Optimal. Leaf size=64 \[ -\frac{2 \left (b x+c x^2\right )^{3/2}}{x^2}+3 c \sqrt{b x+c x^2}+3 b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \]
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Rubi [A] time = 0.027478, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {662, 664, 620, 206} \[ -\frac{2 \left (b x+c x^2\right )^{3/2}}{x^2}+3 c \sqrt{b x+c x^2}+3 b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 662
Rule 664
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^3} \, dx &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{x^2}+(3 c) \int \frac{\sqrt{b x+c x^2}}{x} \, dx\\ &=3 c \sqrt{b x+c x^2}-\frac{2 \left (b x+c x^2\right )^{3/2}}{x^2}+\frac{1}{2} (3 b c) \int \frac{1}{\sqrt{b x+c x^2}} \, dx\\ &=3 c \sqrt{b x+c x^2}-\frac{2 \left (b x+c x^2\right )^{3/2}}{x^2}+(3 b c) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )\\ &=3 c \sqrt{b x+c x^2}-\frac{2 \left (b x+c x^2\right )^{3/2}}{x^2}+3 b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0120287, size = 46, normalized size = 0.72 \[ -\frac{2 b \sqrt{x (b+c x)} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x}{b}\right )}{x \sqrt{\frac{c x}{b}+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 124, normalized size = 1.9 \begin{align*} -2\,{\frac{ \left ( c{x}^{2}+bx \right ) ^{5/2}}{b{x}^{3}}}+8\,{\frac{c \left ( c{x}^{2}+bx \right ) ^{5/2}}{{b}^{2}{x}^{2}}}-8\,{\frac{{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}{{b}^{2}}}-6\,{\frac{{c}^{2}\sqrt{c{x}^{2}+bx}x}{b}}-3\,c\sqrt{c{x}^{2}+bx}+{\frac{3\,b}{2}\sqrt{c}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \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 = 1.97435, size = 270, normalized size = 4.22 \begin{align*} \left [\frac{3 \, b \sqrt{c} x \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (c x - 2 \, b\right )}}{2 \, x}, -\frac{3 \, b \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) - \sqrt{c x^{2} + b x}{\left (c x - 2 \, b\right )}}{x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.45613, size = 103, normalized size = 1.61 \begin{align*} -\frac{3}{2} \, b \sqrt{c} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \sqrt{c x^{2} + b x} c + \frac{2 \, b^{2}}{\sqrt{c} x - \sqrt{c x^{2} + b x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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