Optimal. Leaf size=161 \[ -\frac{512 a^5 \left (a x^2+b x^3\right )^{5/2}}{45045 b^6 x^5}+\frac{256 a^4 \left (a x^2+b x^3\right )^{5/2}}{9009 b^5 x^4}-\frac{64 a^3 \left (a x^2+b x^3\right )^{5/2}}{1287 b^4 x^3}+\frac{32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^2}-\frac{4 a \left (a x^2+b x^3\right )^{5/2}}{39 b^2 x}+\frac{2 \left (a x^2+b x^3\right )^{5/2}}{15 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23122, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \[ -\frac{512 a^5 \left (a x^2+b x^3\right )^{5/2}}{45045 b^6 x^5}+\frac{256 a^4 \left (a x^2+b x^3\right )^{5/2}}{9009 b^5 x^4}-\frac{64 a^3 \left (a x^2+b x^3\right )^{5/2}}{1287 b^4 x^3}+\frac{32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^2}-\frac{4 a \left (a x^2+b x^3\right )^{5/2}}{39 b^2 x}+\frac{2 \left (a x^2+b x^3\right )^{5/2}}{15 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2016
Rule 2002
Rule 2014
Rubi steps
\begin{align*} \int x^2 \left (a x^2+b x^3\right )^{3/2} \, dx &=\frac{2 \left (a x^2+b x^3\right )^{5/2}}{15 b}-\frac{(2 a) \int x \left (a x^2+b x^3\right )^{3/2} \, dx}{3 b}\\ &=\frac{2 \left (a x^2+b x^3\right )^{5/2}}{15 b}-\frac{4 a \left (a x^2+b x^3\right )^{5/2}}{39 b^2 x}+\frac{\left (16 a^2\right ) \int \left (a x^2+b x^3\right )^{3/2} \, dx}{39 b^2}\\ &=\frac{2 \left (a x^2+b x^3\right )^{5/2}}{15 b}+\frac{32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^2}-\frac{4 a \left (a x^2+b x^3\right )^{5/2}}{39 b^2 x}-\frac{\left (32 a^3\right ) \int \frac{\left (a x^2+b x^3\right )^{3/2}}{x} \, dx}{143 b^3}\\ &=\frac{2 \left (a x^2+b x^3\right )^{5/2}}{15 b}-\frac{64 a^3 \left (a x^2+b x^3\right )^{5/2}}{1287 b^4 x^3}+\frac{32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^2}-\frac{4 a \left (a x^2+b x^3\right )^{5/2}}{39 b^2 x}+\frac{\left (128 a^4\right ) \int \frac{\left (a x^2+b x^3\right )^{3/2}}{x^2} \, dx}{1287 b^4}\\ &=\frac{2 \left (a x^2+b x^3\right )^{5/2}}{15 b}+\frac{256 a^4 \left (a x^2+b x^3\right )^{5/2}}{9009 b^5 x^4}-\frac{64 a^3 \left (a x^2+b x^3\right )^{5/2}}{1287 b^4 x^3}+\frac{32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^2}-\frac{4 a \left (a x^2+b x^3\right )^{5/2}}{39 b^2 x}-\frac{\left (256 a^5\right ) \int \frac{\left (a x^2+b x^3\right )^{3/2}}{x^3} \, dx}{9009 b^5}\\ &=\frac{2 \left (a x^2+b x^3\right )^{5/2}}{15 b}-\frac{512 a^5 \left (a x^2+b x^3\right )^{5/2}}{45045 b^6 x^5}+\frac{256 a^4 \left (a x^2+b x^3\right )^{5/2}}{9009 b^5 x^4}-\frac{64 a^3 \left (a x^2+b x^3\right )^{5/2}}{1287 b^4 x^3}+\frac{32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^2}-\frac{4 a \left (a x^2+b x^3\right )^{5/2}}{39 b^2 x}\\ \end{align*}
Mathematica [A] time = 0.0414666, size = 80, normalized size = 0.5 \[ \frac{2 x (a+b x)^3 \left (-1120 a^3 b^2 x^2+1680 a^2 b^3 x^3+640 a^4 b x-256 a^5-2310 a b^4 x^4+3003 b^5 x^5\right )}{45045 b^6 \sqrt{x^2 (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 79, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -3003\,{x}^{5}{b}^{5}+2310\,a{x}^{4}{b}^{4}-1680\,{x}^{3}{a}^{2}{b}^{3}+1120\,{x}^{2}{a}^{3}{b}^{2}-640\,{a}^{4}xb+256\,{a}^{5} \right ) }{45045\,{b}^{6}{x}^{3}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00878, size = 116, normalized size = 0.72 \begin{align*} \frac{2 \,{\left (3003 \, b^{7} x^{7} + 3696 \, a b^{6} x^{6} + 63 \, a^{2} b^{5} x^{5} - 70 \, a^{3} b^{4} x^{4} + 80 \, a^{4} b^{3} x^{3} - 96 \, a^{5} b^{2} x^{2} + 128 \, a^{6} b x - 256 \, a^{7}\right )} \sqrt{b x + a}}{45045 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.868384, size = 217, normalized size = 1.35 \begin{align*} \frac{2 \,{\left (3003 \, b^{7} x^{7} + 3696 \, a b^{6} x^{6} + 63 \, a^{2} b^{5} x^{5} - 70 \, a^{3} b^{4} x^{4} + 80 \, a^{4} b^{3} x^{3} - 96 \, a^{5} b^{2} x^{2} + 128 \, a^{6} b x - 256 \, a^{7}\right )} \sqrt{b x^{3} + a x^{2}}}{45045 \, b^{6} x} \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^{2} \left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13546, size = 242, normalized size = 1.5 \begin{align*} \frac{512 \, a^{\frac{15}{2}} \mathrm{sgn}\left (x\right )}{45045 \, b^{6}} + \frac{2 \,{\left (\frac{5 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5}\right )} a \mathrm{sgn}\left (x\right )}{b^{5}} + \frac{{\left (3003 \,{\left (b x + a\right )}^{\frac{15}{2}} - 20790 \,{\left (b x + a\right )}^{\frac{13}{2}} a + 61425 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{2} - 100100 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{3} + 96525 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{4} - 54054 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{5} + 15015 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{6}\right )} \mathrm{sgn}\left (x\right )}{b^{5}}\right )}}{45045 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]