Optimal. Leaf size=80 \[ -\frac {a^3 \left (a+b x^3\right )^{2/3}}{2 b^4}+\frac {3 a^2 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac {\left (a+b x^3\right )^{11/3}}{11 b^4}-\frac {3 a \left (a+b x^3\right )^{8/3}}{8 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, 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} \[ \frac {3 a^2 \left (a+b x^3\right )^{5/3}}{5 b^4}-\frac {a^3 \left (a+b x^3\right )^{2/3}}{2 b^4}+\frac {\left (a+b x^3\right )^{11/3}}{11 b^4}-\frac {3 a \left (a+b x^3\right )^{8/3}}{8 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{11}}{\sqrt [3]{a+b x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{a+b x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {a^3}{b^3 \sqrt [3]{a+b x}}+\frac {3 a^2 (a+b x)^{2/3}}{b^3}-\frac {3 a (a+b x)^{5/3}}{b^3}+\frac {(a+b x)^{8/3}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^3 \left (a+b x^3\right )^{2/3}}{2 b^4}+\frac {3 a^2 \left (a+b x^3\right )^{5/3}}{5 b^4}-\frac {3 a \left (a+b x^3\right )^{8/3}}{8 b^4}+\frac {\left (a+b x^3\right )^{11/3}}{11 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 50, normalized size = 0.62 \[ \frac {\left (a+b x^3\right )^{2/3} \left (-81 a^3+54 a^2 b x^3-45 a b^2 x^6+40 b^3 x^9\right )}{440 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 46, normalized size = 0.58 \[ \frac {{\left (40 \, b^{3} x^{9} - 45 \, a b^{2} x^{6} + 54 \, a^{2} b x^{3} - 81 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{440 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 61, normalized size = 0.76 \[ -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{3}}{2 \, b^{4}} + \frac {40 \, {\left (b x^{3} + a\right )}^{\frac {11}{3}} - 165 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} a + 264 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{440 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 47, normalized size = 0.59 \[ -\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (-40 b^{3} x^{9}+45 a \,b^{2} x^{6}-54 a^{2} b \,x^{3}+81 a^{3}\right )}{440 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.38, size = 64, normalized size = 0.80 \[ \frac {{\left (b x^{3} + a\right )}^{\frac {11}{3}}}{11 \, b^{4}} - \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} a}{8 \, b^{4}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{5 \, b^{4}} - \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{3}}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.11, size = 48, normalized size = 0.60 \[ -{\left (b\,x^3+a\right )}^{2/3}\,\left (\frac {81\,a^3}{440\,b^4}-\frac {x^9}{11\,b}+\frac {9\,a\,x^6}{88\,b^2}-\frac {27\,a^2\,x^3}{220\,b^3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.31, size = 92, normalized size = 1.15 \[ \begin {cases} - \frac {81 a^{3} \left (a + b x^{3}\right )^{\frac {2}{3}}}{440 b^{4}} + \frac {27 a^{2} x^{3} \left (a + b x^{3}\right )^{\frac {2}{3}}}{220 b^{3}} - \frac {9 a x^{6} \left (a + b x^{3}\right )^{\frac {2}{3}}}{88 b^{2}} + \frac {x^{9} \left (a + b x^{3}\right )^{\frac {2}{3}}}{11 b} & \text {for}\: b \neq 0 \\\frac {x^{12}}{12 \sqrt [3]{a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________