} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{140 i a^{2} b^{\frac{5}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{a}}}{30 i a^{6} \sqrt{b} x \sqrt{\frac{1}{a}} + 30 i a^{5} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{210 i a b^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{a}}}{30 i a^{6} \sqrt{b} x \sqrt{\frac{1}{a}} + 30 i a^{5} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{105 a b^{3} x \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{30 i a^{6} \sqrt{b} x \sqrt{\frac{1}{a}} + 30 i a^{5} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{105 a b^{3} x \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{30 i a^{6} \sqrt{b} x \sqrt{\frac{1}{a}} + 30 i a^{5} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{105 b^{4} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{30 i a^{6} \sqrt{b} x \sqrt{\frac{1}{a}} + 30 i a^{5} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{105 b^{4} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{30 i a^{6} \sqrt{b} x \sqrt{\frac{1}{a}} + 30 i a^{5} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09478, size = 103, normalized size = 1.21 \begin{align*} -\frac{7 \, b^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{b^{3} \sqrt{x}}{{\left (a x + b\right )} a^{4}} + \frac{2 \,{\left (3 \, a^{8} x^{\frac{5}{2}} - 10 \, a^{7} b x^{\frac{3}{2}} + 45 \, a^{6} b^{2} \sqrt{x}\right )}}{15 \, a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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