Optimal. Leaf size=89 \[ -\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0232197, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {612, 620, 206} \[ -\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \left (b x+c x^2\right )^{3/2} \, dx &=\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}-\frac{\left (3 b^2\right ) \int \sqrt{b x+c x^2} \, dx}{16 c}\\ &=-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}+\frac{\left (3 b^4\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{128 c^2}\\ &=-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}+\frac{\left (3 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^2}\\ &=-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2}}{64 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2}}{8 c}+\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.130844, size = 98, normalized size = 1.1 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (2 b^2 c x-3 b^3+24 b c^2 x^2+16 c^3 x^3\right )+\frac{3 b^{7/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{64 c^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.048, size = 95, normalized size = 1.1 \begin{align*}{\frac{2\,cx+b}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}x}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{4}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{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 = 2.02213, size = 393, normalized size = 4.42 \begin{align*} \left [\frac{3 \, b^{4} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (16 \, c^{4} x^{3} + 24 \, b c^{3} x^{2} + 2 \, b^{2} c^{2} x - 3 \, b^{3} c\right )} \sqrt{c x^{2} + b x}}{128 \, c^{3}}, -\frac{3 \, b^{4} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (16 \, c^{4} x^{3} + 24 \, b c^{3} x^{2} + 2 \, b^{2} c^{2} x - 3 \, b^{3} c\right )} \sqrt{c x^{2} + b x}}{64 \, c^{3}}\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 \left (b x + c x^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.39659, size = 112, normalized size = 1.26 \begin{align*} -\frac{3 \, 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{5}{2}}} + \frac{1}{64} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \, c x + 3 \, b\right )} x + \frac{b^{2}}{c}\right )} x - \frac{3 \, b^{3}}{c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]