Optimal. Leaf size=28 \[ \frac {\sqrt [3]{x^3-1} \left (11 x^6-15 x^3+4\right )}{28 x^7} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {453, 264} \begin {gather*} \frac {11 \left (x^3-1\right )^{4/3}}{28 x^4}-\frac {\left (x^3-1\right )^{4/3}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^8} \, dx &=-\frac {\left (-1+x^3\right )^{4/3}}{7 x^7}+\frac {11}{7} \int \frac {\sqrt [3]{-1+x^3}}{x^5} \, dx\\ &=-\frac {\left (-1+x^3\right )^{4/3}}{7 x^7}+\frac {11 \left (-1+x^3\right )^{4/3}}{28 x^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (x^3-1\right )^{4/3} \left (11 x^3-4\right )}{28 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.10, size = 23, normalized size = 0.82 \begin {gather*} \frac {\left (x^3-1\right )^{4/3} \left (11 x^3-4\right )}{28 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 24, normalized size = 0.86 \begin {gather*} \frac {{\left (11 \, x^{6} - 15 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{8}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 29, normalized size = 1.04 \begin {gather*} \frac {\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (11 x^{3}-4\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{28 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.57, size = 25, normalized size = 0.89 \begin {gather*} \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} + \frac {{\left (x^{3} - 1\right )}^{\frac {7}{3}}}{7 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.27, size = 39, normalized size = 1.39 \begin {gather*} \frac {4\,{\left (x^3-1\right )}^{1/3}-15\,x^3\,{\left (x^3-1\right )}^{1/3}+11\,x^6\,{\left (x^3-1\right )}^{1/3}}{28\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 2.48, size = 418, normalized size = 14.93 \begin {gather*} 2 \left (\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases}\right ) - \begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {4 \sqrt [3]{-1 + \frac {1}{x^{3}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {3 x^{6} \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} - \frac {2 x^{3} \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{9} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {5 \sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{6} \Gamma \left (- \frac {1}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________