Optimal. Leaf size=128 \[ -\frac{35 b^3 \sqrt{b x+c x^2}}{64 c^4}+\frac{35 b^2 x \sqrt{b x+c x^2}}{96 c^3}+\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{9/2}}-\frac{7 b x^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{x^3 \sqrt{b x+c x^2}}{4 c} \]
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Rubi [A] time = 0.0576066, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {670, 640, 620, 206} \[ -\frac{35 b^3 \sqrt{b x+c x^2}}{64 c^4}+\frac{35 b^2 x \sqrt{b x+c x^2}}{96 c^3}+\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{9/2}}-\frac{7 b x^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{x^3 \sqrt{b x+c x^2}}{4 c} \]
Antiderivative was successfully verified.
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Rule 670
Rule 640
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt{b x+c x^2}} \, dx &=\frac{x^3 \sqrt{b x+c x^2}}{4 c}-\frac{(7 b) \int \frac{x^3}{\sqrt{b x+c x^2}} \, dx}{8 c}\\ &=-\frac{7 b x^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{x^3 \sqrt{b x+c x^2}}{4 c}+\frac{\left (35 b^2\right ) \int \frac{x^2}{\sqrt{b x+c x^2}} \, dx}{48 c^2}\\ &=\frac{35 b^2 x \sqrt{b x+c x^2}}{96 c^3}-\frac{7 b x^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{x^3 \sqrt{b x+c x^2}}{4 c}-\frac{\left (35 b^3\right ) \int \frac{x}{\sqrt{b x+c x^2}} \, dx}{64 c^3}\\ &=-\frac{35 b^3 \sqrt{b x+c x^2}}{64 c^4}+\frac{35 b^2 x \sqrt{b x+c x^2}}{96 c^3}-\frac{7 b x^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{x^3 \sqrt{b x+c x^2}}{4 c}+\frac{\left (35 b^4\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{128 c^4}\\ &=-\frac{35 b^3 \sqrt{b x+c x^2}}{64 c^4}+\frac{35 b^2 x \sqrt{b x+c x^2}}{96 c^3}-\frac{7 b x^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{x^3 \sqrt{b x+c x^2}}{4 c}+\frac{\left (35 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{64 c^4}\\ &=-\frac{35 b^3 \sqrt{b x+c x^2}}{64 c^4}+\frac{35 b^2 x \sqrt{b x+c x^2}}{96 c^3}-\frac{7 b x^2 \sqrt{b x+c x^2}}{24 c^2}+\frac{x^3 \sqrt{b x+c x^2}}{4 c}+\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.213593, size = 98, normalized size = 0.77 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (70 b^2 c x-105 b^3-56 b c^2 x^2+48 c^3 x^3\right )+\frac{105 b^{7/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{192 c^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 112, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{7\,b{x}^{2}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{35\,{b}^{2}x}{96\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{35\,{b}^{3}}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{35\,{b}^{4}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{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 = 1.97581, size = 408, normalized size = 3.19 \begin{align*} \left [\frac{105 \, b^{4} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (48 \, c^{4} x^{3} - 56 \, b c^{3} x^{2} + 70 \, b^{2} c^{2} x - 105 \, b^{3} c\right )} \sqrt{c x^{2} + b x}}{384 \, c^{5}}, -\frac{105 \, b^{4} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (48 \, c^{4} x^{3} - 56 \, b c^{3} x^{2} + 70 \, b^{2} c^{2} x - 105 \, b^{3} c\right )} \sqrt{c x^{2} + b x}}{192 \, c^{5}}\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^{4}}{\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 = 1.22022, size = 120, normalized size = 0.94 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \, x{\left (\frac{6 \, x}{c} - \frac{7 \, b}{c^{2}}\right )} + \frac{35 \, b^{2}}{c^{3}}\right )} x - \frac{105 \, b^{3}}{c^{4}}\right )} - \frac{35 \, b^{4} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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