Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^3\right )^{4/3}}{4 b^3}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3}-\frac{2 a \left (a+b x^3\right )^{7/3}}{7 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0321434, 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} \[ \frac{a^2 \left (a+b x^3\right )^{4/3}}{4 b^3}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3}-\frac{2 a \left (a+b x^3\right )^{7/3}}{7 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^8 \sqrt [3]{a+b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 \sqrt [3]{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2 \sqrt [3]{a+b x}}{b^2}-\frac{2 a (a+b x)^{4/3}}{b^2}+\frac{(a+b x)^{7/3}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{a^2 \left (a+b x^3\right )^{4/3}}{4 b^3}-\frac{2 a \left (a+b x^3\right )^{7/3}}{7 b^3}+\frac{\left (a+b x^3\right )^{10/3}}{10 b^3}\\ \end{align*}
Mathematica [A] time = 0.0168128, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^3\right )^{4/3} \left (9 a^2-12 a b x^3+14 b^2 x^6\right )}{140 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 36, normalized size = 0.6 \begin{align*}{\frac{14\,{b}^{2}{x}^{6}-12\,{x}^{3}ab+9\,{a}^{2}}{140\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03995, size = 63, normalized size = 1.07 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{10}{3}}}{10 \, b^{3}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a}{7 \, b^{3}} + \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{2}}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65881, size = 105, normalized size = 1.78 \begin{align*} \frac{{\left (14 \, b^{3} x^{9} + 2 \, a b^{2} x^{6} - 3 \, a^{2} b x^{3} + 9 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{140 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.44823, size = 87, normalized size = 1.47 \begin{align*} \begin{cases} \frac{9 a^{3} \sqrt [3]{a + b x^{3}}}{140 b^{3}} - \frac{3 a^{2} x^{3} \sqrt [3]{a + b x^{3}}}{140 b^{2}} + \frac{a x^{6} \sqrt [3]{a + b x^{3}}}{70 b} + \frac{x^{9} \sqrt [3]{a + b x^{3}}}{10} & \text{for}\: b \neq 0 \\\frac{\sqrt [3]{a} x^{9}}{9} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11542, size = 58, normalized size = 0.98 \begin{align*} \frac{14 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} - 40 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a + 35 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a^{2}}{140 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]