Optimal. Leaf size=36 \[ \frac{\left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{18 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0289644, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1352, 609} \[ \frac{\left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{18 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1352
Rule 609
Rubi steps
\begin{align*} \int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx,x,x^3\right )\\ &=\frac{\left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{18 b}\\ \end{align*}
Mathematica [B] time = 0.0214581, size = 82, normalized size = 2.28 \[ \frac{x^3 \sqrt{\left (a+b x^3\right )^2} \left (15 a^2 b^3 x^9+20 a^3 b^2 x^6+15 a^4 b x^3+6 a^5+6 a b^4 x^{12}+b^5 x^{15}\right )}{18 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.009, size = 79, normalized size = 2.2 \begin{align*}{\frac{{x}^{3} \left ({b}^{5}{x}^{15}+6\,a{b}^{4}{x}^{12}+15\,{a}^{2}{b}^{3}{x}^{9}+20\,{a}^{3}{b}^{2}{x}^{6}+15\,{a}^{4}b{x}^{3}+6\,{a}^{5} \right ) }{18\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\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 = 1.7066, size = 135, normalized size = 3.75 \begin{align*} \frac{1}{18} \, b^{5} x^{18} + \frac{1}{3} \, a b^{4} x^{15} + \frac{5}{6} \, a^{2} b^{3} x^{12} + \frac{10}{9} \, a^{3} b^{2} x^{9} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{3} \, a^{5} x^{3} \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 (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.1011, size = 89, normalized size = 2.47 \begin{align*} \frac{1}{18} \,{\left (b^{5} x^{18} + 6 \, a b^{4} x^{15} + 15 \, a^{2} b^{3} x^{12} + 20 \, a^{3} b^{2} x^{9} + 15 \, a^{4} b x^{6} + 6 \, a^{5} x^{3}\right )} \mathrm{sgn}\left (b x^{3} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]