Optimal. Leaf size=102 \[ \frac{5 b^2 \sqrt{b x+c x^2}}{8 c^3}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c} \]
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Rubi [A] time = 0.0425747, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {670, 640, 620, 206} \[ \frac{5 b^2 \sqrt{b x+c x^2}}{8 c^3}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 670
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{b x+c x^2}} \, dx &=\frac{x^2 \sqrt{b x+c x^2}}{3 c}-\frac{(5 b) \int \frac{x^2}{\sqrt{b x+c x^2}} \, dx}{6 c}\\ &=-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c}+\frac{\left (5 b^2\right ) \int \frac{x}{\sqrt{b x+c x^2}} \, dx}{8 c^2}\\ &=\frac{5 b^2 \sqrt{b x+c x^2}}{8 c^3}-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c}-\frac{\left (5 b^3\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{16 c^3}\\ &=\frac{5 b^2 \sqrt{b x+c x^2}}{8 c^3}-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c}-\frac{\left (5 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{8 c^3}\\ &=\frac{5 b^2 \sqrt{b x+c x^2}}{8 c^3}-\frac{5 b x \sqrt{b x+c x^2}}{12 c^2}+\frac{x^2 \sqrt{b x+c x^2}}{3 c}-\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.173059, size = 87, normalized size = 0.85 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (15 b^2-10 b c x+8 c^2 x^2\right )-\frac{15 b^{5/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{24 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 90, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx}}-{\frac{5\,bx}{12\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{2}}{8\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{3}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{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 = 1.96853, size = 350, normalized size = 3.43 \begin{align*} \left [\frac{15 \, 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} - 10 \, b c^{2} x + 15 \, b^{2} c\right )} \sqrt{c x^{2} + b x}}{48 \, c^{4}}, \frac{15 \, b^{3} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (8 \, c^{3} x^{2} - 10 \, b c^{2} x + 15 \, b^{2} c\right )} \sqrt{c x^{2} + b x}}{24 \, c^{4}}\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{x^{3}}{\sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26409, size = 104, normalized size = 1.02 \begin{align*} \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \, x{\left (\frac{4 \, x}{c} - \frac{5 \, b}{c^{2}}\right )} + \frac{15 \, b^{2}}{c^{3}}\right )} + \frac{5 \, 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{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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