Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b x^2\right )^{2/3}}{4 b^3}+\frac{3 \left (a+b x^2\right )^{8/3}}{16 b^3}-\frac{3 a \left (a+b x^2\right )^{5/3}}{5 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0343872, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{3 a^2 \left (a+b x^2\right )^{2/3}}{4 b^3}+\frac{3 \left (a+b x^2\right )^{8/3}}{16 b^3}-\frac{3 a \left (a+b x^2\right )^{5/3}}{5 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt [3]{a+b x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt [3]{a+b x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt [3]{a+b x}}-\frac{2 a (a+b x)^{2/3}}{b^2}+\frac{(a+b x)^{5/3}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a^2 \left (a+b x^2\right )^{2/3}}{4 b^3}-\frac{3 a \left (a+b x^2\right )^{5/3}}{5 b^3}+\frac{3 \left (a+b x^2\right )^{8/3}}{16 b^3}\\ \end{align*}
Mathematica [A] time = 0.0222005, size = 39, normalized size = 0.66 \[ \frac{3 \left (a+b x^2\right )^{2/3} \left (9 a^2-6 a b x^2+5 b^2 x^4\right )}{80 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 36, normalized size = 0.6 \begin{align*}{\frac{15\,{b}^{2}{x}^{4}-18\,ab{x}^{2}+27\,{a}^{2}}{80\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.66832, size = 63, normalized size = 1.07 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}}}{16 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a}{5 \, b^{3}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{2}}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.68285, size = 81, normalized size = 1.37 \begin{align*} \frac{3 \,{\left (5 \, b^{2} x^{4} - 6 \, a b x^{2} + 9 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{80 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.57727, size = 631, normalized size = 10.69 \begin{align*} \frac{27 a^{\frac{32}{3}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{80 a^{8} b^{3} + 240 a^{7} b^{4} x^{2} + 240 a^{6} b^{5} x^{4} + 80 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{32}{3}}}{80 a^{8} b^{3} + 240 a^{7} b^{4} x^{2} + 240 a^{6} b^{5} x^{4} + 80 a^{5} b^{6} x^{6}} + \frac{63 a^{\frac{29}{3}} b x^{2} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{80 a^{8} b^{3} + 240 a^{7} b^{4} x^{2} + 240 a^{6} b^{5} x^{4} + 80 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{29}{3}} b x^{2}}{80 a^{8} b^{3} + 240 a^{7} b^{4} x^{2} + 240 a^{6} b^{5} x^{4} + 80 a^{5} b^{6} x^{6}} + \frac{42 a^{\frac{26}{3}} b^{2} x^{4} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{80 a^{8} b^{3} + 240 a^{7} b^{4} x^{2} + 240 a^{6} b^{5} x^{4} + 80 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{26}{3}} b^{2} x^{4}}{80 a^{8} b^{3} + 240 a^{7} b^{4} x^{2} + 240 a^{6} b^{5} x^{4} + 80 a^{5} b^{6} x^{6}} + \frac{18 a^{\frac{23}{3}} b^{3} x^{6} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{80 a^{8} b^{3} + 240 a^{7} b^{4} x^{2} + 240 a^{6} b^{5} x^{4} + 80 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{23}{3}} b^{3} x^{6}}{80 a^{8} b^{3} + 240 a^{7} b^{4} x^{2} + 240 a^{6} b^{5} x^{4} + 80 a^{5} b^{6} x^{6}} + \frac{27 a^{\frac{20}{3}} b^{4} x^{8} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{80 a^{8} b^{3} + 240 a^{7} b^{4} x^{2} + 240 a^{6} b^{5} x^{4} + 80 a^{5} b^{6} x^{6}} + \frac{15 a^{\frac{17}{3}} b^{5} x^{10} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{2}{3}}}{80 a^{8} b^{3} + 240 a^{7} b^{4} x^{2} + 240 a^{6} b^{5} x^{4} + 80 a^{5} b^{6} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.47091, size = 58, normalized size = 0.98 \begin{align*} \frac{3 \,{\left (5 \,{\left (b x^{2} + a\right )}^{\frac{8}{3}} - 16 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} a + 20 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{2}\right )}}{80 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]