Optimal. Leaf size=78 \[ \frac{15}{4} b^2 \sqrt{a+b x}-\frac{15}{4} \sqrt{a} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{(a+b x)^{5/2}}{2 x^2}-\frac{5 b (a+b x)^{3/2}}{4 x} \]
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Rubi [A] time = 0.0211749, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 50, 63, 208} \[ \frac{15}{4} b^2 \sqrt{a+b x}-\frac{15}{4} \sqrt{a} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{(a+b x)^{5/2}}{2 x^2}-\frac{5 b (a+b x)^{3/2}}{4 x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2}}{x^3} \, dx &=-\frac{(a+b x)^{5/2}}{2 x^2}+\frac{1}{4} (5 b) \int \frac{(a+b x)^{3/2}}{x^2} \, dx\\ &=-\frac{5 b (a+b x)^{3/2}}{4 x}-\frac{(a+b x)^{5/2}}{2 x^2}+\frac{1}{8} \left (15 b^2\right ) \int \frac{\sqrt{a+b x}}{x} \, dx\\ &=\frac{15}{4} b^2 \sqrt{a+b x}-\frac{5 b (a+b x)^{3/2}}{4 x}-\frac{(a+b x)^{5/2}}{2 x^2}+\frac{1}{8} \left (15 a b^2\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=\frac{15}{4} b^2 \sqrt{a+b x}-\frac{5 b (a+b x)^{3/2}}{4 x}-\frac{(a+b x)^{5/2}}{2 x^2}+\frac{1}{4} (15 a b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )\\ &=\frac{15}{4} b^2 \sqrt{a+b x}-\frac{5 b (a+b x)^{3/2}}{4 x}-\frac{(a+b x)^{5/2}}{2 x^2}-\frac{15}{4} \sqrt{a} b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0207901, size = 35, normalized size = 0.45 \[ -\frac{2 b^2 (a+b x)^{7/2} \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};\frac{b x}{a}+1\right )}{7 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 61, normalized size = 0.8 \begin{align*} 2\,{b}^{2} \left ( \sqrt{bx+a}+a \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( -{\frac{9\, \left ( bx+a \right ) ^{3/2}}{8}}+{\frac{7\,a\sqrt{bx+a}}{8}} \right ) }-{\frac{15}{8\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \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 = 1.54592, size = 320, normalized size = 4.1 \begin{align*} \left [\frac{15 \, \sqrt{a} b^{2} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (8 \, b^{2} x^{2} - 9 \, a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{8 \, x^{2}}, \frac{15 \, \sqrt{-a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (8 \, b^{2} x^{2} - 9 \, a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{4 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.71615, size = 126, normalized size = 1.62 \begin{align*} - \frac{15 \sqrt{a} b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4} - \frac{a^{3}}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{11 a^{2} \sqrt{b}}{4 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{a b^{\frac{3}{2}}}{4 \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{2 b^{\frac{5}{2}} \sqrt{x}}{\sqrt{\frac{a}{b x} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15919, size = 108, normalized size = 1.38 \begin{align*} \frac{\frac{15 \, a b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 8 \, \sqrt{b x + a} b^{3} - \frac{9 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{3} - 7 \, \sqrt{b x + a} a^{2} b^{3}}{b^{2} x^{2}}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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