Optimal. Leaf size=101 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{64 b^{3/2}}+\frac{5 a^2 (a+2 b x) \sqrt{a x+b x^2}}{64 b}+\frac{5}{24} a \left (a x+b x^2\right )^{3/2}+\frac{\left (a x+b x^2\right )^{5/2}}{4 x} \]
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Rubi [A] time = 0.041171, antiderivative size = 101, 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 = {664, 612, 620, 206} \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{64 b^{3/2}}+\frac{5 a^2 (a+2 b x) \sqrt{a x+b x^2}}{64 b}+\frac{5}{24} a \left (a x+b x^2\right )^{3/2}+\frac{\left (a x+b x^2\right )^{5/2}}{4 x} \]
Antiderivative was successfully verified.
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Rule 664
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^2\right )^{5/2}}{x^2} \, dx &=\frac{\left (a x+b x^2\right )^{5/2}}{4 x}+\frac{1}{8} (5 a) \int \frac{\left (a x+b x^2\right )^{3/2}}{x} \, dx\\ &=\frac{5}{24} a \left (a x+b x^2\right )^{3/2}+\frac{\left (a x+b x^2\right )^{5/2}}{4 x}+\frac{1}{16} \left (5 a^2\right ) \int \sqrt{a x+b x^2} \, dx\\ &=\frac{5 a^2 (a+2 b x) \sqrt{a x+b x^2}}{64 b}+\frac{5}{24} a \left (a x+b x^2\right )^{3/2}+\frac{\left (a x+b x^2\right )^{5/2}}{4 x}-\frac{\left (5 a^4\right ) \int \frac{1}{\sqrt{a x+b x^2}} \, dx}{128 b}\\ &=\frac{5 a^2 (a+2 b x) \sqrt{a x+b x^2}}{64 b}+\frac{5}{24} a \left (a x+b x^2\right )^{3/2}+\frac{\left (a x+b x^2\right )^{5/2}}{4 x}-\frac{\left (5 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )}{64 b}\\ &=\frac{5 a^2 (a+2 b x) \sqrt{a x+b x^2}}{64 b}+\frac{5}{24} a \left (a x+b x^2\right )^{3/2}+\frac{\left (a x+b x^2\right )^{5/2}}{4 x}-\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{64 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.131444, size = 98, normalized size = 0.97 \[ \frac{\sqrt{x (a+b x)} \left (\sqrt{b} \left (118 a^2 b x+15 a^3+136 a b^2 x^2+48 b^3 x^3\right )-\frac{15 a^{7/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{x} \sqrt{\frac{b x}{a}+1}}\right )}{192 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 135, normalized size = 1.3 \begin{align*}{\frac{2}{3\,a{x}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{2\,b}{3\,a} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}-{\frac{5\,bx}{12} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,a}{24} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}x}{32}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{3}}{64\,b}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{4}}{128}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{3}{2}}}} \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 = 2.11876, size = 406, normalized size = 4.02 \begin{align*} \left [\frac{15 \, a^{4} \sqrt{b} \log \left (2 \, b x + a - 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (48 \, b^{4} x^{3} + 136 \, a b^{3} x^{2} + 118 \, a^{2} b^{2} x + 15 \, a^{3} b\right )} \sqrt{b x^{2} + a x}}{384 \, b^{2}}, \frac{15 \, a^{4} \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) +{\left (48 \, b^{4} x^{3} + 136 \, a b^{3} x^{2} + 118 \, a^{2} b^{2} x + 15 \, a^{3} b\right )} \sqrt{b x^{2} + a x}}{192 \, b^{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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18341, size = 113, normalized size = 1.12 \begin{align*} \frac{5 \, a^{4} \log \left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{128 \, b^{\frac{3}{2}}} + \frac{1}{192} \, \sqrt{b x^{2} + a x}{\left (\frac{15 \, a^{3}}{b} + 2 \,{\left (59 \, a^{2} + 4 \,{\left (6 \, b^{2} x + 17 \, a b\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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