Optimal. Leaf size=78 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{3/2}}+\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c}+\frac{1}{3} \left (b x+c x^2\right )^{3/2} \]
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Rubi [A] time = 0.0265645, antiderivative size = 78, 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 = {664, 612, 620, 206} \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{3/2}}+\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c}+\frac{1}{3} \left (b x+c x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 664
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x} \, dx &=\frac{1}{3} \left (b x+c x^2\right )^{3/2}+\frac{1}{2} b \int \sqrt{b x+c x^2} \, dx\\ &=\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c}+\frac{1}{3} \left (b x+c x^2\right )^{3/2}-\frac{b^3 \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{16 c}\\ &=\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c}+\frac{1}{3} \left (b x+c x^2\right )^{3/2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{8 c}\\ &=\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c}+\frac{1}{3} \left (b x+c x^2\right )^{3/2}-\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.117933, size = 87, normalized size = 1.12 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (3 b^2+14 b c x+8 c^2 x^2\right )-\frac{3 b^{5/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{24 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 81, normalized size = 1. \begin{align*}{\frac{1}{3} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{bx}{4}\sqrt{c{x}^{2}+bx}}+{\frac{{b}^{2}}{8\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{3}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \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 = 2.01422, size = 344, normalized size = 4.41 \begin{align*} \left [\frac{3 \, b^{3} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (8 \, c^{3} x^{2} + 14 \, b c^{2} x + 3 \, b^{2} c\right )} \sqrt{c x^{2} + b x}}{48 \, c^{2}}, \frac{3 \, b^{3} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (8 \, c^{3} x^{2} + 14 \, b c^{2} x + 3 \, b^{2} c\right )} \sqrt{c x^{2} + b x}}{24 \, c^{2}}\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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.45398, size = 97, normalized size = 1.24 \begin{align*} \frac{b^{3} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{3}{2}}} + \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \, c x + 7 \, b\right )} x + \frac{3 \, b^{2}}{c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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