Optimal. Leaf size=112 \[ \frac {3 \sqrt [3]{x^8+2 x^3-1}}{x}+\log \left (\sqrt [3]{x^8+2 x^3-1}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^8+2 x^3-1}+x}\right )-\frac {1}{2} \log \left (x^2+\sqrt [3]{x^8+2 x^3-1} x+\left (x^8+2 x^3-1\right )^{2/3}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 0.99, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^3+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^3+x^8\right )} \, dx &=\int \left (-\frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x^2}+\frac {x \left (3+8 x^5\right ) \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+\int \frac {x \left (3+8 x^5\right ) \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+\int \left (\frac {3 x \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}+\frac {8 x^6 \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+3 \int \frac {x \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx+8 \int \frac {x^6 \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^3+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.74, size = 112, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{x^8+2 x^3-1}}{x}+\log \left (\sqrt [3]{x^8+2 x^3-1}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^8+2 x^3-1}+x}\right )-\frac {1}{2} \log \left (x^2+\sqrt [3]{x^8+2 x^3-1} x+\left (x^8+2 x^3-1\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 17.21, size = 152, normalized size = 1.36 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (\frac {23155756059884469826063290091369873601204942180224 \, \sqrt {3} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 61059012875773331838678659685174425801373874951458 \, \sqrt {3} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (35248398304721470575821713544519821387080907584081 \, x^{8} + 77355782772550371408192688432791971088370316149922 \, x^{3} - 35248398304721470575821713544519821387080907584081\right )}}{3 \, {\left (20044909029062956675424368815298850195325332161233 \, x^{8} + 38996537437007387681732053612201126295409798546850 \, x^{3} - 20044909029062956675424368815298850195325332161233\right )}}\right ) + x \log \left (\frac {x^{8} + x^{3} + 3 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{8} + x^{3} - 1}\right ) + 6 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{2 \, x} \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 (5 \, x^{8} + 3\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} + x^{3} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 6.58, size = 1136, normalized size = 10.14
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{8} + 3\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} + x^{3} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (5\,x^8+3\right )\,{\left (x^8+2\,x^3-1\right )}^{1/3}}{x^2\,\left (x^8+x^3-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (5 x^{8} + 3\right ) \sqrt [3]{x^{8} + 2 x^{3} - 1}}{x^{2} \left (x^{8} + x^{3} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________