Optimal. Leaf size=131 \[ -\frac{7 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}+\frac{7 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{9/2}}-\frac{7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{x^2 \left (b x+c x^2\right )^{3/2}}{5 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0597608, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {670, 640, 612, 620, 206} \[ -\frac{7 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}+\frac{7 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{9/2}}-\frac{7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{x^2 \left (b x+c x^2\right )^{3/2}}{5 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 670
Rule 640
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int x^3 \sqrt{b x+c x^2} \, dx &=\frac{x^2 \left (b x+c x^2\right )^{3/2}}{5 c}-\frac{(7 b) \int x^2 \sqrt{b x+c x^2} \, dx}{10 c}\\ &=-\frac{7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac{\left (7 b^2\right ) \int x \sqrt{b x+c x^2} \, dx}{16 c^2}\\ &=\frac{7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac{7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{x^2 \left (b x+c x^2\right )^{3/2}}{5 c}-\frac{\left (7 b^3\right ) \int \sqrt{b x+c x^2} \, dx}{32 c^3}\\ &=-\frac{7 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac{7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac{\left (7 b^5\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{256 c^4}\\ &=-\frac{7 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac{7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac{\left (7 b^5\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{128 c^4}\\ &=-\frac{7 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^4}+\frac{7 b^2 \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac{7 b x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac{x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac{7 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.154078, size = 109, normalized size = 0.83 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-56 b^2 c^2 x^2+70 b^3 c x-105 b^4+48 b c^3 x^3+384 c^4 x^4\right )+\frac{105 b^{9/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{1920 c^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 129, normalized size = 1. \begin{align*}{\frac{{x}^{2}}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,bx}{40\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}}{48\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{3}x}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,{b}^{4}}{128\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,{b}^{5}}{256}\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.07286, size = 459, normalized size = 3.5 \begin{align*} \left [\frac{105 \, b^{5} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 56 \, b^{2} c^{3} x^{2} + 70 \, b^{3} c^{2} x - 105 \, b^{4} c\right )} \sqrt{c x^{2} + b x}}{3840 \, c^{5}}, -\frac{105 \, b^{5} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 56 \, b^{2} c^{3} x^{2} + 70 \, b^{3} c^{2} x - 105 \, b^{4} c\right )} \sqrt{c x^{2} + b x}}{1920 \, c^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{x \left (b + c x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30265, size = 131, normalized size = 1. \begin{align*} \frac{1}{1920} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, x + \frac{b}{c}\right )} x - \frac{7 \, b^{2}}{c^{2}}\right )} x + \frac{35 \, b^{3}}{c^{3}}\right )} x - \frac{105 \, b^{4}}{c^{4}}\right )} - \frac{7 \, b^{5} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]