Optimal. Leaf size=118 \[ \frac{5 a^4 (a+2 b x) \sqrt{a x+b x^2}}{512 b^3}-\frac{5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}-\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{512 b^{7/2}}+\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0359948, 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} \[ \frac{5 a^4 (a+2 b x) \sqrt{a x+b x^2}}{512 b^3}-\frac{5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}-\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{512 b^{7/2}}+\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \left (a x+b x^2\right )^{5/2} \, dx &=\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac{\left (5 a^2\right ) \int \left (a x+b x^2\right )^{3/2} \, dx}{24 b}\\ &=-\frac{5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}+\frac{\left (5 a^4\right ) \int \sqrt{a x+b x^2} \, dx}{128 b^2}\\ &=\frac{5 a^4 (a+2 b x) \sqrt{a x+b x^2}}{512 b^3}-\frac{5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac{\left (5 a^6\right ) \int \frac{1}{\sqrt{a x+b x^2}} \, dx}{1024 b^3}\\ &=\frac{5 a^4 (a+2 b x) \sqrt{a x+b x^2}}{512 b^3}-\frac{5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac{\left (5 a^6\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a x+b x^2}}\right )}{512 b^3}\\ &=\frac{5 a^4 (a+2 b x) \sqrt{a x+b x^2}}{512 b^3}-\frac{5 a^2 (a+2 b x) \left (a x+b x^2\right )^{3/2}}{192 b^2}+\frac{(a+2 b x) \left (a x+b x^2\right )^{5/2}}{12 b}-\frac{5 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{512 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.183326, size = 120, normalized size = 1.02 \[ \frac{\sqrt{x (a+b x)} \left (\sqrt{b} \left (8 a^3 b^2 x^2+432 a^2 b^3 x^3-10 a^4 b x+15 a^5+640 a b^4 x^4+256 b^5 x^5\right )-\frac{15 a^{11/2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{\sqrt{x} \sqrt{\frac{b x}{a}+1}}\right )}{1536 b^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 134, normalized size = 1.1 \begin{align*}{\frac{2\,bx+a}{12\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}x}{96\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{3}}{192\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{5}}{512\,{b}^{3}}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{6}}{1024}\ln \left ({ \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\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 = 1.93928, size = 501, normalized size = 4.25 \begin{align*} \left [\frac{15 \, a^{6} \sqrt{b} \log \left (2 \, b x + a - 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt{b x^{2} + a x}}{3072 \, b^{4}}, \frac{15 \, a^{6} \sqrt{-b} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) +{\left (256 \, b^{6} x^{5} + 640 \, a b^{5} x^{4} + 432 \, a^{2} b^{4} x^{3} + 8 \, a^{3} b^{3} x^{2} - 10 \, a^{4} b^{2} x + 15 \, a^{5} b\right )} \sqrt{b x^{2} + a x}}{1536 \, b^{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 (a x + b x^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27501, size = 144, normalized size = 1.22 \begin{align*} \frac{5 \, a^{6} \log \left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{1024 \, b^{\frac{7}{2}}} + \frac{1}{1536} \, \sqrt{b x^{2} + a x}{\left (\frac{15 \, a^{5}}{b^{3}} - 2 \,{\left (\frac{5 \, a^{4}}{b^{2}} - 4 \,{\left (\frac{a^{3}}{b} + 2 \,{\left (27 \, a^{2} + 8 \,{\left (2 \, b^{2} x + 5 \, a b\right )} x\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]