Optimal. Leaf size=89 \[ 5 b^2 \sqrt{a x+b x^2}+5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )-\frac{2 \left (a x+b x^2\right )^{5/2}}{3 x^4}-\frac{10 b \left (a x+b x^2\right )^{3/2}}{3 x^2} \]
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Rubi [A] time = 0.0405703, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {662, 664, 620, 206} \[ 5 b^2 \sqrt{a x+b x^2}+5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )-\frac{2 \left (a x+b x^2\right )^{5/2}}{3 x^4}-\frac{10 b \left (a x+b x^2\right )^{3/2}}{3 x^2} \]
Antiderivative was successfully verified.
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Rule 662
Rule 664
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^2\right )^{5/2}}{x^5} \, dx &=-\frac{2 \left (a x+b x^2\right )^{5/2}}{3 x^4}+\frac{1}{3} (5 b) \int \frac{\left (a x+b x^2\right )^{3/2}}{x^3} \, dx\\ &=-\frac{10 b \left (a x+b x^2\right )^{3/2}}{3 x^2}-\frac{2 \left (a x+b x^2\right )^{5/2}}{3 x^4}+\left (5 b^2\right ) \int \frac{\sqrt{a x+b x^2}}{x} \, dx\\ &=5 b^2 \sqrt{a x+b x^2}-\frac{10 b \left (a x+b x^2\right )^{3/2}}{3 x^2}-\frac{2 \left (a x+b x^2\right )^{5/2}}{3 x^4}+\frac{1}{2} \left (5 a b^2\right ) \int \frac{1}{\sqrt{a x+b x^2}} \, dx\\ &=5 b^2 \sqrt{a x+b x^2}-\frac{10 b \left (a x+b x^2\right )^{3/2}}{3 x^2}-\frac{2 \left (a x+b x^2\right )^{5/2}}{3 x^4}+\left (5 a b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )\\ &=5 b^2 \sqrt{a x+b x^2}-\frac{10 b \left (a x+b x^2\right )^{3/2}}{3 x^2}-\frac{2 \left (a x+b x^2\right )^{5/2}}{3 x^4}+5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0128478, size = 50, normalized size = 0.56 \[ -\frac{2 a^2 \sqrt{x (a+b x)} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x}{a}\right )}{3 x^2 \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 209, normalized size = 2.4 \begin{align*} -{\frac{2}{3\,a{x}^{5}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{8\,b}{3\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}+16\,{\frac{{b}^{2} \left ( b{x}^{2}+ax \right ) ^{7/2}}{{a}^{3}{x}^{3}}}-{\frac{128\,{b}^{3}}{3\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}+{\frac{128\,{b}^{4}}{3\,{a}^{4}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{80\,{b}^{4}x}{3\,{a}^{3}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{40\,{b}^{3}}{3\,{a}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-10\,{\frac{{b}^{3}\sqrt{b{x}^{2}+ax}x}{a}}-5\,{b}^{2}\sqrt{b{x}^{2}+ax}+{\frac{5\,a}{2}{b}^{{\frac{3}{2}}}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \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 = 1.9333, size = 343, normalized size = 3.85 \begin{align*} \left [\frac{15 \, a b^{\frac{3}{2}} x^{2} \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (3 \, b^{2} x^{2} - 14 \, a b x - 2 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{6 \, x^{2}}, -\frac{15 \, a \sqrt{-b} b x^{2} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (3 \, b^{2} x^{2} - 14 \, a b x - 2 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{3 \, 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 \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21262, size = 180, normalized size = 2.02 \begin{align*} -\frac{5}{2} \, a b^{\frac{3}{2}} \log \left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right ) + \sqrt{b x^{2} + a x} b^{2} + \frac{2 \,{\left (9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{2} b + 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{3} \sqrt{b} + a^{4}\right )}}{3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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