3.23.38 \(\int \frac {(-1+x^4)^{2/3} (3+x^4) (-2-x^3+2 x^4)}{x^6 (-2+3 x^3+2 x^4)} \, dx\)

Optimal. Leaf size=167 \[ -\sqrt [3]{2} 3^{2/3} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^4-1}+3 x\right )+3 \sqrt [3]{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x-2 \sqrt [3]{2} \sqrt [3]{x^4-1}}\right )+\left (\frac {3}{2}\right )^{2/3} \log \left (-\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^4-1} x+2^{2/3} \sqrt [3]{3} \left (x^4-1\right )^{2/3}+3 x^2\right )+\frac {3 \left (x^4-1\right )^{2/3} \left (x^4-5 x^3-1\right )}{5 x^5} \]

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Rubi [F]  time = 1.49, 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 (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-3*(-1 + x^4)^(2/3))/x^2 + (12*x^2)/(1 + Sqrt[3] + (-1 + x^4)^(1/3)) - (6*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + (-1
+ x^4)^(1/3))*Sqrt[(1 - (-1 + x^4)^(1/3) + (-1 + x^4)^(2/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))^2]*EllipticE[Arc
Sin[(1 - Sqrt[3] + (-1 + x^4)^(1/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))], -7 - 4*Sqrt[3]])/(x^2*Sqrt[(1 + (-1 +
x^4)^(1/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))^2]) + (4*Sqrt[2]*3^(3/4)*(1 + (-1 + x^4)^(1/3))*Sqrt[(1 - (-1 + x
^4)^(1/3) + (-1 + x^4)^(2/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] + (-1 + x^4)^(
1/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))], -7 - 4*Sqrt[3]])/(x^2*Sqrt[(1 + (-1 + x^4)^(1/3))/(1 + Sqrt[3] + (-1
+ x^4)^(1/3))^2]) - (3*(-1 + x^4)^(2/3)*Hypergeometric2F1[-5/4, -2/3, -1/4, x^4])/(5*x^5*(1 - x^4)^(2/3)) - ((
-1 + x^4)^(2/3)*Hypergeometric2F1[-2/3, -1/4, 3/4, x^4])/(x*(1 - x^4)^(2/3)) - 18*Defer[Int][(-1 + x^4)^(2/3)/
(-2 + 3*x^3 + 2*x^4), x] - 16*Defer[Int][(x*(-1 + x^4)^(2/3))/(-2 + 3*x^3 + 2*x^4), x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx &=\int \left (\frac {3 \left (-1+x^4\right )^{2/3}}{x^6}+\frac {6 \left (-1+x^4\right )^{2/3}}{x^3}+\frac {\left (-1+x^4\right )^{2/3}}{x^2}-\frac {2 (9+8 x) \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4}\right ) \, dx\\ &=-\left (2 \int \frac {(9+8 x) \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx\right )+3 \int \frac {\left (-1+x^4\right )^{2/3}}{x^6} \, dx+6 \int \frac {\left (-1+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-1+x^4\right )^{2/3}}{x^2} \, dx\\ &=-\left (2 \int \left (\frac {9 \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4}+\frac {8 x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4}\right ) \, dx\right )+3 \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )+\frac {\left (-1+x^4\right )^{2/3} \int \frac {\left (1-x^4\right )^{2/3}}{x^2} \, dx}{\left (1-x^4\right )^{2/3}}+\frac {\left (3 \left (-1+x^4\right )^{2/3}\right ) \int \frac {\left (1-x^4\right )^{2/3}}{x^6} \, dx}{\left (1-x^4\right )^{2/3}}\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}+4 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx,x,x^2\right )-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx+\frac {\left (6 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx+\frac {\left (6 \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}+\frac {\left (6 \sqrt {2 \left (2-\sqrt {3}\right )} \sqrt {x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^4}\right )}{x^2}\\ &=-\frac {3 \left (-1+x^4\right )^{2/3}}{x^2}+\frac {12 x^2}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}-\frac {6 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right )|-7-4 \sqrt {3}\right )}{x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}+\frac {4 \sqrt {2} 3^{3/4} \left (1+\sqrt [3]{-1+x^4}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^4}}{1+\sqrt {3}+\sqrt [3]{-1+x^4}}\right )|-7-4 \sqrt {3}\right )}{x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}-\frac {3 \left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};x^4\right )}{5 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};x^4\right )}{x \left (1-x^4\right )^{2/3}}-16 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx-18 \int \frac {\left (-1+x^4\right )^{2/3}}{-2+3 x^3+2 x^4} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^6 \left (-2+3 x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.59, size = 167, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+x^4\right )^{2/3} \left (-1-5 x^3+x^4\right )}{5 x^5}+3 \sqrt [3]{2} \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x-2 \sqrt [3]{2} \sqrt [3]{-1+x^4}}\right )-\sqrt [3]{2} 3^{2/3} \log \left (3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^4}\right )+\left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2-\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^4}+2^{2/3} \sqrt [3]{3} \left (-1+x^4\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-1 + x^4)^(2/3)*(-1 - 5*x^3 + x^4))/(5*x^5) + 3*2^(1/3)*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x - 2*2^(1/3)*
(-1 + x^4)^(1/3))] - 2^(1/3)*3^(2/3)*Log[3*x + 2^(1/3)*3^(2/3)*(-1 + x^4)^(1/3)] + (3/2)^(2/3)*Log[3*x^2 - 2^(
1/3)*3^(2/3)*x*(-1 + x^4)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + x^4)^(2/3)]

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fricas [B]  time = 142.17, size = 418, normalized size = 2.50 \begin {gather*} \frac {10 \, \sqrt {3} \left (-18\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {4 \, \sqrt {3} \left (-18\right )^{\frac {2}{3}} {\left (2 \, x^{9} - 3 \, x^{8} - 9 \, x^{7} - 4 \, x^{5} + 3 \, x^{4} + 2 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-18\right )^{\frac {1}{3}} {\left (4 \, x^{10} - 42 \, x^{9} + 9 \, x^{8} - 8 \, x^{6} + 42 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (8 \, x^{12} - 180 \, x^{11} + 216 \, x^{10} + 27 \, x^{9} - 24 \, x^{8} + 360 \, x^{7} - 216 \, x^{6} + 24 \, x^{4} - 180 \, x^{3} - 8\right )}}{3 \, {\left (8 \, x^{12} + 36 \, x^{11} - 432 \, x^{10} + 27 \, x^{9} - 24 \, x^{8} - 72 \, x^{7} + 432 \, x^{6} + 24 \, x^{4} + 36 \, x^{3} - 8\right )}}\right ) + 10 \, \left (-18\right )^{\frac {1}{3}} x^{5} \log \left (\frac {3 \, \left (-18\right )^{\frac {2}{3}} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} + 18 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - \left (-18\right )^{\frac {1}{3}} {\left (2 \, x^{4} + 3 \, x^{3} - 2\right )}}{2 \, x^{4} + 3 \, x^{3} - 2}\right ) - 5 \, \left (-18\right )^{\frac {1}{3}} x^{5} \log \left (\frac {36 \, \left (-18\right )^{\frac {1}{3}} {\left (x^{5} - 3 \, x^{4} - x\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}} + \left (-18\right )^{\frac {2}{3}} {\left (4 \, x^{8} - 42 \, x^{7} + 9 \, x^{6} - 8 \, x^{4} + 42 \, x^{3} + 4\right )} + 54 \, {\left (4 \, x^{6} - 3 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{4 \, x^{8} + 12 \, x^{7} + 9 \, x^{6} - 8 \, x^{4} - 12 \, x^{3} + 4}\right ) + 18 \, {\left (x^{4} - 5 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)^(2/3)*(x^4+3)*(2*x^4-x^3-2)/x^6/(2*x^4+3*x^3-2),x, algorithm="fricas")

[Out]

1/30*(10*sqrt(3)*(-18)^(1/3)*x^5*arctan(1/3*(4*sqrt(3)*(-18)^(2/3)*(2*x^9 - 3*x^8 - 9*x^7 - 4*x^5 + 3*x^4 + 2*
x)*(x^4 - 1)^(2/3) + 6*sqrt(3)*(-18)^(1/3)*(4*x^10 - 42*x^9 + 9*x^8 - 8*x^6 + 42*x^5 + 4*x^2)*(x^4 - 1)^(1/3)
- sqrt(3)*(8*x^12 - 180*x^11 + 216*x^10 + 27*x^9 - 24*x^8 + 360*x^7 - 216*x^6 + 24*x^4 - 180*x^3 - 8))/(8*x^12
+ 36*x^11 - 432*x^10 + 27*x^9 - 24*x^8 - 72*x^7 + 432*x^6 + 24*x^4 + 36*x^3 - 8)) + 10*(-18)^(1/3)*x^5*log((3
*(-18)^(2/3)*(x^4 - 1)^(1/3)*x^2 + 18*(x^4 - 1)^(2/3)*x - (-18)^(1/3)*(2*x^4 + 3*x^3 - 2))/(2*x^4 + 3*x^3 - 2)
) - 5*(-18)^(1/3)*x^5*log((36*(-18)^(1/3)*(x^5 - 3*x^4 - x)*(x^4 - 1)^(2/3) + (-18)^(2/3)*(4*x^8 - 42*x^7 + 9*
x^6 - 8*x^4 + 42*x^3 + 4) + 54*(4*x^6 - 3*x^5 - 4*x^2)*(x^4 - 1)^(1/3))/(4*x^8 + 12*x^7 + 9*x^6 - 8*x^4 - 12*x
^3 + 4)) + 18*(x^4 - 5*x^3 - 1)*(x^4 - 1)^(2/3))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)^(2/3)*(x^4+3)*(2*x^4-x^3-2)/x^6/(2*x^4+3*x^3-2),x, algorithm="giac")

[Out]

integrate((2*x^4 - x^3 - 2)*(x^4 + 3)*(x^4 - 1)^(2/3)/((2*x^4 + 3*x^3 - 2)*x^6), x)

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maple [C]  time = 123.73, size = 808, normalized size = 4.84

method	result	size
risch	\(\frac {\frac {3}{5} x^{8}-3 x^{7}-\frac {6}{5} x^{4}+3 x^{3}+\frac {3}{5}}{x^{5} \left (x^{4}-1\right )^{\frac {1}{3}}}-\ln \left (-\frac {-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+2 \left (x^{4}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x -\left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}-12 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{2}-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) x^{4}+18 x^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )-12 \left (x^{4}-1\right )^{\frac {2}{3}} x +12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )}{2 x^{4}+3 x^{3}-2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )-6 \ln \left (-\frac {-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+2 \left (x^{4}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x -\left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}-12 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{2}-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) x^{4}+18 x^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )-12 \left (x^{4}-1\right )^{\frac {2}{3}} x +12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )}{2 x^{4}+3 x^{3}-2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )+\RootOf \left (\textit {\_Z}^{3}+18\right ) \ln \left (\frac {-3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+\left (x^{4}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x +\left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}+3 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{2}+6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right ) x^{4}+3 \left (x^{4}-1\right )^{\frac {2}{3}} x -6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+36 \textit {\_Z}^{2}\right )}{2 x^{4}+3 x^{3}-2}\right )\)	\(808\)
trager	\(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}} \left (x^{4}-5 x^{3}-1\right )}{5 x^{5}}+576 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) \ln \left (-\frac {86999040 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{4}-96896 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{4}-163123200 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}+181680 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}+96192 \left (x^{4}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x +3712608 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{2}-501 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}+679680 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) x^{4}-757 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{4}-86999040 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+96896 \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right )+4078080 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) x^{3}-4542 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}-41679 \left (x^{4}-1\right )^{\frac {2}{3}} x -679680 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right )+757 \RootOf \left (\textit {\_Z}^{3}+18\right )}{2 x^{4}+3 x^{3}-2}\right )+\RootOf \left (\textit {\_Z}^{3}+18\right ) \ln \left (\frac {-55812096 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{4}+151040 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{4}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{3}-283200 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} x^{3}+96192 \left (x^{4}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x -4001184 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{2}-501 \left (x^{4}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2} x^{2}+2180160 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) x^{4}-5900 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{4}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+18\right )^{2}-151040 \RootOf \left (\textit {\_Z}^{3}+18\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right )-654048 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right ) x^{3}+1770 \RootOf \left (\textit {\_Z}^{3}+18\right ) x^{3}+38673 \left (x^{4}-1\right )^{\frac {2}{3}} x -2180160 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+18\right )^{2}+576 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+18\right )+331776 \textit {\_Z}^{2}\right )+5900 \RootOf \left (\textit {\_Z}^{3}+18\right )}{2 x^{4}+3 x^{3}-2}\right )\)	\(992\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)^(2/3)*(x^4+3)*(2*x^4-x^3-2)/x^6/(2*x^4+3*x^3-2),x,method=_RETURNVERBOSE)

[Out]

3/5*(x^8-5*x^7-2*x^4+5*x^3+1)/x^5/(x^4-1)^(1/3)-ln(-(-6*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)
^2*RootOf(_Z^3+18)^2*x^3+2*(x^4-1)^(2/3)*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)*RootOf(_Z^3+18
)^2*x-(x^4-1)^(1/3)*RootOf(_Z^3+18)^2*x^2-12*(x^4-1)^(1/3)*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z
^2)*RootOf(_Z^3+18)*x^2-12*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)*x^4+18*x^3*RootOf(RootOf(_Z^
3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)-12*(x^4-1)^(2/3)*x+12*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_
Z^2))/(2*x^4+3*x^3-2))*RootOf(_Z^3+18)-6*ln(-(-6*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)^2*Root
Of(_Z^3+18)^2*x^3+2*(x^4-1)^(2/3)*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)*RootOf(_Z^3+18)^2*x-(
x^4-1)^(1/3)*RootOf(_Z^3+18)^2*x^2-12*(x^4-1)^(1/3)*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)*Roo
tOf(_Z^3+18)*x^2-12*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)*x^4+18*x^3*RootOf(RootOf(_Z^3+18)^2
+6*_Z*RootOf(_Z^3+18)+36*_Z^2)-12*(x^4-1)^(2/3)*x+12*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2))/(
2*x^4+3*x^3-2))*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)+RootOf(_Z^3+18)*ln((-3*RootOf(RootOf(_Z
^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)^2*RootOf(_Z^3+18)^2*x^3+(x^4-1)^(2/3)*RootOf(RootOf(_Z^3+18)^2+6*_Z*Roo
tOf(_Z^3+18)+36*_Z^2)*RootOf(_Z^3+18)^2*x+(x^4-1)^(1/3)*RootOf(_Z^3+18)^2*x^2+3*(x^4-1)^(1/3)*RootOf(RootOf(_Z
^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2)*RootOf(_Z^3+18)*x^2+6*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*
_Z^2)*x^4+3*(x^4-1)^(2/3)*x-6*RootOf(RootOf(_Z^3+18)^2+6*_Z*RootOf(_Z^3+18)+36*_Z^2))/(2*x^4+3*x^3-2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)^(2/3)*(x^4+3)*(2*x^4-x^3-2)/x^6/(2*x^4+3*x^3-2),x, algorithm="maxima")

[Out]

integrate((2*x^4 - x^3 - 2)*(x^4 + 3)*(x^4 - 1)^(2/3)/((2*x^4 + 3*x^3 - 2)*x^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (x^4-1\right )}^{2/3}\,\left (x^4+3\right )\,\left (-2\,x^4+x^3+2\right )}{x^6\,\left (2\,x^4+3\,x^3-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^4 - 1)^(2/3)*(x^4 + 3)*(x^3 - 2*x^4 + 2))/(x^6*(3*x^3 + 2*x^4 - 2)),x)

[Out]

int(-((x^4 - 1)^(2/3)*(x^4 + 3)*(x^3 - 2*x^4 + 2))/(x^6*(3*x^3 + 2*x^4 - 2)), 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**4-1)**(2/3)*(x**4+3)*(2*x**4-x**3-2)/x**6/(2*x**4+3*x**3-2),x)

[Out]

Timed out

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