Optimal. Leaf size=51 \[ -\frac{(a+b x)^{3/2}}{x}+3 b \sqrt{a+b x}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0147685, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 50, 63, 208} \[ -\frac{(a+b x)^{3/2}}{x}+3 b \sqrt{a+b x}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]
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)^{3/2}}{x^2} \, dx &=-\frac{(a+b x)^{3/2}}{x}+\frac{1}{2} (3 b) \int \frac{\sqrt{a+b x}}{x} \, dx\\ &=3 b \sqrt{a+b x}-\frac{(a+b x)^{3/2}}{x}+\frac{1}{2} (3 a b) \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=3 b \sqrt{a+b x}-\frac{(a+b x)^{3/2}}{x}+(3 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )\\ &=3 b \sqrt{a+b x}-\frac{(a+b x)^{3/2}}{x}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0118689, size = 33, normalized size = 0.65 \[ \frac{2 b (a+b x)^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x}{a}+1\right )}{5 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 47, normalized size = 0.9 \begin{align*} 2\,b \left ( \sqrt{bx+a}+a \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-3/2\,{\frac{1}{\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 = 1.66102, size = 247, normalized size = 4.84 \begin{align*} \left [\frac{3 \, \sqrt{a} b x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (2 \, b x - a\right )} \sqrt{b x + a}}{2 \, x}, \frac{3 \, \sqrt{-a} b x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (2 \, b x - a\right )} \sqrt{b x + a}}{x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.38634, size = 92, normalized size = 1.8 \begin{align*} - 3 \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} - \frac{a^{2}}{\sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{a \sqrt{b}}{\sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{2 b^{\frac{3}{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 = 1.21026, size = 76, normalized size = 1.49 \begin{align*} \frac{\frac{3 \, a b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{b x + a} b^{2} - \frac{\sqrt{b x + a} a b}{x}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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