3.11.25 \(\int \frac {2+3 x^5}{(-1+x^2+x^5) \sqrt [3]{-x+x^6}} \, dx\)

Optimal. Leaf size=85 \[ -\log \left (\sqrt [3]{x^6-x}+x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-x}-x}\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^6-x} x+\left (x^6-x\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 1.19, 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 {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(9*x*(1 - x^5)^(1/3)*Hypergeometric2F1[2/15, 1/3, 17/15, x^5])/(2*(-x + x^6)^(1/3)) + (15*x^(1/3)*(-1 + x^5)^(
1/3)*Defer[Subst][Defer[Int][x/((-1 + x^15)^(1/3)*(-1 + x^6 + x^15)), x], x, x^(1/3)])/(-x + x^6)^(1/3) - (9*x
^(1/3)*(-1 + x^5)^(1/3)*Defer[Subst][Defer[Int][x^7/((-1 + x^15)^(1/3)*(-1 + x^6 + x^15)), x], x, x^(1/3)])/(-
x + x^6)^(1/3)

Rubi steps

\begin {align*} \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \int \frac {2+3 x^5}{\sqrt [3]{x} \sqrt [3]{-1+x^5} \left (-1+x^2+x^5\right )} \, dx}{\sqrt [3]{-x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x \left (2+3 x^{15}\right )}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {3 x}{\sqrt [3]{-1+x^{15}}}+\frac {x \left (5-3 x^6\right )}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x \left (5-3 x^6\right )}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}+\frac {\left (9 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^{15}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ &=\frac {\left (9 \sqrt [3]{x} \sqrt [3]{1-x^5}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^{15}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {5 x}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )}-\frac {3 x^7}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ &=\frac {9 x \sqrt [3]{1-x^5} \, _2F_1\left (\frac {2}{15},\frac {1}{3};\frac {17}{15};x^5\right )}{2 \sqrt [3]{-x+x^6}}-\frac {\left (9 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}+\frac {\left (15 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^{15}} \left (-1+x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+3 x^5}{\left (-1+x^2+x^5\right ) \sqrt [3]{-x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(2 + 3*x^5)/((-1 + x^2 + x^5)*(-x + x^6)^(1/3)),x]

[Out]

Integrate[(2 + 3*x^5)/((-1 + x^2 + x^5)*(-x + x^6)^(1/3)), x]

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IntegrateAlgebraic [A]  time = 2.69, size = 85, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{x^6-x}+x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6-x}-x}\right )+\frac {1}{2} \log \left (-\sqrt [3]{x^6-x} x+\left (x^6-x\right )^{2/3}+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + 3*x^5)/((-1 + x^2 + x^5)*(-x + x^6)^(1/3)),x]

[Out]

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

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fricas [A]  time = 2.07, size = 104, normalized size = 1.22 \begin {gather*} -\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{6} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} - 1\right )} + 2 \, \sqrt {3} {\left (x^{6} - x\right )}^{\frac {2}{3}}}{x^{5} - 8 \, x^{2} - 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{5} + x^{2} + 3 \, {\left (x^{6} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{6} - x\right )}^{\frac {2}{3}} - 1}{x^{5} + x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5+2)/(x^5+x^2-1)/(x^6-x)^(1/3),x, algorithm="fricas")

[Out]

-sqrt(3)*arctan((4*sqrt(3)*(x^6 - x)^(1/3)*x + sqrt(3)*(x^5 - 1) + 2*sqrt(3)*(x^6 - x)^(2/3))/(x^5 - 8*x^2 - 1
)) - 1/2*log((x^5 + x^2 + 3*(x^6 - x)^(1/3)*x + 3*(x^6 - x)^(2/3) - 1)/(x^5 + x^2 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} + 2}{{\left (x^{6} - x\right )}^{\frac {1}{3}} {\left (x^{5} + x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5+2)/(x^5+x^2-1)/(x^6-x)^(1/3),x, algorithm="giac")

[Out]

integrate((3*x^5 + 2)/((x^6 - x)^(1/3)*(x^5 + x^2 - 1)), x)

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maple [C]  time = 7.78, size = 360, normalized size = 4.24 \begin {gather*} -\ln \left (-\frac {8327084306326444968 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}-10176976877377096586 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-421273930349059602349 x^{5}-64534903374029948502 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-433300799797487350553 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {2}{3}}-861051921881822746252 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {1}{3}} x -365066111281356098815 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-8327084306326444968 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-427751122084335395699 \left (x^{6}-x \right )^{\frac {2}{3}}+433300799797487350553 x \left (x^{6}-x \right )^{\frac {1}{3}}+366916003852406750433 x^{2}+10176976877377096586 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+421273930349059602349}{x^{5}+x^{2}-1}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {54357926496652851916 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}+375243088158733195401 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}+56207819067703503534 x^{5}-421273930349059602349 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+433300799797487350553 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {2}{3}}-427751122084335395699 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-x \right )^{\frac {1}{3}} x -10176976877377096586 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-54357926496652851916 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-861051921881822746252 \left (x^{6}-x \right )^{\frac {2}{3}}-433300799797487350553 x \left (x^{6}-x \right )^{\frac {1}{3}}+8327084306326444968 x^{2}-375243088158733195401 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-56207819067703503534}{x^{5}+x^{2}-1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^5+2)/(x^5+x^2-1)/(x^6-x)^(1/3),x)

[Out]

-ln(-(8327084306326444968*RootOf(_Z^2-_Z+1)^2*x^5-10176976877377096586*RootOf(_Z^2-_Z+1)*x^5-42127393034905960
2349*x^5-64534903374029948502*RootOf(_Z^2-_Z+1)^2*x^2-433300799797487350553*RootOf(_Z^2-_Z+1)*(x^6-x)^(2/3)-86
1051921881822746252*RootOf(_Z^2-_Z+1)*(x^6-x)^(1/3)*x-365066111281356098815*RootOf(_Z^2-_Z+1)*x^2-832708430632
6444968*RootOf(_Z^2-_Z+1)^2-427751122084335395699*(x^6-x)^(2/3)+433300799797487350553*x*(x^6-x)^(1/3)[phone]
3852406750433*x^2+10176976877377096586*RootOf(_Z^2-_Z+1)+421273930349059602349)/(x^5+x^2-1))+RootOf(_Z^2-_Z+1)
*ln(-(54357926496652851916*RootOf(_Z^2-_Z+1)^2*x^5+375243088158733195401*RootOf(_Z^2-_Z+1)*x^5+562078190677035
03534*x^5-421273930349059602349*RootOf(_Z^2-_Z+1)^2*x^2+433300799797487350553*RootOf(_Z^2-_Z+1)*(x^6-x)^(2/3)-
427751122084335395699*RootOf(_Z^2-_Z+1)*(x^6-x)^(1/3)*x-10176976877377096586*RootOf(_Z^2-_Z+1)*x^2-54357926496
652851916*RootOf(_Z^2-_Z+1)^2-861051921881822746252*(x^6-x)^(2/3)-433300799797487350553*x*(x^6-x)^(1/3)+832708
4306326444968*x^2-375243088158733195401*RootOf(_Z^2-_Z+1)-56207819067703503534)/(x^5+x^2-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} + 2}{{\left (x^{6} - x\right )}^{\frac {1}{3}} {\left (x^{5} + x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^5+2)/(x^5+x^2-1)/(x^6-x)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*x^5 + 2)/((x^6 - x)^(1/3)*(x^5 + x^2 - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,x^5+2}{{\left (x^6-x\right )}^{1/3}\,\left (x^5+x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^5 + 2)/((x^6 - x)^(1/3)*(x^2 + x^5 - 1)),x)

[Out]

int((3*x^5 + 2)/((x^6 - x)^(1/3)*(x^2 + x^5 - 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**5+2)/(x**5+x**2-1)/(x**6-x)**(1/3),x)

[Out]

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

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