3.22.22 \(\int \frac {(-3+x^4) (1-x^3+x^4)^{2/3}}{x^3 (1+x^3+x^4)} \, dx\)

Optimal. Leaf size=154 \[ 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^4-x^3+1}+2 x\right )+2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4-x^3+1}-x}\right )+\frac {3 \left (x^4-x^3+1\right )^{2/3}}{2 x^2}-\frac {\log \left (-2 x^2+2^{2/3} \sqrt [3]{x^4-x^3+1} x-\sqrt [3]{2} \left (x^4-x^3+1\right )^{2/3}\right )}{\sqrt [3]{2}} \]

________________________________________________________________________________________

Rubi [F]  time = 0.82, 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 {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx &=\int \left (-\frac {3 \left (1-x^3+x^4\right )^{2/3}}{x^3}+\frac {(3+4 x) \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \frac {(3+4 x) \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx\\ &=-\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \left (\frac {3 \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}+\frac {4 x \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx+4 \int \frac {x \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.80, size = 154, normalized size = 1.00 \begin {gather*} \frac {3 \left (1-x^3+x^4\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1-x^3+x^4}}\right )+2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3+x^4}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3+x^4}-\sqrt [3]{2} \left (1-x^3+x^4\right )^{2/3}\right )}{\sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 - x^3 + x^4)^(2/3))/(2*x^2) + 2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 - x^3 + x^4)^(1/3))] +
2^(2/3)*Log[2*x + 2^(2/3)*(1 - x^3 + x^4)^(1/3)] - Log[-2*x^2 + 2^(2/3)*x*(1 - x^3 + x^4)^(1/3) - 2^(1/3)*(1
- x^3 + x^4)^(2/3)]/2^(1/3)

________________________________________________________________________________________

fricas [B]  time = 74.09, size = 413, normalized size = 2.68 \begin {gather*} \frac {2 \cdot 4^{\frac {1}{3}} \sqrt {3} x^{2} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{9} - 4 \, x^{8} - 5 \, x^{7} + 2 \, x^{5} - 4 \, x^{4} + x\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} - 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (x^{10} - 16 \, x^{9} + 19 \, x^{8} + 2 \, x^{6} - 16 \, x^{5} + x^{2}\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (x^{12} - 33 \, x^{11} + 111 \, x^{10} - 71 \, x^{9} + 3 \, x^{8} - 66 \, x^{7} + 111 \, x^{6} + 3 \, x^{4} - 33 \, x^{3} + 1\right )}}{3 \, {\left (x^{12} + 3 \, x^{11} - 105 \, x^{10} + 109 \, x^{9} + 3 \, x^{8} + 6 \, x^{7} - 105 \, x^{6} + 3 \, x^{4} + 3 \, x^{3} + 1\right )}}\right ) + 2 \cdot 4^{\frac {1}{3}} x^{2} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 6 \, {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + 4^{\frac {1}{3}} {\left (x^{4} + x^{3} + 1\right )}}{x^{4} + x^{3} + 1}\right ) - 4^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{5} - 5 \, x^{4} + x\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {2}{3}} {\left (x^{8} - 16 \, x^{7} + 19 \, x^{6} + 2 \, x^{4} - 16 \, x^{3} + 1\right )} - 24 \, {\left (x^{6} - 2 \, x^{5} + x^{2}\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}}}{x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} + 1}\right ) + 9 \, {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3)*(x^4-x^3+1)^(2/3)/x^3/(x^4+x^3+1),x, algorithm="fricas")

[Out]

1/6*(2*4^(1/3)*sqrt(3)*x^2*arctan(1/3*(3*4^(2/3)*sqrt(3)*(x^9 - 4*x^8 - 5*x^7 + 2*x^5 - 4*x^4 + x)*(x^4 - x^3
+ 1)^(2/3) - 6*4^(1/3)*sqrt(3)*(x^10 - 16*x^9 + 19*x^8 + 2*x^6 - 16*x^5 + x^2)*(x^4 - x^3 + 1)^(1/3) - sqrt(3)
*(x^12 - 33*x^11 + 111*x^10 - 71*x^9 + 3*x^8 - 66*x^7 + 111*x^6 + 3*x^4 - 33*x^3 + 1))/(x^12 + 3*x^11 - 105*x^
10 + 109*x^9 + 3*x^8 + 6*x^7 - 105*x^6 + 3*x^4 + 3*x^3 + 1)) + 2*4^(1/3)*x^2*log(-(3*4^(2/3)*(x^4 - x^3 + 1)^(
1/3)*x^2 + 6*(x^4 - x^3 + 1)^(2/3)*x + 4^(1/3)*(x^4 + x^3 + 1))/(x^4 + x^3 + 1)) - 4^(1/3)*x^2*log(-(6*4^(1/3)
*(x^5 - 5*x^4 + x)*(x^4 - x^3 + 1)^(2/3) - 4^(2/3)*(x^8 - 16*x^7 + 19*x^6 + 2*x^4 - 16*x^3 + 1) - 24*(x^6 - 2*
x^5 + x^2)*(x^4 - x^3 + 1)^(1/3))/(x^8 + 2*x^7 + x^6 + 2*x^4 + 2*x^3 + 1)) + 9*(x^4 - x^3 + 1)^(2/3))/x^2

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3)*(x^4-x^3+1)^(2/3)/x^3/(x^4+x^3+1),x, algorithm="giac")

[Out]

integrate((x^4 - x^3 + 1)^(2/3)*(x^4 - 3)/((x^4 + x^3 + 1)*x^3), x)

________________________________________________________________________________________

maple [C]  time = 68.68, size = 1089, normalized size = 7.07

method	result	size
risch	\(\frac {3 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}-\ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+4 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x +\RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-8 \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{4}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}-4\right )-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )}{x^{4}+x^{3}+1}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+4 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -5 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-8 \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{4}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{4}+9 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}+6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{3}+10 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{3}-4\right )-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )}{x^{4}+x^{3}+1}\right )-2 \ln \left (\frac {2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}+4 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x +\RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-8 \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{4}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}-4\right )-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )}{x^{4}+x^{3}+1}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \textit {\_Z}^{2}\right )\)	\(1089\)
trager	\(\text {Expression too large to display}\)	\(1509\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-3)*(x^4-x^3+1)^(2/3)/x^3/(x^4+x^3+1),x,method=_RETURNVERBOSE)

[Out]

3/2*(x^4-x^3+1)^(2/3)/x^2-ln((2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^3-2*Roo
tOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^3+4*(x^4-x^3+1)^(2/3)*RootOf(RootOf(_Z^3
-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2*x+RootOf(_Z^3-4)^2*(x^4-x^3+1)^(1/3)*x^2-8*(x^4-x^3+1)^(1/3
)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x^2+2*RootOf(_Z^3-4)*x^4-2*RootOf(RootOf(
_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^4-2*RootOf(_Z^3-4)*x^3+2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4
*_Z^2)*x^3-2*(x^4-x^3+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))/(x^4+
x^3+1))*RootOf(_Z^3-4)+2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*ln(-(3*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+4*(x^4-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2*x-5*Roo
tOf(_Z^3-4)^2*(x^4-x^3+1)^(1/3)*x^2-8*(x^4-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*Ro
otOf(_Z^3-4)*x^2-3*RootOf(_Z^3-4)*x^4-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^4+9*RootOf(_Z^3-
4)*x^3+6*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^3+10*(x^4-x^3+1)^(2/3)*x-3*RootOf(_Z^3-4)-2*Roo
tOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/(x^4+x^3+1))-2*ln((2*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
+4*(x^4-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^2*x+RootOf(_Z^3-4)^2*(
x^4-x^3+1)^(1/3)*x^2-8*(x^4-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x^
2+2*RootOf(_Z^3-4)*x^4-2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^4-2*RootOf(_Z^3-4)*x^3+2*RootOf
(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^3-2*(x^4-x^3+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))/(x^4+x^3+1))*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3)*(x^4-x^3+1)^(2/3)/x^3/(x^4+x^3+1),x, algorithm="maxima")

[Out]

integrate((x^4 - x^3 + 1)^(2/3)*(x^4 - 3)/((x^4 + x^3 + 1)*x^3), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^4-3\right )\,{\left (x^4-x^3+1\right )}^{2/3}}{x^3\,\left (x^4+x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 - 3)*(x^4 - x^3 + 1)^(2/3))/(x^3*(x^3 + x^4 + 1)),x)

[Out]

int(((x^4 - 3)*(x^4 - x^3 + 1)^(2/3))/(x^3*(x^3 + x^4 + 1)), x)

________________________________________________________________________________________

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**4-3)*(x**4-x**3+1)**(2/3)/x**3/(x**4+x**3+1),x)

[Out]

Timed out

________________________________________________________________________________________