3.32.25 \(\int \frac {x (2 x^3 c_3-c_4)}{(-x+x^3 c_3+c_4) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2)} \, dx\)

Optimal. Leaf size=723 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6+\text {$\#$1}^6 c_1{}^2-\text {$\#$1}^6 c_1-2 \text {$\#$1}^3+\text {$\#$1}^3 c_0-2 \text {$\#$1}^3 c_0 c_1+\text {$\#$1}^3 c_1+1+c_0{}^2-c_0\& ,\frac {\text {$\#$1}^3 \left (-c_1{}^2\right ) \log \left (-\text {$\#$1}+\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}}\right )+\text {$\#$1}^3 c_1 \log \left (-\text {$\#$1}+\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}}\right )-\text {$\#$1}^3 \log \left (-\text {$\#$1}+\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}}\right )+c_0 \log \left (-\text {$\#$1}+\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}}\right )+c_0 c_1 \log \left (-\text {$\#$1}+\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}}\right )-2 c_1 \log \left (-\text {$\#$1}+\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}}\right )+\log \left (-\text {$\#$1}+\sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}}\right )}{2 \text {$\#$1}^4+2 \text {$\#$1}^4 c_1{}^2-2 \text {$\#$1}^4 c_1-2 \text {$\#$1}+\text {$\#$1} c_0-2 \text {$\#$1} c_0 c_1+\text {$\#$1} c_1}\& \right ]-\frac {\sqrt [3]{-1-c_1} \log \left (\sqrt [3]{-1-c_1} \sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}}+\sqrt [3]{1+c_0}\right )}{3 \sqrt [3]{1+c_0}}+\frac {\sqrt [3]{-1-c_1} \log \left (-\sqrt [3]{-1-c_1} \sqrt [3]{1+c_0} \sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}}+(-1-c_1){}^{2/3} \left (\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}\right ){}^{2/3}+(1+c_0){}^{2/3}\right )}{6 \sqrt [3]{1+c_0}}-\frac {\sqrt [3]{-1-c_1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1-c_1} \sqrt [3]{\frac {c_3 x^3+c_0 x+c_4}{c_3 x^3+c_1 x+c_4}}}{\sqrt {3} \sqrt [3]{1+c_0}}\right )}{\sqrt {3} \sqrt [3]{1+c_0}} \]

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Rubi [F]  time = 27.78, 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 {x \left (2 x^3 c_3-c_4\right )}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x*(2*x^3*C[3] - C[4]))/((-x + x^3*C[3] + C[4])*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1
/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*C[4] + C[4]^2)),x]

[Out]

((x*C[0] + x^3*C[3] + C[4])^(1/3)*Defer[Int][(x*C[1] + x^3*C[3] + C[4])^(1/3)/((x - x^3*C[3] - C[4])*(x*C[0] +
x^3*C[3] + C[4])^(1/3)), x])/(3*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)*(x*C[1] + x^3*C
[3] + C[4])^(1/3)) + (C[3]*(x*C[0] + x^3*C[3] + C[4])^(1/3)*Defer[Int][(x^2*(x*C[1] + x^3*C[3] + C[4])^(1/3))/
((-x + x^3*C[3] + C[4])*(x*C[0] + x^3*C[3] + C[4])^(1/3)), x])/(((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3]
+ C[4]))^(1/3)*(x*C[1] + x^3*C[3] + C[4])^(1/3)) + (C[4]*(x*C[0] + x^3*C[3] + C[4])^(1/3)*Defer[Int][(x*C[1]
+ x^3*C[3] + C[4])^(1/3)/((x*C[0] + x^3*C[3] + C[4])^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*
C[4] + C[4]^2)), x])/(3*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)*(x*C[1] + x^3*C[3] + C[4
])^(1/3)) - ((x*C[0] + x^3*C[3] + C[4])^(1/3)*Defer[Int][(x*(x*C[1] + x^3*C[3] + C[4])^(1/3))/((x*C[0] + x^3*C
[3] + C[4])^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*C[4] + C[4]^2)), x])/(3*((x*C[0] + x^3*C[
3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)*(x*C[1] + x^3*C[3] + C[4])^(1/3)) - (C[3]*C[4]*(x*C[0] + x^3*C[3]
+ C[4])^(1/3)*Defer[Int][(x^2*(x*C[1] + x^3*C[3] + C[4])^(1/3))/((x*C[0] + x^3*C[3] + C[4])^(1/3)*(x^2 + x^4*
C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*C[4] + C[4]^2)), x])/(((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] +
C[4]))^(1/3)*(x*C[1] + x^3*C[3] + C[4])^(1/3)) - (5*C[3]*(x*C[0] + x^3*C[3] + C[4])^(1/3)*Defer[Int][(x^3*(x*
C[1] + x^3*C[3] + C[4])^(1/3))/((x*C[0] + x^3*C[3] + C[4])^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3
*C[3]*C[4] + C[4]^2)), x])/(3*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)*(x*C[1] + x^3*C[3]
+ C[4])^(1/3)) - (C[3]^2*(x*C[0] + x^3*C[3] + C[4])^(1/3)*Defer[Int][(x^5*(x*C[1] + x^3*C[3] + C[4])^(1/3))/(
(x*C[0] + x^3*C[3] + C[4])^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*C[4] + C[4]^2)), x])/(((x*
C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)*(x*C[1] + x^3*C[3] + C[4])^(1/3))

Rubi steps

\begin {align*} \int \frac {x \left (2 x^3 c_3-c_4\right )}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx &=\frac {\sqrt [3]{x c_0+x^3 c_3+c_4} \int \frac {x \left (2 x^3 c_3-c_4\right ) \sqrt [3]{x c_1+x^3 c_3+c_4}}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx}{\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}\\ &=\frac {\sqrt [3]{x c_0+x^3 c_3+c_4} \int \left (\frac {\left (-1+3 x^2 c_3\right ) \sqrt [3]{x c_1+x^3 c_3+c_4}}{3 \left (-x+x^3 c_3+c_4\right ) \sqrt [3]{x c_0+x^3 c_3+c_4}}+\frac {\sqrt [3]{x c_1+x^3 c_3+c_4} \left (-x-5 x^3 c_3-3 x^5 c_3{}^2+c_4-3 x^2 c_3 c_4\right )}{3 \sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )}\right ) \, dx}{\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}\\ &=\frac {\sqrt [3]{x c_0+x^3 c_3+c_4} \int \frac {\left (-1+3 x^2 c_3\right ) \sqrt [3]{x c_1+x^3 c_3+c_4}}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{x c_0+x^3 c_3+c_4}} \, dx}{3 \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}+\frac {\sqrt [3]{x c_0+x^3 c_3+c_4} \int \frac {\sqrt [3]{x c_1+x^3 c_3+c_4} \left (-x-5 x^3 c_3-3 x^5 c_3{}^2+c_4-3 x^2 c_3 c_4\right )}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx}{3 \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}\\ &=\frac {\sqrt [3]{x c_0+x^3 c_3+c_4} \int \left (\frac {\sqrt [3]{x c_1+x^3 c_3+c_4}}{\left (x-x^3 c_3-c_4\right ) \sqrt [3]{x c_0+x^3 c_3+c_4}}+\frac {3 x^2 c_3 \sqrt [3]{x c_1+x^3 c_3+c_4}}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{x c_0+x^3 c_3+c_4}}\right ) \, dx}{3 \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}+\frac {\sqrt [3]{x c_0+x^3 c_3+c_4} \int \left (-\frac {x \sqrt [3]{x c_1+x^3 c_3+c_4}}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )}-\frac {5 x^3 c_3 \sqrt [3]{x c_1+x^3 c_3+c_4}}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )}-\frac {3 x^5 c_3{}^2 \sqrt [3]{x c_1+x^3 c_3+c_4}}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )}+\frac {c_4 \sqrt [3]{x c_1+x^3 c_3+c_4}}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )}-\frac {3 x^2 c_3 c_4 \sqrt [3]{x c_1+x^3 c_3+c_4}}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )}\right ) \, dx}{3 \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}\\ &=\frac {\sqrt [3]{x c_0+x^3 c_3+c_4} \int \frac {\sqrt [3]{x c_1+x^3 c_3+c_4}}{\left (x-x^3 c_3-c_4\right ) \sqrt [3]{x c_0+x^3 c_3+c_4}} \, dx}{3 \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}-\frac {\sqrt [3]{x c_0+x^3 c_3+c_4} \int \frac {x \sqrt [3]{x c_1+x^3 c_3+c_4}}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx}{3 \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}+\frac {\left (c_3 \sqrt [3]{x c_0+x^3 c_3+c_4}\right ) \int \frac {x^2 \sqrt [3]{x c_1+x^3 c_3+c_4}}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{x c_0+x^3 c_3+c_4}} \, dx}{\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}-\frac {\left (5 c_3 \sqrt [3]{x c_0+x^3 c_3+c_4}\right ) \int \frac {x^3 \sqrt [3]{x c_1+x^3 c_3+c_4}}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx}{3 \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}-\frac {\left (c_3{}^2 \sqrt [3]{x c_0+x^3 c_3+c_4}\right ) \int \frac {x^5 \sqrt [3]{x c_1+x^3 c_3+c_4}}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx}{\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}+\frac {\left (c_4 \sqrt [3]{x c_0+x^3 c_3+c_4}\right ) \int \frac {\sqrt [3]{x c_1+x^3 c_3+c_4}}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx}{3 \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}-\frac {\left (c_3 c_4 \sqrt [3]{x c_0+x^3 c_3+c_4}\right ) \int \frac {x^2 \sqrt [3]{x c_1+x^3 c_3+c_4}}{\sqrt [3]{x c_0+x^3 c_3+c_4} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx}{\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \sqrt [3]{x c_1+x^3 c_3+c_4}}\\ \end {align*}

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Mathematica [F]  time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (2 x^3 c_3-c_4\right )}{\left (-x+x^3 c_3+c_4\right ) \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}} \left (x^2+x^4 c_3+x^6 c_3{}^2+x c_4+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x*(2*x^3*C[3] - C[4]))/((-x + x^3*C[3] + C[4])*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4
]))^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*C[4] + C[4]^2)),x]

[Out]

Integrate[(x*(2*x^3*C[3] - C[4]))/((-x + x^3*C[3] + C[4])*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4
]))^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*C[4] + C[4]^2)), x]

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IntegrateAlgebraic [A]  time = 6.78, size = 723, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1-c_1} \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}}{\sqrt {3} \sqrt [3]{1+c_0}}\right ) \sqrt [3]{-1-c_1}}{\sqrt {3} \sqrt [3]{1+c_0}}-\frac {\sqrt [3]{-1-c_1} \log \left (\sqrt [3]{1+c_0}+\sqrt [3]{-1-c_1} \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}\right )}{3 \sqrt [3]{1+c_0}}+\frac {\sqrt [3]{-1-c_1} \log \left ((1+c_0){}^{2/3}-\sqrt [3]{1+c_0} \sqrt [3]{-1-c_1} \sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}+(-1-c_1){}^{2/3} \left (\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}\right ){}^{2/3}\right )}{6 \sqrt [3]{1+c_0}}+\frac {1}{3} \text {RootSum}\left [1-c_0+c_0{}^2-2 \text {$\#$1}^3+c_0 \text {$\#$1}^3+c_1 \text {$\#$1}^3-2 c_0 c_1 \text {$\#$1}^3+\text {$\#$1}^6-c_1 \text {$\#$1}^6+c_1{}^2 \text {$\#$1}^6\&,\frac {\log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right )+c_0 \log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right )-2 c_1 \log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right )+c_0 c_1 \log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right )-\log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right ) \text {$\#$1}^3+c_1 \log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right ) \text {$\#$1}^3-c_1{}^2 \log \left (\sqrt [3]{\frac {x c_0+x^3 c_3+c_4}{x c_1+x^3 c_3+c_4}}-\text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}+c_0 \text {$\#$1}+c_1 \text {$\#$1}-2 c_0 c_1 \text {$\#$1}+2 \text {$\#$1}^4-2 c_1 \text {$\#$1}^4+2 c_1{}^2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x*(2*x^3*C[3] - C[4]))/((-x + x^3*C[3] + C[4])*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C
[3] + C[4]))^(1/3)*(x^2 + x^4*C[3] + x^6*C[3]^2 + x*C[4] + 2*x^3*C[3]*C[4] + C[4]^2)),x]

[Out]

-((ArcTan[1/Sqrt[3] - (2*(-1 - C[1])^(1/3)*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3))/(Sqr
t[3]*(1 + C[0])^(1/3))]*(-1 - C[1])^(1/3))/(Sqrt[3]*(1 + C[0])^(1/3))) - ((-1 - C[1])^(1/3)*Log[(1 + C[0])^(1/
3) + (-1 - C[1])^(1/3)*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3)])/(3*(1 + C[0])^(1/3)) +
((-1 - C[1])^(1/3)*Log[(1 + C[0])^(2/3) - (1 + C[0])^(1/3)*(-1 - C[1])^(1/3)*((x*C[0] + x^3*C[3] + C[4])/(x*C[
1] + x^3*C[3] + C[4]))^(1/3) + (-1 - C[1])^(2/3)*((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(2/3)
])/(6*(1 + C[0])^(1/3)) + RootSum[1 - C[0] + C[0]^2 - 2*#1^3 + C[0]*#1^3 + C[1]*#1^3 - 2*C[0]*C[1]*#1^3 + #1^6
- C[1]*#1^6 + C[1]^2*#1^6 & , (Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1] + C[0]
*Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1] - 2*C[1]*Log[((x*C[0] + x^3*C[3] + C[
4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1] + C[0]*C[1]*Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C
[4]))^(1/3) - #1] - Log[((x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1]*#1^3 + C[1]*Log[((
x*C[0] + x^3*C[3] + C[4])/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1]*#1^3 - C[1]^2*Log[((x*C[0] + x^3*C[3] + C[4]
)/(x*C[1] + x^3*C[3] + C[4]))^(1/3) - #1]*#1^3)/(-2*#1 + C[0]*#1 + C[1]*#1 - 2*C[0]*C[1]*#1 + 2*#1^4 - 2*C[1]*
#1^4 + 2*C[1]^2*#1^4) & ]/3

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*_C3*x^3-_C4)/(_C3*x^3+_C4-x)/((_C3*x^3+_C0*x+_C4)/(_C3*x^3+_C1*x+_C4))^(1/3)/(_C3^2*x^6+2*_C3*_
C4*x^3+_C3*x^4+_C4^2+_C4*x+x^2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, \_{C_{3}} x^{3} - \_{C_{4}}\right )} x}{{\left (\_{C_{3}}^{2} x^{6} + 2 \, \_{C_{3}} \_{C_{4}} x^{3} + \_{C_{3}} x^{4} + \_{C_{4}}^{2} + \_{C_{4}} x + x^{2}\right )} {\left (\_{C_{3}} x^{3} + \_{C_{4}} - x\right )} \left (\frac {\_{C_{3}} x^{3} + \_{C_{0}} x + \_{C_{4}}}{\_{C_{3}} x^{3} + \_{C_{1}} x + \_{C_{4}}}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*_C3*x^3-_C4)/(_C3*x^3+_C4-x)/((_C3*x^3+_C0*x+_C4)/(_C3*x^3+_C1*x+_C4))^(1/3)/(_C3^2*x^6+2*_C3*_
C4*x^3+_C3*x^4+_C4^2+_C4*x+x^2),x, algorithm="giac")

[Out]

integrate((2*_C3*x^3 - _C4)*x/((_C3^2*x^6 + 2*_C3*_C4*x^3 + _C3*x^4 + _C4^2 + _C4*x + x^2)*(_C3*x^3 + _C4 - x)
*((_C3*x^3 + _C0*x + _C4)/(_C3*x^3 + _C1*x + _C4))^(1/3)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x \left (2 \textit {\_C3} \,x^{3}-\textit {\_C4} \right )}{\left (\textit {\_C3} \,x^{3}+\textit {\_C4} -x \right ) \left (\frac {\textit {\_C3} \,x^{3}+\textit {\_C0} x +\textit {\_C4}}{\textit {\_C3} \,x^{3}+\textit {\_C1} x +\textit {\_C4}}\right )^{\frac {1}{3}} \left (\textit {\_C3}^{2} x^{6}+2 \textit {\_C3} \textit {\_C4} \,x^{3}+\textit {\_C3} \,x^{4}+\textit {\_C4}^{2}+\textit {\_C4} x +x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*_C3*x^3-_C4)/(_C3*x^3+_C4-x)/((_C3*x^3+_C0*x+_C4)/(_C3*x^3+_C1*x+_C4))^(1/3)/(_C3^2*x^6+2*_C3*_C4*x^3
+_C3*x^4+_C4^2+_C4*x+x^2),x)

[Out]

int(x*(2*_C3*x^3-_C4)/(_C3*x^3+_C4-x)/((_C3*x^3+_C0*x+_C4)/(_C3*x^3+_C1*x+_C4))^(1/3)/(_C3^2*x^6+2*_C3*_C4*x^3
+_C3*x^4+_C4^2+_C4*x+x^2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, \_{C_{3}} x^{3} - \_{C_{4}}\right )} x}{{\left (\_{C_{3}}^{2} x^{6} + 2 \, \_{C_{3}} \_{C_{4}} x^{3} + \_{C_{3}} x^{4} + \_{C_{4}}^{2} + \_{C_{4}} x + x^{2}\right )} {\left (\_{C_{3}} x^{3} + \_{C_{4}} - x\right )} \left (\frac {\_{C_{3}} x^{3} + \_{C_{0}} x + \_{C_{4}}}{\_{C_{3}} x^{3} + \_{C_{1}} x + \_{C_{4}}}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*_C3*x^3-_C4)/(_C3*x^3+_C4-x)/((_C3*x^3+_C0*x+_C4)/(_C3*x^3+_C1*x+_C4))^(1/3)/(_C3^2*x^6+2*_C3*_
C4*x^3+_C3*x^4+_C4^2+_C4*x+x^2),x, algorithm="maxima")

[Out]

integrate((2*_C3*x^3 - _C4)*x/((_C3^2*x^6 + 2*_C3*_C4*x^3 + _C3*x^4 + _C4^2 + _C4*x + x^2)*(_C3*x^3 + _C4 - x)
*((_C3*x^3 + _C0*x + _C4)/(_C3*x^3 + _C1*x + _C4))^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {x\,\left (_{\mathrm {C4}}-2\,_{\mathrm {C3}}\,x^3\right )}{{\left (\frac {_{\mathrm {C3}}\,x^3+_{\mathrm {C0}}\,x+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^3+_{\mathrm {C1}}\,x+_{\mathrm {C4}}}\right )}^{1/3}\,\left (_{\mathrm {C3}}\,x^3-x+_{\mathrm {C4}}\right )\,\left ({_{\mathrm {C3}}}^2\,x^6+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^3+_{\mathrm {C3}}\,x^4+{_{\mathrm {C4}}}^2+_{\mathrm {C4}}\,x+x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*(_C4 - 2*_C3*x^3))/(((_C4 + _C0*x + _C3*x^3)/(_C4 + _C1*x + _C3*x^3))^(1/3)*(_C4 - x + _C3*x^3)*(_C4*x
+ _C3*x^4 + _C4^2 + x^2 + _C3^2*x^6 + 2*_C3*_C4*x^3)),x)

[Out]

int(-(x*(_C4 - 2*_C3*x^3))/(((_C4 + _C0*x + _C3*x^3)/(_C4 + _C1*x + _C3*x^3))^(1/3)*(_C4 - x + _C3*x^3)*(_C4*x
+ _C3*x^4 + _C4^2 + x^2 + _C3^2*x^6 + 2*_C3*_C4*x^3)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(2*_C3*x**3-_C4)/(_C3*x**3+_C4-x)/((_C3*x**3+_C0*x+_C4)/(_C3*x**3+_C1*x+_C4))**(1/3)/(_C3**2*x**6+
2*_C3*_C4*x**3+_C3*x**4+_C4**2+_C4*x+x**2),x)

[Out]

Timed out

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