Optimal. Leaf size=91 \[ -\frac{2 b^2 \sqrt{a x+b x^2}}{x}+2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )-\frac{2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac{2 \left (a x+b x^2\right )^{5/2}}{5 x^5} \]
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Rubi [A] time = 0.0406987, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {662, 620, 206} \[ -\frac{2 b^2 \sqrt{a x+b x^2}}{x}+2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )-\frac{2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac{2 \left (a x+b x^2\right )^{5/2}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 662
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^2\right )^{5/2}}{x^6} \, dx &=-\frac{2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+b \int \frac{\left (a x+b x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac{2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac{2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+b^2 \int \frac{\sqrt{a x+b x^2}}{x^2} \, dx\\ &=-\frac{2 b^2 \sqrt{a x+b x^2}}{x}-\frac{2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac{2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+b^3 \int \frac{1}{\sqrt{a x+b x^2}} \, dx\\ &=-\frac{2 b^2 \sqrt{a x+b x^2}}{x}-\frac{2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac{2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )\\ &=-\frac{2 b^2 \sqrt{a x+b x^2}}{x}-\frac{2 b \left (a x+b x^2\right )^{3/2}}{3 x^3}-\frac{2 \left (a x+b x^2\right )^{5/2}}{5 x^5}+2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.013242, size = 50, normalized size = 0.55 \[ -\frac{2 a^2 \sqrt{x (a+b x)} \, _2F_1\left (-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};-\frac{b x}{a}\right )}{5 x^3 \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 232, normalized size = 2.6 \begin{align*} -{\frac{2}{5\,a{x}^{6}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{4\,b}{15\,{a}^{2}{x}^{5}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{16\,{b}^{2}}{15\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}+{\frac{32\,{b}^{3}}{5\,{a}^{4}{x}^{3}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{256\,{b}^{4}}{15\,{a}^{5}{x}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}+{\frac{256\,{b}^{5}}{15\,{a}^{5}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{32\,{b}^{5}x}{3\,{a}^{4}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{16\,{b}^{4}}{3\,{a}^{3}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-4\,{\frac{{b}^{4}\sqrt{b{x}^{2}+ax}x}{{a}^{2}}}-2\,{\frac{{b}^{3}\sqrt{b{x}^{2}+ax}}{a}}+{b}^{{\frac{5}{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 = 2.03784, size = 346, normalized size = 3.8 \begin{align*} \left [\frac{15 \, b^{\frac{5}{2}} x^{3} \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) - 2 \,{\left (23 \, b^{2} x^{2} + 11 \, a b x + 3 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{15 \, x^{3}}, -\frac{2 \,{\left (15 \, \sqrt{-b} b^{2} x^{3} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) +{\left (23 \, b^{2} x^{2} + 11 \, a b x + 3 \, a^{2}\right )} \sqrt{b x^{2} + a x}\right )}}{15 \, x^{3}}\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^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29556, size = 236, normalized size = 2.59 \begin{align*} -b^{\frac{5}{2}} \log \left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right ) + \frac{2 \,{\left (45 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a b^{2} + 45 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{2} b^{\frac{3}{2}} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{3} b + 15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{4} \sqrt{b} + 3 \, a^{5}\right )}}{15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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