Optimal. Leaf size=94 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{8 \sqrt{b}}-\frac{5}{8} a (a+2 b x) \sqrt{a x+b x^2}+\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}-\frac{5}{3} b \left (a x+b x^2\right )^{3/2} \]
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Rubi [A] time = 0.0391494, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {662, 664, 612, 620, 206} \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{8 \sqrt{b}}-\frac{5}{8} a (a+2 b x) \sqrt{a x+b x^2}+\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}-\frac{5}{3} b \left (a x+b x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 662
Rule 664
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^2\right )^{5/2}}{x^3} \, dx &=\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}-(5 b) \int \frac{\left (a x+b x^2\right )^{3/2}}{x} \, dx\\ &=-\frac{5}{3} b \left (a x+b x^2\right )^{3/2}+\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}-\frac{1}{2} (5 a b) \int \sqrt{a x+b x^2} \, dx\\ &=-\frac{5}{8} a (a+2 b x) \sqrt{a x+b x^2}-\frac{5}{3} b \left (a x+b x^2\right )^{3/2}+\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}+\frac{1}{16} \left (5 a^3\right ) \int \frac{1}{\sqrt{a x+b x^2}} \, dx\\ &=-\frac{5}{8} a (a+2 b x) \sqrt{a x+b x^2}-\frac{5}{3} b \left (a x+b x^2\right )^{3/2}+\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}+\frac{1}{8} \left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )\\ &=-\frac{5}{8} a (a+2 b x) \sqrt{a x+b x^2}-\frac{5}{3} b \left (a x+b x^2\right )^{3/2}+\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{8 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.124836, size = 80, normalized size = 0.85 \[ \frac{1}{24} \sqrt{x (a+b x)} \left (\frac{15 a^{5/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{x} \sqrt{\frac{b x}{a}+1}}+33 a^2+26 a b x+8 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 158, normalized size = 1.7 \begin{align*} 2\,{\frac{ \left ( b{x}^{2}+ax \right ) ^{7/2}}{a{x}^{3}}}-{\frac{16\,b}{3\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}+{\frac{16\,{b}^{2}}{3\,{a}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{10\,{b}^{2}x}{3\,a} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,b}{3} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,abx}{4}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{2}}{8}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{3}}{16}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{b}}}} \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 = 1.92348, size = 346, normalized size = 3.68 \begin{align*} \left [\frac{15 \, a^{3} \sqrt{b} \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (8 \, b^{3} x^{2} + 26 \, a b^{2} x + 33 \, a^{2} b\right )} \sqrt{b x^{2} + a x}}{48 \, b}, -\frac{15 \, a^{3} \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (8 \, b^{3} x^{2} + 26 \, a b^{2} x + 33 \, a^{2} b\right )} \sqrt{b x^{2} + a x}}{24 \, b}\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^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24978, size = 97, normalized size = 1.03 \begin{align*} -\frac{5 \, a^{3} \log \left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{16 \, \sqrt{b}} + \frac{1}{24} \, \sqrt{b x^{2} + a x}{\left (33 \, a^{2} + 2 \,{\left (4 \, b^{2} x + 13 \, a b\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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