Optimal. Leaf size=110 \[ \frac{63 b^2 \sqrt{x}}{4 a^5}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{11/2}}-\frac{9 x^{7/2}}{4 a^2 (a x+b)}-\frac{21 b x^{3/2}}{4 a^4}+\frac{63 x^{5/2}}{20 a^3}-\frac{x^{9/2}}{2 a (a x+b)^2} \]
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Rubi [A] time = 0.0413503, antiderivative size = 110, 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} \[ \frac{63 b^2 \sqrt{x}}{4 a^5}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{11/2}}-\frac{9 x^{7/2}}{4 a^2 (a x+b)}-\frac{21 b x^{3/2}}{4 a^4}+\frac{63 x^{5/2}}{20 a^3}-\frac{x^{9/2}}{2 a (a x+b)^2} \]
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 )^3} \, dx &=\int \frac{x^{9/2}}{(b+a x)^3} \, dx\\ &=-\frac{x^{9/2}}{2 a (b+a x)^2}+\frac{9 \int \frac{x^{7/2}}{(b+a x)^2} \, dx}{4 a}\\ &=-\frac{x^{9/2}}{2 a (b+a x)^2}-\frac{9 x^{7/2}}{4 a^2 (b+a x)}+\frac{63 \int \frac{x^{5/2}}{b+a x} \, dx}{8 a^2}\\ &=\frac{63 x^{5/2}}{20 a^3}-\frac{x^{9/2}}{2 a (b+a x)^2}-\frac{9 x^{7/2}}{4 a^2 (b+a x)}-\frac{(63 b) \int \frac{x^{3/2}}{b+a x} \, dx}{8 a^3}\\ &=-\frac{21 b x^{3/2}}{4 a^4}+\frac{63 x^{5/2}}{20 a^3}-\frac{x^{9/2}}{2 a (b+a x)^2}-\frac{9 x^{7/2}}{4 a^2 (b+a x)}+\frac{\left (63 b^2\right ) \int \frac{\sqrt{x}}{b+a x} \, dx}{8 a^4}\\ &=\frac{63 b^2 \sqrt{x}}{4 a^5}-\frac{21 b x^{3/2}}{4 a^4}+\frac{63 x^{5/2}}{20 a^3}-\frac{x^{9/2}}{2 a (b+a x)^2}-\frac{9 x^{7/2}}{4 a^2 (b+a x)}-\frac{\left (63 b^3\right ) \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{8 a^5}\\ &=\frac{63 b^2 \sqrt{x}}{4 a^5}-\frac{21 b x^{3/2}}{4 a^4}+\frac{63 x^{5/2}}{20 a^3}-\frac{x^{9/2}}{2 a (b+a x)^2}-\frac{9 x^{7/2}}{4 a^2 (b+a x)}-\frac{\left (63 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{4 a^5}\\ &=\frac{63 b^2 \sqrt{x}}{4 a^5}-\frac{21 b x^{3/2}}{4 a^4}+\frac{63 x^{5/2}}{20 a^3}-\frac{x^{9/2}}{2 a (b+a x)^2}-\frac{9 x^{7/2}}{4 a^2 (b+a x)}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{4 a^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0059522, size = 27, normalized size = 0.25 \[ \frac{2 x^{11/2} \, _2F_1\left (3,\frac{11}{2};\frac{13}{2};-\frac{a x}{b}\right )}{11 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 90, normalized size = 0.8 \begin{align*}{\frac{2}{5\,{a}^{3}}{x}^{{\frac{5}{2}}}}-2\,{\frac{b{x}^{3/2}}{{a}^{4}}}+12\,{\frac{{b}^{2}\sqrt{x}}{{a}^{5}}}+{\frac{17\,{b}^{3}}{4\,{a}^{4} \left ( ax+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{15\,{b}^{4}}{4\,{a}^{5} \left ( ax+b \right ) ^{2}}\sqrt{x}}-{\frac{63\,{b}^{3}}{4\,{a}^{5}}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.86749, size = 564, normalized size = 5.13 \begin{align*} \left [\frac{315 \,{\left (a^{2} b^{2} x^{2} + 2 \, 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 (8 \, a^{4} x^{4} - 24 \, a^{3} b x^{3} + 168 \, a^{2} b^{2} x^{2} + 525 \, a b^{3} x + 315 \, b^{4}\right )} \sqrt{x}}{40 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}, -\frac{315 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{x} \sqrt{\frac{b}{a}}}{b}\right ) -{\left (8 \, a^{4} x^{4} - 24 \, a^{3} b x^{3} + 168 \, a^{2} b^{2} x^{2} + 525 \, a b^{3} x + 315 \, b^{4}\right )} \sqrt{x}}{20 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{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 = 1.08539, size = 119, normalized size = 1.08 \begin{align*} -\frac{63 \, b^{3} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{5}} + \frac{17 \, a b^{3} x^{\frac{3}{2}} + 15 \, b^{4} \sqrt{x}}{4 \,{\left (a x + b\right )}^{2} a^{5}} + \frac{2 \,{\left (a^{12} x^{\frac{5}{2}} - 5 \, a^{11} b x^{\frac{3}{2}} + 30 \, a^{10} b^{2} \sqrt{x}\right )}}{5 \, a^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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