Optimal. Leaf size=65 \[ 2 a^2 \sqrt{a+b x}-2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2}{3} a (a+b x)^{3/2}+\frac{2}{5} (a+b x)^{5/2} \]
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Rubi [A] time = 0.0208395, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {50, 63, 208} \[ 2 a^2 \sqrt{a+b x}-2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2}{3} a (a+b x)^{3/2}+\frac{2}{5} (a+b x)^{5/2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2}}{x} \, dx &=\frac{2}{5} (a+b x)^{5/2}+a \int \frac{(a+b x)^{3/2}}{x} \, dx\\ &=\frac{2}{3} a (a+b x)^{3/2}+\frac{2}{5} (a+b x)^{5/2}+a^2 \int \frac{\sqrt{a+b x}}{x} \, dx\\ &=2 a^2 \sqrt{a+b x}+\frac{2}{3} a (a+b x)^{3/2}+\frac{2}{5} (a+b x)^{5/2}+a^3 \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=2 a^2 \sqrt{a+b x}+\frac{2}{3} a (a+b x)^{3/2}+\frac{2}{5} (a+b x)^{5/2}+\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{b}\\ &=2 a^2 \sqrt{a+b x}+\frac{2}{3} a (a+b x)^{3/2}+\frac{2}{5} (a+b x)^{5/2}-2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0887927, size = 58, normalized size = 0.89 \[ -2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2}{3} a \sqrt{a+b x} (4 a+b x)+\frac{2}{5} (a+b x)^{5/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 50, normalized size = 0.8 \begin{align*}{\frac{2\,a}{3} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{2}{5} \left ( bx+a \right ) ^{{\frac{5}{2}}}}-2\,{a}^{5/2}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) +2\,{a}^{2}\sqrt{bx+a} \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 = 1.57475, size = 288, normalized size = 4.43 \begin{align*} \left [a^{\frac{5}{2}} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \frac{2}{15} \,{\left (3 \, b^{2} x^{2} + 11 \, a b x + 23 \, a^{2}\right )} \sqrt{b x + a}, 2 \, \sqrt{-a} a^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) + \frac{2}{15} \,{\left (3 \, b^{2} x^{2} + 11 \, a b x + 23 \, a^{2}\right )} \sqrt{b x + a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.68259, size = 97, normalized size = 1.49 \begin{align*} \frac{46 a^{\frac{5}{2}} \sqrt{1 + \frac{b x}{a}}}{15} + a^{\frac{5}{2}} \log{\left (\frac{b x}{a} \right )} - 2 a^{\frac{5}{2}} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )} + \frac{22 a^{\frac{3}{2}} b x \sqrt{1 + \frac{b x}{a}}}{15} + \frac{2 \sqrt{a} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18573, size = 76, normalized size = 1.17 \begin{align*} \frac{2 \, a^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2}{5} \,{\left (b x + a\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (b x + a\right )}^{\frac{3}{2}} a + 2 \, \sqrt{b x + a} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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