Optimal. Leaf size=73 \[ -\frac{\left (a x^2+b x^3\right )^{3/2}}{x^4}+\frac{3 b \sqrt{a x^2+b x^3}}{x}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right ) \]
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Rubi [A] time = 0.0931158, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2020, 2021, 2008, 206} \[ -\frac{\left (a x^2+b x^3\right )^{3/2}}{x^4}+\frac{3 b \sqrt{a x^2+b x^3}}{x}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right ) \]
Antiderivative was successfully verified.
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Rule 2020
Rule 2021
Rule 2008
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx &=-\frac{\left (a x^2+b x^3\right )^{3/2}}{x^4}+\frac{1}{2} (3 b) \int \frac{\sqrt{a x^2+b x^3}}{x^2} \, dx\\ &=\frac{3 b \sqrt{a x^2+b x^3}}{x}-\frac{\left (a x^2+b x^3\right )^{3/2}}{x^4}+\frac{1}{2} (3 a b) \int \frac{1}{\sqrt{a x^2+b x^3}} \, dx\\ &=\frac{3 b \sqrt{a x^2+b x^3}}{x}-\frac{\left (a x^2+b x^3\right )^{3/2}}{x^4}-(3 a b) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{x}{\sqrt{a x^2+b x^3}}\right )\\ &=\frac{3 b \sqrt{a x^2+b x^3}}{x}-\frac{\left (a x^2+b x^3\right )^{3/2}}{x^4}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )\\ \end{align*}
Mathematica [C] time = 0.0134288, size = 40, normalized size = 0.55 \[ \frac{2 b \left (x^2 (a+b x)\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x}{a}+1\right )}{5 a^2 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 72, normalized size = 1. \begin{align*} -{\frac{1}{{x}^{4}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ({a}^{{\frac{3}{2}}}\sqrt{bx+a}-2\,\sqrt{bx+a}bx\sqrt{a}+3\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) xab \right ) \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.898792, size = 304, normalized size = 4.16 \begin{align*} \left [\frac{3 \, \sqrt{a} b x^{2} \log \left (\frac{b x^{2} + 2 \, a x - 2 \, \sqrt{b x^{3} + a x^{2}} \sqrt{a}}{x^{2}}\right ) + 2 \, \sqrt{b x^{3} + a x^{2}}{\left (2 \, b x - a\right )}}{2 \, x^{2}}, \frac{3 \, \sqrt{-a} b x^{2} \arctan \left (\frac{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}{a x}\right ) + \sqrt{b x^{3} + a x^{2}}{\left (2 \, b x - a\right )}}{x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.45642, size = 84, normalized size = 1.15 \begin{align*} \frac{\frac{3 \, a b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) \mathrm{sgn}\left (x\right )}{\sqrt{-a}} + 2 \, \sqrt{b x + a} b^{2} \mathrm{sgn}\left (x\right ) - \frac{\sqrt{b x + a} a b \mathrm{sgn}\left (x\right )}{x}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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