3.13.88 \(\int \frac {1+2 x^3}{(-1+x+x^3) \sqrt [3]{-x^2+x^5}} \, dx\) [1288]

Optimal. Leaf size=93 \[ -\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x^2+x^5}}\right )-\log \left (x+\sqrt [3]{-x^2+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x^2+x^5}+\left (-x^2+x^5\right )^{2/3}\right ) \]

[Out]

-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^5-x^2)^(1/3)))-ln(x+(x^5-x^2)^(1/3))+1/2*ln(x^2-x*(x^5-x^2)^(1/3)+(x^5-x^2)
^(2/3))

________________________________________________________________________________________

Rubi [F]
time = 0.74, 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 {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(6*x*(1 - x^3)^(1/3)*Hypergeometric2F1[1/9, 1/3, 10/9, x^3])/(-x^2 + x^5)^(1/3) + (9*x^(2/3)*(-1 + x^3)^(1/3)*
Defer[Subst][Defer[Int][1/((-1 + x^9)^(1/3)*(-1 + x^3 + x^9)), x], x, x^(1/3)])/(-x^2 + x^5)^(1/3) - (6*x^(2/3
)*(-1 + x^3)^(1/3)*Defer[Subst][Defer[Int][x^3/((-1 + x^9)^(1/3)*(-1 + x^3 + x^9)), x], x, x^(1/3)])/(-x^2 + x
^5)^(1/3)

Rubi steps

\begin {align*} \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^3}\right ) \int \frac {1+2 x^3}{x^{2/3} \sqrt [3]{-1+x^3} \left (-1+x+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1+2 x^9}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \left (\frac {2}{\sqrt [3]{-1+x^9}}+\frac {3-2 x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \frac {3-2 x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}+\frac {\left (6 x^{2/3} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^9}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (6 x^{2/3} \sqrt [3]{1-x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^9}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}+\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \left (\frac {3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )}-\frac {2 x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {6 x \sqrt [3]{1-x^3} \, _2F_1\left (\frac {1}{9},\frac {1}{3};\frac {10}{9};x^3\right )}{\sqrt [3]{-x^2+x^5}}-\frac {\left (6 x^{2/3} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}+\frac {\left (9 x^{2/3} \sqrt [3]{-1+x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 6.63, size = 120, normalized size = 1.29 \begin {gather*} \frac {x^{2/3} \sqrt [3]{-1+x^3} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{-1+x^3}}\right )-2 \log \left (\sqrt [3]{x}+\sqrt [3]{-1+x^3}\right )+\log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right )}{2 \sqrt [3]{x^2 \left (-1+x^3\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.37, size = 580, normalized size = 6.24

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {3175724374 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-28078997418 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-11115035309 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}[phone] x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {2}{3}}-35847343722 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {1}{3}} x -3175724374 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x [phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-13024082868 \left (x^{5}-x^{2}\right )^{\frac {2}{3}}[phone] x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -29244634000 x^{2}-40942487600 x}{\left (x^{3}+x -1\right ) x}\right )}{2}-\frac {\ln \left (\frac {-3175724374 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-15376099922 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}[phone] x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {2}{3}}-35847343722 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {1}{3}} x [phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -8110275566 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-58670604576 \left (x^{5}-x^{2}\right )^{\frac {2}{3}}[phone] x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x [phone] x^{2}-2512609740 x}{\left (x^{3}+x -1\right ) x}\right ) \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )}{2}+\ln \left (\frac {-3175724374 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-15376099922 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}[phone] x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {2}{3}}-35847343722 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {1}{3}} x [phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -8110275566 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-58670604576 \left (x^{5}-x^{2}\right )^{\frac {2}{3}}[phone] x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}[phone] \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x [phone] x^{2}-2512609740 x}{\left (x^{3}+x -1\right ) x}\right )\)	\(580\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(_Z^2-2*_Z+4)*ln(-(3175724374*RootOf(_Z^2-2*_Z+4)^2*x^4-28078997418*RootOf(_Z^2-2*_Z+4)*x^4-11115035
309*RootOf(_Z^2-2*_Z+4)^2*x^2+40942487600*x^4+35847343722*RootOf(_Z^2-2*_Z+4)*(x^5-x^2)^(2/3)-35847343722*Root
Of(_Z^2-2*_Z+4)*(x^5-x^2)^(1/3)*x-3175724374*RootOf(_Z^2-2*_Z+4)^2*x+36349865670*RootOf(_Z^2-2*_Z+4)*x^2-13024
082868*(x^5-x^2)^(2/3)[phone]*x*(x^5-x^2)^(1/3)[phone]*RootOf(_Z^2-2*_Z+4)*x-29244634000*x^2-4094248
7600*x)/(x^3+x-1)/x)-1/2*ln((-3175724374*RootOf(_Z^2-2*_Z+4)^2*x^4-15376099922*RootOf(_Z^2-2*_Z+4)*x^4+1111503
5309*RootOf(_Z^2-2*_Z+4)^2*x^2+2512609740*x^4+35847343722*RootOf(_Z^2-2*_Z+4)*(x^5-x^2)^(2/3)-35847343722*Root
Of(_Z^2-2*_Z+4)*(x^5-x^2)^(1/3)*x+3175724374*RootOf(_Z^2-2*_Z+4)^2*x-8110275566*RootOf(_Z^2-2*_Z+4)*x^2-586706
04576*(x^5-x^2)^(2/3)[phone]*x*(x^5-x^2)^(1/3)[phone]*RootOf(_Z^2-2*_Z+4)*x+1005043896*x^2-251260974
0*x)/(x^3+x-1)/x)*RootOf(_Z^2-2*_Z+4)+ln((-3175724374*RootOf(_Z^2-2*_Z+4)^2*x^4-15376099922*RootOf(_Z^2-2*_Z+4
)*x^4+11115035309*RootOf(_Z^2-2*_Z+4)^2*x^2+2512609740*x^4+35847343722*RootOf(_Z^2-2*_Z+4)*(x^5-x^2)^(2/3)-358
47343722*RootOf(_Z^2-2*_Z+4)*(x^5-x^2)^(1/3)*x+3175724374*RootOf(_Z^2-2*_Z+4)^2*x-8110275566*RootOf(_Z^2-2*_Z+
4)*x^2-58670604576*(x^5-x^2)^(2/3)[phone]*x*(x^5-x^2)^(1/3)[phone]*RootOf(_Z^2-2*_Z+4)*x+1005043896*
x^2-2512609740*x)/(x^3+x-1)/x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 1.00, size = 124, normalized size = 1.33 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{4} + x^{2} - x\right )} + 2 \, \sqrt {3} {\left (x^{5} - x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{4} - x^{2} - x\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{4} + x^{2} + 3 \, {\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} x - x + 3 \, {\left (x^{5} - x^{2}\right )}^{\frac {2}{3}}}{x^{4} + x^{2} - x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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((2*x**3+1)/(x**3+x-1)/(x**5-x**2)**(1/3),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________