3.20.17 \(\int \frac {\sqrt [3]{1+x^5} (-3+2 x^5)}{x^2 (2-x^3+2 x^5)} \, dx\)

Optimal. Leaf size=133 \[ \frac {3 \sqrt [3]{x^5+1}}{2 x}+\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1}-x\right )}{2 \sqrt [3]{2}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^5+1}+x}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1} x+2^{2/3} \left (x^5+1\right )^{2/3}+x^2\right )}{4 \sqrt [3]{2}} \]

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Rubi [F]  time = 0.85, 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+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - x^3 + 2*x^5)),x]

[Out]

(3*Hypergeometric2F1[-1/3, -1/5, 4/5, -x^5])/(2*x) - (3*Defer[Int][(x*(1 + x^5)^(1/3))/(2 - x^3 + 2*x^5), x])/
2 + 5*Defer[Int][(x^3*(1 + x^5)^(1/3))/(2 - x^3 + 2*x^5), x]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx &=\int \left (-\frac {3 \sqrt [3]{1+x^5}}{2 x^2}+\frac {x \left (-3+10 x^2\right ) \sqrt [3]{1+x^5}}{2 \left (2-x^3+2 x^5\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x \left (-3+10 x^2\right ) \sqrt [3]{1+x^5}}{2-x^3+2 x^5} \, dx-\frac {3}{2} \int \frac {\sqrt [3]{1+x^5}}{x^2} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{5};\frac {4}{5};-x^5\right )}{2 x}+\frac {1}{2} \int \left (-\frac {3 x \sqrt [3]{1+x^5}}{2-x^3+2 x^5}+\frac {10 x^3 \sqrt [3]{1+x^5}}{2-x^3+2 x^5}\right ) \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{5};\frac {4}{5};-x^5\right )}{2 x}-\frac {3}{2} \int \frac {x \sqrt [3]{1+x^5}}{2-x^3+2 x^5} \, dx+5 \int \frac {x^3 \sqrt [3]{1+x^5}}{2-x^3+2 x^5} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - x^3 + 2*x^5)),x]

[Out]

Integrate[((1 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - x^3 + 2*x^5)), x]

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IntegrateAlgebraic [A]  time = 2.46, size = 133, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{1+x^5}}{2 x}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^5}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^5}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^5}+2^{2/3} \left (1+x^5\right )^{2/3}\right )}{4 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - x^3 + 2*x^5)),x]

[Out]

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

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fricas [B]  time = 101.44, size = 384, normalized size = 2.89 \begin {gather*} \frac {2 \, \sqrt {3} 2^{\frac {2}{3}} x \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (24 \, \sqrt {2} {\left (2 \, x^{11} + x^{9} - x^{7} + 4 \, x^{6} + x^{4} + 2 \, x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 2^{\frac {5}{6}} {\left (8 \, x^{15} + 60 \, x^{13} + 24 \, x^{11} + 24 \, x^{10} - x^{9} + 120 \, x^{8} + 24 \, x^{6} + 24 \, x^{5} + 60 \, x^{3} + 8\right )} + 12 \cdot 2^{\frac {1}{6}} {\left (4 \, x^{12} + 14 \, x^{10} + x^{8} + 8 \, x^{7} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{15} - 12 \, x^{13} - 48 \, x^{11} + 24 \, x^{10} - x^{9} - 24 \, x^{8} - 48 \, x^{6} + 24 \, x^{5} - 12 \, x^{3} + 8\right )}}\right ) + 2 \cdot 2^{\frac {2}{3}} x \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}} x + 2^{\frac {1}{3}} {\left (2 \, x^{5} - x^{3} + 2\right )}}{2 \, x^{5} - x^{3} + 2}\right ) - 2^{\frac {2}{3}} x \log \left (\frac {12 \cdot 2^{\frac {1}{3}} {\left (x^{6} + x^{4} + x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (4 \, x^{10} + 14 \, x^{8} + x^{6} + 8 \, x^{5} + 14 \, x^{3} + 4\right )} + 6 \, {\left (4 \, x^{7} + x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{4 \, x^{10} - 4 \, x^{8} + x^{6} + 8 \, x^{5} - 4 \, x^{3} + 4}\right ) + 36 \, {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{24 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)^(1/3)*(2*x^5-3)/x^2/(2*x^5-x^3+2),x, algorithm="fricas")

[Out]

1/24*(2*sqrt(3)*2^(2/3)*x*arctan(1/6*sqrt(3)*2^(1/6)*(24*sqrt(2)*(2*x^11 + x^9 - x^7 + 4*x^6 + x^4 + 2*x)*(x^5
+ 1)^(2/3) + 2^(5/6)*(8*x^15 + 60*x^13 + 24*x^11 + 24*x^10 - x^9 + 120*x^8 + 24*x^6 + 24*x^5 + 60*x^3 + 8) +
12*2^(1/6)*(4*x^12 + 14*x^10 + x^8 + 8*x^7 + 14*x^5 + 4*x^2)*(x^5 + 1)^(1/3))/(8*x^15 - 12*x^13 - 48*x^11 + 24
*x^10 - x^9 - 24*x^8 - 48*x^6 + 24*x^5 - 12*x^3 + 8)) + 2*2^(2/3)*x*log((3*2^(2/3)*(x^5 + 1)^(1/3)*x^2 - 6*(x^
5 + 1)^(2/3)*x + 2^(1/3)*(2*x^5 - x^3 + 2))/(2*x^5 - x^3 + 2)) - 2^(2/3)*x*log((12*2^(1/3)*(x^6 + x^4 + x)*(x^
5 + 1)^(2/3) + 2^(2/3)*(4*x^10 + 14*x^8 + x^6 + 8*x^5 + 14*x^3 + 4) + 6*(4*x^7 + x^5 + 4*x^2)*(x^5 + 1)^(1/3))
/(4*x^10 - 4*x^8 + x^6 + 8*x^5 - 4*x^3 + 4)) + 36*(x^5 + 1)^(1/3))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{5} - x^{3} + 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)^(1/3)*(2*x^5-3)/x^2/(2*x^5-x^3+2),x, algorithm="giac")

[Out]

integrate((2*x^5 - 3)*(x^5 + 1)^(1/3)/((2*x^5 - x^3 + 2)*x^2), x)

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maple [C]  time = 83.51, size = 1015, normalized size = 7.63

method	result	size
trager	\(\frac {3 \left (x^{5}+1\right )^{\frac {1}{3}}}{2 x}+\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {76247796 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{4} x^{5}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{5}-152495592 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{4} x^{3}-1752775326 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}-203327456 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{5}-2337033768 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{5}[phone] \left (x^{5}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x [phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{4}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{3}-127079660 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}-1460646105 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}[phone] \left (x^{5}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}[phone] \left (x^{5}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{2}-1142792958 \left (x^{5}+1\right )^{\frac {2}{3}} x -203327456 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-2337033768 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )}{2 x^{5}-x^{3}+2}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{3}-4\right ) \ln \left (-\frac {-76247796 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{4} x^{5}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{5}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{4} x^{3}-837801774 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{3} x^{3}-254159320 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{5}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{5}[phone] \left (x^{5}+1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x -76247796 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{4}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{3}-25415932 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} x^{3}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}-571396479 \left (x^{5}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{2}-4420524204 \left (x^{5}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) x^{2}[phone] \left (x^{5}+1\right )^{\frac {2}{3}} x -254159320 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}[phone] \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )}{2 x^{5}-x^{3}+2}\right )}{4}\)	\(1015\)
risch	\(\frac {3 \left (x^{5}+1\right )^{\frac {1}{3}}}{2 x}+\frac {\left (\frac {\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 {\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^{8}-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^{8}+4 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{10}-8 \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^{10}-6 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{10}+2 x^{5}+1\right )^{\frac {1}{3}} x^{6}-12 \left (x^{10}+2 x^{5}+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^{6}+2 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{8}-4 \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^{8}+6 \left (x^{10}+2 x^{5}+1\right )^{\frac {2}{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^{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}+8 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{5}-16 \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^{5}-6 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{10}+2 x^{5}+1\right )^{\frac {1}{3}} x -12 \left (x^{10}+2 x^{5}+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 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{3}-4 \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}+4 \RootOf \left (\textit {\_Z}^{3}-4\right )-8 \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 )}{\left (2 x^{5}-x^{3}+2\right ) \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right )}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{3}-4\right ) \ln \left (\frac {-\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^{8}+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^{8}+4 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{10}-8 \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^{10}-6 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{10}+2 x^{5}+1\right )^{\frac {1}{3}} x^{6}+6 \left (x^{10}+2 x^{5}+1\right )^{\frac {2}{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^{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}+8 \RootOf \left (\textit {\_Z}^{3}-4\right ) x^{5}-16 \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^{5}+12 \left (x^{10}+2 x^{5}+1\right )^{\frac {2}{3}} x^{2}-6 \RootOf \left (\textit {\_Z}^{3}-4\right )^{2} \left (x^{10}+2 x^{5}+1\right )^{\frac {1}{3}} x +4 \RootOf \left (\textit {\_Z}^{3}-4\right )-8 \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 )}{\left (2 x^{5}-x^{3}+2\right ) \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right )}\right )}{4}\right ) \left (\left (x^{5}+1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{5}+1\right )^{\frac {2}{3}}}\)	\(1060\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+1)^(1/3)*(2*x^5-3)/x^2/(2*x^5-x^3+2),x,method=_RETURNVERBOSE)

[Out]

3/2*(x^5+1)^(1/3)/x+3/2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*ln((76247796*RootOf(RootOf(_Z^3-4
)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^4*x^5+876387663*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36
*_Z^2)^2*RootOf(_Z^3-4)^3*x^5-152495592*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^4*
x^3-1752775326*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^3*x^3-203327456*RootOf(_Z
^3-4)^2*x^5-2337033768*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^5+496072665*(x^5+
1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x+76247796*RootOf(RootOf(_Z^3-4
)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^4+876387663*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^
2)^2*RootOf(_Z^3-4)^3-127079660*RootOf(_Z^3-4)^2*x^3-1460646105*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36
*_Z^2)*RootOf(_Z^3-4)*x^3+736754034*(x^5+1)^(1/3)*RootOf(_Z^3-4)*x^2+3428378874*(x^5+1)^(1/3)*RootOf(RootOf(_Z
^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^2-1142792958*(x^5+1)^(2/3)*x-203327456*RootOf(_Z^3-4)^2-2337033768*Root
Of(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4))/(2*x^5-x^3+2))+1/4*RootOf(_Z^3-4)*ln(-(-76247
796*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^4*x^5+418900887*RootOf(RootOf(_Z^3-4)^
2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^3*x^5+152495592*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36
*_Z^2)*RootOf(_Z^3-4)^4*x^3-837801774*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^3*
x^3-254159320*RootOf(_Z^3-4)^2*x^5+1396336290*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3
-4)*x^5+496072665*(x^5+1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x-762477
96*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^4+418900887*RootOf(RootOf(_Z^3-4)^2+6*_
Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^3-25415932*RootOf(_Z^3-4)^2*x^3+139633629*RootOf(RootOf(_Z^3-4)^2+6
*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^3-571396479*(x^5+1)^(1/3)*RootOf(_Z^3-4)*x^2-4420524204*(x^5+1)^(
1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^2+1473508068*(x^5+1)^(2/3)*x-254159320*RootOf(_Z^3
-4)^2+1396336290*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4))/(2*x^5-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^{5} - 3\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{5} - x^{3} + 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)^(1/3)*(2*x^5-3)/x^2/(2*x^5-x^3+2),x, algorithm="maxima")

[Out]

integrate((2*x^5 - 3)*(x^5 + 1)^(1/3)/((2*x^5 - x^3 + 2)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^5+1\right )}^{1/3}\,\left (2\,x^5-3\right )}{x^2\,\left (2\,x^5-x^3+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5 + 1)^(1/3)*(2*x^5 - 3))/(x^2*(2*x^5 - x^3 + 2)),x)

[Out]

int(((x^5 + 1)^(1/3)*(2*x^5 - 3))/(x^2*(2*x^5 - x^3 + 2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (2 x^{5} - 3\right )}{x^{2} \left (2 x^{5} - x^{3} + 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5+1)**(1/3)*(2*x**5-3)/x**2/(2*x**5-x**3+2),x)

[Out]

Integral(((x + 1)*(x**4 - x**3 + x**2 - x + 1))**(1/3)*(2*x**5 - 3)/(x**2*(2*x**5 - x**3 + 2)), x)

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