Optimal. Leaf size=64 \[ 2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}-\frac{2 b \sqrt{a+b x}}{\sqrt{x}} \]
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Rubi [A] time = 0.0214081, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {47, 63, 217, 206} \[ 2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}-\frac{2 b \sqrt{a+b x}}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{x^{5/2}} \, dx &=-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}+b \int \frac{\sqrt{a+b x}}{x^{3/2}} \, dx\\ &=-\frac{2 b \sqrt{a+b x}}{\sqrt{x}}-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}+b^2 \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx\\ &=-\frac{2 b \sqrt{a+b x}}{\sqrt{x}}-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 b \sqrt{a+b x}}{\sqrt{x}}-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )\\ &=-\frac{2 b \sqrt{a+b x}}{\sqrt{x}}-\frac{2 (a+b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0100274, size = 48, normalized size = 0.75 \[ -\frac{2 a \sqrt{a+b x} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};-\frac{b x}{a}\right )}{3 x^{3/2} \sqrt{\frac{b x}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 67, normalized size = 1.1 \begin{align*} -{\frac{8\,bx+2\,a}{3}\sqrt{bx+a}{x}^{-{\frac{3}{2}}}}+{{b}^{{\frac{3}{2}}}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) \sqrt{x \left ( bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\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 = 1.72707, size = 302, normalized size = 4.72 \begin{align*} \left [\frac{3 \, b^{\frac{3}{2}} x^{2} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (4 \, b x + a\right )} \sqrt{b x + a} \sqrt{x}}{3 \, x^{2}}, -\frac{2 \,{\left (3 \, \sqrt{-b} b x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (4 \, b x + a\right )} \sqrt{b x + a} \sqrt{x}\right )}}{3 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.60244, size = 71, normalized size = 1.11 \begin{align*} - \frac{2 a \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{8 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3} - b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )} + 2 b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{b x} + 1} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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