Optimal. Leaf size=118 \[ \frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2}}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.036886, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {612, 620, 206} \[ \frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2}}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \left (b x+c x^2\right )^{5/2} \, dx &=\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac{\left (5 b^2\right ) \int \left (b x+c x^2\right )^{3/2} \, dx}{24 c}\\ &=-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}+\frac{\left (5 b^4\right ) \int \sqrt{b x+c x^2} \, dx}{128 c^2}\\ &=\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac{\left (5 b^6\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{1024 c^3}\\ &=\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac{\left (5 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{512 c^3}\\ &=\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2}}{12 c}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.121959, size = 120, normalized size = 1.02 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (8 b^3 c^2 x^2+432 b^2 c^3 x^3-10 b^4 c x+15 b^5+640 b c^4 x^4+256 c^5 x^5\right )-\frac{15 b^{11/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{1536 c^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 134, normalized size = 1.1 \begin{align*}{\frac{2\,cx+b}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}x}{96\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{3}}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{4}x}{256\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{5}}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{6}}{1024}\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.
[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 = 2.02157, size = 501, normalized size = 4.25 \begin{align*} \left [\frac{15 \, b^{6} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (256 \, c^{6} x^{5} + 640 \, b c^{5} x^{4} + 432 \, b^{2} c^{4} x^{3} + 8 \, b^{3} c^{3} x^{2} - 10 \, b^{4} c^{2} x + 15 \, b^{5} c\right )} \sqrt{c x^{2} + b x}}{3072 \, c^{4}}, \frac{15 \, b^{6} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (256 \, c^{6} x^{5} + 640 \, b c^{5} x^{4} + 432 \, b^{2} c^{4} x^{3} + 8 \, b^{3} c^{3} x^{2} - 10 \, b^{4} c^{2} x + 15 \, b^{5} c\right )} \sqrt{c x^{2} + b x}}{1536 \, c^{4}}\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{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.35832, size = 144, normalized size = 1.22 \begin{align*} \frac{5 \, b^{6} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{7}{2}}} + \frac{1}{1536} \, \sqrt{c x^{2} + b x}{\left (\frac{15 \, b^{5}}{c^{3}} - 2 \,{\left (\frac{5 \, b^{4}}{c^{2}} - 4 \,{\left (\frac{b^{3}}{c} + 2 \,{\left (27 \, b^{2} + 8 \,{\left (2 \, c^{2} x + 5 \, b c\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]