Optimal. Leaf size=59 \[ \frac{3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}+\frac{3 \left (a+b x^2\right )^{7/3}}{14 b^3}-\frac{3 a \left (a+b x^2\right )^{4/3}}{4 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0335988, 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} \[ \frac{3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}+\frac{3 \left (a+b x^2\right )^{7/3}}{14 b^3}-\frac{3 a \left (a+b x^2\right )^{4/3}}{4 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{2/3}}-\frac{2 a \sqrt [3]{a+b x}}{b^2}+\frac{(a+b x)^{4/3}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{3 a^2 \sqrt [3]{a+b x^2}}{2 b^3}-\frac{3 a \left (a+b x^2\right )^{4/3}}{4 b^3}+\frac{3 \left (a+b x^2\right )^{7/3}}{14 b^3}\\ \end{align*}
Mathematica [A] time = 0.0174606, size = 39, normalized size = 0.66 \[ \frac{3 \sqrt [3]{a+b x^2} \left (9 a^2-3 a b x^2+2 b^2 x^4\right )}{28 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 36, normalized size = 0.6 \begin{align*}{\frac{6\,{b}^{2}{x}^{4}-9\,ab{x}^{2}+27\,{a}^{2}}{28\,{b}^{3}}\sqrt [3]{b{x}^{2}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.19484, size = 63, normalized size = 1.07 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}}}{14 \, b^{3}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a}{4 \, b^{3}} + \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{2}}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.64501, size = 81, normalized size = 1.37 \begin{align*} \frac{3 \,{\left (2 \, b^{2} x^{4} - 3 \, a b x^{2} + 9 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{28 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.61195, size = 631, normalized size = 10.69 \begin{align*} \frac{27 a^{\frac{31}{3}} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{31}{3}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac{72 a^{\frac{28}{3}} b x^{2} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{28}{3}} b x^{2}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac{60 a^{\frac{25}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac{81 a^{\frac{25}{3}} b^{2} x^{4}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac{18 a^{\frac{22}{3}} b^{3} x^{6} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} - \frac{27 a^{\frac{22}{3}} b^{3} x^{6}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac{9 a^{\frac{19}{3}} b^{4} x^{8} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} + \frac{6 a^{\frac{16}{3}} b^{5} x^{10} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{28 a^{8} b^{3} + 84 a^{7} b^{4} x^{2} + 84 a^{6} b^{5} x^{4} + 28 a^{5} b^{6} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.74925, size = 58, normalized size = 0.98 \begin{align*} \frac{3 \,{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a + 14 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{2}\right )}}{28 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]