Optimal. Leaf size=72 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c}}+\frac{3}{4} b \sqrt{b x+c x^2}+\frac{\left (b x+c x^2\right )^{3/2}}{2 x} \]
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Rubi [A] time = 0.0279387, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {664, 620, 206} \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c}}+\frac{3}{4} b \sqrt{b x+c x^2}+\frac{\left (b x+c x^2\right )^{3/2}}{2 x} \]
Antiderivative was successfully verified.
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Rule 664
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^2} \, dx &=\frac{\left (b x+c x^2\right )^{3/2}}{2 x}+\frac{1}{4} (3 b) \int \frac{\sqrt{b x+c x^2}}{x} \, dx\\ &=\frac{3}{4} b \sqrt{b x+c x^2}+\frac{\left (b x+c x^2\right )^{3/2}}{2 x}+\frac{1}{8} \left (3 b^2\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx\\ &=\frac{3}{4} b \sqrt{b x+c x^2}+\frac{\left (b x+c x^2\right )^{3/2}}{2 x}+\frac{1}{4} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )\\ &=\frac{3}{4} b \sqrt{b x+c x^2}+\frac{\left (b x+c x^2\right )^{3/2}}{2 x}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.106998, size = 69, normalized size = 0.96 \[ \frac{1}{4} \sqrt{x (b+c x)} \left (\frac{3 b^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1}}+5 b+2 c x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 99, normalized size = 1.4 \begin{align*} 2\,{\frac{ \left ( c{x}^{2}+bx \right ) ^{5/2}}{b{x}^{2}}}-2\,{\frac{c \left ( c{x}^{2}+bx \right ) ^{3/2}}{b}}-{\frac{3\,cx}{2}\sqrt{c{x}^{2}+bx}}-{\frac{3\,b}{4}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}} \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 = 1.8205, size = 292, normalized size = 4.06 \begin{align*} \left [\frac{3 \, b^{2} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (2 \, c^{2} x + 5 \, b c\right )} \sqrt{c x^{2} + b x}}{8 \, c}, -\frac{3 \, b^{2} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (2 \, c^{2} x + 5 \, b c\right )} \sqrt{c x^{2} + b x}}{4 \, c}\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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38224, size = 81, normalized size = 1.12 \begin{align*} -\frac{3 \, b^{2} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, \sqrt{c}} + \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (2 \, c x + 5 \, b\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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