Optimal. Leaf size=62 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{(a+b x)^{3/2}}{2 x^2}-\frac{3 b \sqrt{a+b x}}{4 x} \]
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Rubi [A] time = 0.0160989, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {47, 63, 208} \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{(a+b x)^{3/2}}{2 x^2}-\frac{3 b \sqrt{a+b x}}{4 x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{x^3} \, dx &=-\frac{(a+b x)^{3/2}}{2 x^2}+\frac{1}{4} (3 b) \int \frac{\sqrt{a+b x}}{x^2} \, dx\\ &=-\frac{3 b \sqrt{a+b x}}{4 x}-\frac{(a+b x)^{3/2}}{2 x^2}+\frac{1}{8} \left (3 b^2\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=-\frac{3 b \sqrt{a+b x}}{4 x}-\frac{(a+b x)^{3/2}}{2 x^2}+\frac{1}{4} (3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )\\ &=-\frac{3 b \sqrt{a+b x}}{4 x}-\frac{(a+b x)^{3/2}}{2 x^2}-\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0531582, size = 68, normalized size = 1.1 \[ -\frac{2 a^2+3 b^2 x^2 \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )+7 a b x+5 b^2 x^2}{4 x^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 51, normalized size = 0.8 \begin{align*} 2\,{b}^{2} \left ({\frac{-5/8\, \left ( bx+a \right ) ^{3/2}+3/8\,a\sqrt{bx+a}}{{b}^{2}{x}^{2}}}-3/8\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \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 = 1.55413, size = 296, normalized size = 4.77 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (5 \, a b x + 2 \, a^{2}\right )} \sqrt{b x + a}}{8 \, a x^{2}}, \frac{3 \, \sqrt{-a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) -{\left (5 \, a b x + 2 \, a^{2}\right )} \sqrt{b x + a}}{4 \, a x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.1397, size = 76, normalized size = 1.23 \begin{align*} - \frac{a \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{2 x^{\frac{3}{2}}} - \frac{5 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{4 \sqrt{x}} - \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23718, size = 86, normalized size = 1.39 \begin{align*} \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3} - 3 \, \sqrt{b x + a} a b^{3}}{b^{2} x^{2}}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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