3.27.22 \(\int \frac {x^3 (3+x^2)}{(1+x^2) \sqrt [3]{1+x^2-x^3} (1+x^2+x^3)} \, dx\)

Optimal. Leaf size=230 \[ \log \left (\sqrt [3]{-x^3+x^2+1}+x\right )-\frac {\log \left (2^{2/3} \sqrt [3]{-x^3+x^2+1}+2 x\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x^2-\sqrt [3]{-x^3+x^2+1} x+\left (-x^3+x^2+1\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} \sqrt [3]{-x^3+x^2+1} x-\sqrt [3]{2} \left (-x^3+x^2+1\right )^{2/3}\right )}{2 \sqrt [3]{2}}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{-x^3+x^2+1}-x}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{-x^3+x^2+1}-x}\right )}{\sqrt [3]{2}} \]

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Rubi [F]  time = 6.35, 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^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^3*(3 + x^2))/((1 + x^2)*(1 + x^2 - x^3)^(1/3)*(1 + x^2 + x^3)),x]

[Out]

-1/4*((2 + 2^(1/3)*(2/(29 + 3*Sqrt[93])^(1/3) + (58 + 6*Sqrt[93])^(1/3)) - 6*x)^(1/3)*(2 + (2*2^(1/3) + (58 +
6*Sqrt[93])^(2/3))/(29 + 3*Sqrt[93])^(1/3) - 6*x)^(2/3)*(1 - (2^(2/3)*(2*2^(1/3) + 2*(29 + 3*Sqrt[93])^(1/3) +
(58 + 6*Sqrt[93])^(2/3) - 6*(29 + 3*Sqrt[93])^(1/3)*x))/(6 + 3*2^(1/3)*(29 + 3*Sqrt[93])^(2/3) + 2^(1/6)*(29
+ 3*Sqrt[93])^(1/3)*Sqrt[3*(4 - 2*(2/(29 + 3*Sqrt[93]))^(2/3) - 2^(1/3)*(29 + 3*Sqrt[93])^(2/3))]))^(1/3)*(1 -
(2^(2/3)*(2*2^(1/3) + 2*(29 + 3*Sqrt[93])^(1/3) + (58 + 6*Sqrt[93])^(2/3) - 6*(29 + 3*Sqrt[93])^(1/3)*x))/(6
+ 3*2^(1/3)*(29 + 3*Sqrt[93])^(2/3) - I*2^(1/6)*(29 + 3*Sqrt[93])^(1/3)*Sqrt[3*(-4 + 2*(2/(29 + 3*Sqrt[93]))^(
2/3) + 2^(1/3)*(29 + 3*Sqrt[93])^(2/3))]))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, ((29 + 3*Sqrt[93])^(1/3)*(2*(2/(
29 + 3*Sqrt[93])^(1/3) + (58 + 6*Sqrt[93])^(1/3)) + 2*2^(2/3)*(1 - 3*x)))/(6 + 3*2^(1/3)*(29 + 3*Sqrt[93])^(2/
3) + 2^(1/6)*(29 + 3*Sqrt[93])^(1/3)*Sqrt[3*(4 - 2*(2/(29 + 3*Sqrt[93]))^(2/3) - 2^(1/3)*(29 + 3*Sqrt[93])^(2/
3))]), ((29 + 3*Sqrt[93])^(1/3)*(2*(2/(29 + 3*Sqrt[93])^(1/3) + (58 + 6*Sqrt[93])^(1/3)) + 2*2^(2/3)*(1 - 3*x)
))/(6 + 3*2^(1/3)*(29 + 3*Sqrt[93])^(2/3) - I*2^(1/6)*(29 + 3*Sqrt[93])^(1/3)*Sqrt[3*(-4 + 2*(2/(29 + 3*Sqrt[9
3]))^(2/3) + 2^(1/3)*(29 + 3*Sqrt[93])^(2/3))])])/(1 + x^2 - x^3)^(1/3) - 3*Defer[Int][1/((1 + x^2 - x^3)^(1/3
)*(1 + x^2 + x^3)), x] - Defer[Int][x^2/((1 + x^2 - x^3)^(1/3)*(1 + x^2 + x^3)), x] + ((I/3)*(-2 + 2*(2/(29 +
3*Sqrt[93]))^(2/3) + 2^(1/3)*(29 + 3*Sqrt[93])^(2/3) - 2*((2/(29 + 3*Sqrt[93]))^(1/3) + ((29 + 3*Sqrt[93])/2)^
(1/3))*(1 - 3*x) + 2*(1 - 3*x)^2)^(1/3)*(2 + 2^(1/3)*(2/(29 + 3*Sqrt[93])^(1/3) + (58 + 6*Sqrt[93])^(1/3)) - 6
*x)^(1/3)*Defer[Subst][Defer[Int][1/(((-1/3 + I) - x)*((2/(29 + 3*Sqrt[93])^(1/3) + (58 + 6*Sqrt[93])^(1/3))/(
3*2^(2/3)) - x)^(1/3)*((-2 + 2*(2/(29 + 3*Sqrt[93]))^(2/3) + 2^(1/3)*(29 + 3*Sqrt[93])^(2/3))/18 + (((2/(29 +
3*Sqrt[93]))^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))*x)/3 + x^2)^(1/3)), x], x, -1/3 + x])/(2^(2/3)*(1 + x^2 - x^
3)^(1/3)) + ((I/3)*(-2 + 2*(2/(29 + 3*Sqrt[93]))^(2/3) + 2^(1/3)*(29 + 3*Sqrt[93])^(2/3) - 2*((2/(29 + 3*Sqrt[
93]))^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))*(1 - 3*x) + 2*(1 - 3*x)^2)^(1/3)*(2 + 2^(1/3)*(2/(29 + 3*Sqrt[93])^
(1/3) + (58 + 6*Sqrt[93])^(1/3)) - 6*x)^(1/3)*Defer[Subst][Defer[Int][1/(((2/(29 + 3*Sqrt[93])^(1/3) + (58 + 6
*Sqrt[93])^(1/3))/(3*2^(2/3)) - x)^(1/3)*((1/3 + I) + x)*((-2 + 2*(2/(29 + 3*Sqrt[93]))^(2/3) + 2^(1/3)*(29 +
3*Sqrt[93])^(2/3))/18 + (((2/(29 + 3*Sqrt[93]))^(1/3) + ((29 + 3*Sqrt[93])/2)^(1/3))*x)/3 + x^2)^(1/3)), x], x
, -1/3 + x])/(2^(2/3)*(1 + x^2 - x^3)^(1/3))

Rubi steps

\begin {align*} \int \frac {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx &=\int \left (\frac {1}{\sqrt [3]{1+x^2-x^3}}+\frac {2}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3}}+\frac {-3-x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )}\right ) \, dx\\ &=2 \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3}} \, dx+\int \frac {1}{\sqrt [3]{1+x^2-x^3}} \, dx+\int \frac {-3-x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx\\ &=2 \int \left (\frac {i}{2 (i-x) \sqrt [3]{1+x^2-x^3}}+\frac {i}{2 (i+x) \sqrt [3]{1+x^2-x^3}}\right ) \, dx+\int \left (-\frac {3}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )}-\frac {x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {29}{27}+\frac {x}{3}-x^3}} \, dx,x,-\frac {1}{3}+x\right )\\ &=i \int \frac {1}{(i-x) \sqrt [3]{1+x^2-x^3}} \, dx+i \int \frac {1}{(i+x) \sqrt [3]{1+x^2-x^3}} \, dx-3 \int \frac {1}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx+\frac {\left (\sqrt [3]{2+2 \sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}-6 x} \sqrt [3]{-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+18 \left (-\frac {1}{3}+x\right )^2+2 \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) (-1+3 x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}}{3\ 2^{2/3}}-x} \sqrt [3]{\frac {1}{18} \left (-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )+\frac {1}{3} \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) x+x^2}} \, dx,x,-\frac {1}{3}+x\right )}{3\ 2^{2/3} \sqrt [3]{1+x^2-x^3}}-\int \frac {x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx\\ &=i \operatorname {Subst}\left (\int \frac {1}{\left (\left (-\frac {1}{3}+i\right )-x\right ) \sqrt [3]{\frac {29}{27}+\frac {x}{3}-x^3}} \, dx,x,-\frac {1}{3}+x\right )+i \operatorname {Subst}\left (\int \frac {1}{\left (\left (\frac {1}{3}+i\right )+x\right ) \sqrt [3]{\frac {29}{27}+\frac {x}{3}-x^3}} \, dx,x,-\frac {1}{3}+x\right )-3 \int \frac {1}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx-\frac {\left (\sqrt [3]{2+2 \sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}-6 x} \sqrt [3]{1+\frac {-2 \left (2+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )+6\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+\sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (4-2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}} \sqrt [3]{1+\frac {-2 \left (2+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )+6\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-i \sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (-4+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt [3]{1-\frac {6\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+\sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (4-2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}} \sqrt [3]{1-\frac {6\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-i \sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (-4+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}}} \, dx,x,\frac {\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}+2^{2/3} (1-3 x)}{3\ 2^{2/3}}\right )}{\sqrt [3]{6} \sqrt [3]{1+x^2-x^3}}-\int \frac {x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx\\ &=-\frac {\left (2+2 \sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}-6 x\right ) \sqrt [3]{1-\frac {2 \left (2+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+\sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (4-2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}} \sqrt [3]{1-\frac {2 \left (2+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-i \sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (-4+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {2 \sqrt [3]{29+3 \sqrt {93}} \left (\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}+2^{2/3} (1-3 x)\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+\sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (4-2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}},\frac {2 \sqrt [3]{29+3 \sqrt {93}} \left (\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}+2^{2/3} (1-3 x)\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-i \sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (-4+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}\right )}{4 \sqrt [3]{1+x^2-x^3}}-3 \int \frac {1}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx+\frac {\left (i \sqrt [3]{2+2 \sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}-6 x} \sqrt [3]{-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+18 \left (-\frac {1}{3}+x\right )^2+2 \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) (-1+3 x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (-\frac {1}{3}+i\right )-x\right ) \sqrt [3]{\frac {\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}}{3\ 2^{2/3}}-x} \sqrt [3]{\frac {1}{18} \left (-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )+\frac {1}{3} \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) x+x^2}} \, dx,x,-\frac {1}{3}+x\right )}{3\ 2^{2/3} \sqrt [3]{1+x^2-x^3}}+\frac {\left (i \sqrt [3]{2+2 \sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}-6 x} \sqrt [3]{-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+18 \left (-\frac {1}{3}+x\right )^2+2 \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) (-1+3 x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}}{3\ 2^{2/3}}-x} \left (\left (\frac {1}{3}+i\right )+x\right ) \sqrt [3]{\frac {1}{18} \left (-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )+\frac {1}{3} \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) x+x^2}} \, dx,x,-\frac {1}{3}+x\right )}{3\ 2^{2/3} \sqrt [3]{1+x^2-x^3}}-\int \frac {x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^3*(3 + x^2))/((1 + x^2)*(1 + x^2 - x^3)^(1/3)*(1 + x^2 + x^3)),x]

[Out]

Integrate[(x^3*(3 + x^2))/((1 + x^2)*(1 + x^2 - x^3)^(1/3)*(1 + x^2 + x^3)), x]

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IntegrateAlgebraic [A]  time = 0.80, size = 230, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^2-x^3}}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^2-x^3}}\right )}{\sqrt [3]{2}}+\log \left (x+\sqrt [3]{1+x^2-x^3}\right )-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{1+x^2-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^2-x^3}+\left (1+x^2-x^3\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^2-x^3}-\sqrt [3]{2} \left (1+x^2-x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^3*(3 + x^2))/((1 + x^2)*(1 + x^2 - x^3)^(1/3)*(1 + x^2 + x^3)),x]

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^2 - x^3)^(1/3))] - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 + x^
2 - x^3)^(1/3))])/2^(1/3) + Log[x + (1 + x^2 - x^3)^(1/3)] - Log[2*x + 2^(2/3)*(1 + x^2 - x^3)^(1/3)]/2^(1/3)
- Log[x^2 - x*(1 + x^2 - x^3)^(1/3) + (1 + x^2 - x^3)^(2/3)]/2 + Log[-2*x^2 + 2^(2/3)*x*(1 + x^2 - x^3)^(1/3)
- 2^(1/3)*(1 + x^2 - x^3)^(2/3)]/(2*2^(1/3))

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fricas [C]  time = 2.91, size = 471, normalized size = 2.05 \begin {gather*} \frac {1}{4} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} \log \left (-\frac {x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{3} - 6 \cdot 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} + 8 \, x - 24 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}}{8 \, x}\right ) - \frac {1}{8} \, {\left (2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} - 2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}}\right )} \log \left (-\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} + 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} - 8 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}\right )}}{8 \, x}\right ) - \frac {1}{8} \, {\left (2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} + 2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}}\right )} \log \left (\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} - 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} + 8 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}\right )}}{8 \, x}\right ) - \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \log \left (\frac {x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{3} + 32 \, x + 24 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}}{8 \, x}\right ) - \frac {1}{2} \, \log \left (\frac {x^{2} - {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + x^{2} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^2+3)/(x^2+1)/(-x^3+x^2+1)^(1/3)/(x^3+x^2+1),x, algorithm="fricas")

[Out]

1/4*2^(2/3)*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))*log(-1/8*(x*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^3 - 6*2^(1/3)*
x*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^2 + 8*x - 24*(-x^3 + x^2 + 1)^(1/3))/x) - 1/8*(2^(2/3)*(I*sqrt(3)*(-1)^(
1/3) - (-1)^(1/3)) - 2*sqrt(3/2)*sqrt(-2^(1/3)*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^2))*log(-3/8*(2^(2/3)*sqrt(
3/2)*sqrt(-2^(1/3)*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^2)*x*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3)) + 2^(1/3)*x*(I
*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^2 - 8*(-x^3 + x^2 + 1)^(1/3))/x) - 1/8*(2^(2/3)*(I*sqrt(3)*(-1)^(1/3) - (-1)
^(1/3)) + 2*sqrt(3/2)*sqrt(-2^(1/3)*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^2))*log(3/8*(2^(2/3)*sqrt(3/2)*sqrt(-2
^(1/3)*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^2)*x*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3)) - 2^(1/3)*x*(I*sqrt(3)*(-1
)^(1/3) - (-1)^(1/3))^2 + 8*(-x^3 + x^2 + 1)^(1/3))/x) - sqrt(3)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^3 + x^
2 + 1)^(1/3))/x) + log(1/8*(x*(I*sqrt(3)*(-1)^(1/3) - (-1)^(1/3))^3 + 32*x + 24*(-x^3 + x^2 + 1)^(1/3))/x) - 1
/2*log((x^2 - (-x^3 + x^2 + 1)^(1/3)*x + (-x^3 + x^2 + 1)^(2/3))/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^2+3)/(x^2+1)/(-x^3+x^2+1)^(1/3)/(x^3+x^2+1),x, algorithm="giac")

[Out]

integrate((x^2 + 3)*x^3/((x^3 + x^2 + 1)*(-x^3 + x^2 + 1)^(1/3)*(x^2 + 1)), x)

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maple [C]  time = 11.04, size = 1845, normalized size = 8.02

method	result	size
trager	\(\text {Expression too large to display}\)	\(1845\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^2+3)/(x^2+1)/(-x^3+x^2+1)^(1/3)/(x^3+x^2+1),x,method=_RETURNVERBOSE)

[Out]

ln((-RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)^2*RootOf(_Z^3+4)^4*x^3-6*RootOf(RootOf(_Z^3+4)^2+2*_Z
*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^2*(-x^3+x^2+1)^(1/3)*x^2-4*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+
4*_Z^2)*RootOf(_Z^3+4)^2*x^3+2*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^2+12*(-x
^3+x^2+1)^(2/3)*x+12*(-x^3+x^2+1)^(1/3)*x^2+2*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_
Z^2)-4*x^3+4*x^2+4)/(x^2+1))+1/2*RootOf(_Z^3+4)*ln((RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf
(_Z^3+4)^3*x^3-2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+3*RootOf(_Z^3+4)^2
*(-x^3+x^2+1)^(1/3)*x^2+6*(-x^3+x^2+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4
)*x^2+RootOf(_Z^3+4)*x^3-2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^3+6*(-x^3+x^2+1)^(2/3)*x-Root
Of(_Z^3+4)*x^2+2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^2-RootOf(_Z^3+4)+2*RootOf(RootOf(_Z^3+4
)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2))/(x^3+x^2+1))-1/2*ln(-(RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*Roo
tOf(_Z^3+4)^3*x^3-4*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+4*RootOf(_Z^3+4
)^2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*(-x^3+x^2+1)^(2/3)*x-RootOf(_Z^3+4)^2*(-x^3+x^2+1)^(1/
3)*x^2-10*(-x^3+x^2+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)*x^2-3*RootOf(_
Z^3+4)*x^3+12*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^3-10*(-x^3+x^2+1)^(2/3)*x+RootOf(_Z^3+4)*x
^2-4*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^2+RootOf(_Z^3+4)-4*RootOf(RootOf(_Z^3+4)^2+2*_Z*Roo
tOf(_Z^3+4)+4*_Z^2))/(x^3+x^2+1))*RootOf(_Z^3+4)-ln(-(RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*Root
Of(_Z^3+4)^3*x^3-4*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)^2*RootOf(_Z^3+4)^2*x^3+4*RootOf(_Z^3+4)
^2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*(-x^3+x^2+1)^(2/3)*x-RootOf(_Z^3+4)^2*(-x^3+x^2+1)^(1/3
)*x^2-10*(-x^3+x^2+1)^(1/3)*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)*x^2-3*RootOf(_Z
^3+4)*x^3+12*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^3-10*(-x^3+x^2+1)^(2/3)*x+RootOf(_Z^3+4)*x^
2-4*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^2+RootOf(_Z^3+4)-4*RootOf(RootOf(_Z^3+4)^2+2*_Z*Root
Of(_Z^3+4)+4*_Z^2))/(x^3+x^2+1))*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)-ln(-(-RootOf(RootOf(_Z^3+
4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)^2*RootOf(_Z^3+4)^4*x^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf
(_Z^3+4)+4*_Z^2)*(-x^3+x^2+1)^(2/3)*x+6*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^2*(
-x^3+x^2+1)^(1/3)*x^2+2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^2*x^3-2*RootOf(_Z^3
+4)^2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^2-12*(-x^3+x^2+1)^(2/3)*x-2*RootOf(_Z^3+4)^2*RootO
f(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)+8*x^3-4*x^2-4)/(x^2+1))+1/2*ln(-(-RootOf(RootOf(_Z^3+4)^2+2*_Z*
RootOf(_Z^3+4)+4*_Z^2)^2*RootOf(_Z^3+4)^4*x^3+6*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4
*_Z^2)*(-x^3+x^2+1)^(2/3)*x+6*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^2*(-x^3+x^2+1
)^(1/3)*x^2+2*RootOf(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*RootOf(_Z^3+4)^2*x^3-2*RootOf(_Z^3+4)^2*Root
Of(RootOf(_Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)*x^2-12*(-x^3+x^2+1)^(2/3)*x-2*RootOf(_Z^3+4)^2*RootOf(RootOf(_
Z^3+4)^2+2*_Z*RootOf(_Z^3+4)+4*_Z^2)+8*x^3-4*x^2-4)/(x^2+1))*RootOf(_Z^3+4)^2*RootOf(RootOf(_Z^3+4)^2+2*_Z*Roo
tOf(_Z^3+4)+4*_Z^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^2+3)/(x^2+1)/(-x^3+x^2+1)^(1/3)/(x^3+x^2+1),x, algorithm="maxima")

[Out]

integrate((x^2 + 3)*x^3/((x^3 + x^2 + 1)*(-x^3 + x^2 + 1)^(1/3)*(x^2 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(x^2 + 3))/((x^2 + 1)*(x^2 + x^3 + 1)*(x^2 - x^3 + 1)^(1/3)),x)

[Out]

int((x^3*(x^2 + 3))/((x^2 + 1)*(x^2 + x^3 + 1)*(x^2 - x^3 + 1)^(1/3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (x^{2} + 3\right )}{\left (x^{2} + 1\right ) \sqrt [3]{- x^{3} + x^{2} + 1} \left (x^{3} + x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(x**2+3)/(x**2+1)/(-x**3+x**2+1)**(1/3)/(x**3+x**2+1),x)

[Out]

Integral(x**3*(x**2 + 3)/((x**2 + 1)*(-x**3 + x**2 + 1)**(1/3)*(x**3 + x**2 + 1)), x)

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