\(\int \frac {2-3 x^5}{(1-x^2+x^5) \sqrt [3]{x+x^6}} \, dx\) [993]

Optimal result

Integrand size = 29, antiderivative size = 75 \[ \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^6}}\right )-\log \left (-x+\sqrt [3]{x+x^6}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+x^6}+\left (x+x^6\right )^{2/3}\right ) \]

[Out]

3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^6+x)^(1/3)))-ln(-x+(x^6+x)^(1/3))+1/2*ln(x^2+x*(x^6+x)^(1/3)+(x^6+x)^(2/3))

Rubi [F]

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

[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*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^{15}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}} \\ & = -\frac {9 x \sqrt [3]{1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {2}{15},\frac {1}{3},\frac {17}{15},-x^5\right )}{2 \sqrt [3]{x+x^6}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \text {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} \operatorname {Hypergeometric2F1}\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 ) \text {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 ) \text {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*}

Mathematica [F]

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

[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]

Maple [A] (verified)

Time = 6.54 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(-\ln \left (\frac {-x +\left (x^{6}+x \right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {x^{2}+x \left (x^{6}+x \right )^{\frac {1}{3}}+\left (x^{6}+x \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{6}+x \right )^{\frac {1}{3}}\right )}{3 x}\right )\)	\(72\)
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {523628198354 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}[phone] x^{5}-1047256396708 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {2}{3}}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {1}{3}} x [phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}[phone] \left (x^{6}+x \right )^{\frac {2}{3}}[phone] x \left (x^{6}+x \right )^{\frac {1}{3}}[phone] x^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )[phone]}{x^{5}-x^{2}+1}\right )-\ln \left (\frac {8961517 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}-1439545 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-76335018 x^{5}-17923034 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-70252591 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {2}{3}}-70252591 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {1}{3}} x -59851529 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+8961517 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-92818507 \left (x^{6}+x \right )^{\frac {2}{3}}-92818507 x \left (x^{6}+x \right )^{\frac {1}{3}}-25445006 x^{2}-1439545 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-76335018}{x^{5}-x^{2}+1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {8961517 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}-1439545 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-76335018 x^{5}-17923034 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-70252591 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {2}{3}}-70252591 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {1}{3}} x -59851529 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+8961517 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-92818507 \left (x^{6}+x \right )^{\frac {2}{3}}-92818507 x \left (x^{6}+x \right )^{\frac {1}{3}}-25445006 x^{2}-1439545 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-76335018}{x^{5}-x^{2}+1}\right )\)	\(522\)

[In]

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

[Out]

-ln((-x+(x^6+x)^(1/3))/x)+1/2*ln((x^2+x*(x^6+x)^(1/3)+(x^6+x)^(2/3))/x^2)-3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x
^6+x)^(1/3)))

Fricas [A] (verification not implemented)

none

Time = 1.28 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx=\sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{6} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} + 1\right )} - 2 \, \sqrt {3} {\left (x^{6} + x\right )}^{\frac {2}{3}}}{x^{5} + 8 \, x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{5} - x^{2} + 3 \, {\left (x^{6} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{6} + x\right )}^{\frac {2}{3}} + 1}{x^{5} - x^{2} + 1}\right ) \]

[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))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx=-\int \frac {3\,x^5-2}{{\left (x^6+x\right )}^{1/3}\,\left (x^5-x^2+1\right )} \,d x \]

[In]

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

[Out]

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