\(\int \frac {\sqrt [3]{1+2 x^7} (-3+8 x^7)}{x^2 (1+x^3+2 x^7)} \, dx\) [1399]

Optimal result

Integrand size = 34, antiderivative size = 100 \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\frac {3 \sqrt [3]{1+2 x^7}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+2 x^7}}\right )-\log \left (x+\sqrt [3]{1+2 x^7}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+2 x^7}+\left (1+2 x^7\right )^{2/3}\right ) \]

[Out]

3*(2*x^7+1)^(1/3)/x+3^(1/2)*arctan(3^(1/2)*x/(-x+2*(2*x^7+1)^(1/3)))-ln(x+(2*x^7+1)^(1/3))+1/2*ln(x^2-x*(2*x^7
+1)^(1/3)+(2*x^7+1)^(2/3))

Rubi [F]

\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx \]

[In]

Int[((1 + 2*x^7)^(1/3)*(-3 + 8*x^7))/(x^2*(1 + x^3 + 2*x^7)),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \sqrt [3]{1+2 x^7}}{x^2}+\frac {x \left (3+14 x^4\right ) \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}\right ) \, dx \\ & = -\left (3 \int \frac {\sqrt [3]{1+2 x^7}}{x^2} \, dx\right )+\int \frac {x \left (3+14 x^4\right ) \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{7},\frac {6}{7},-2 x^7\right )}{x}+\int \left (\frac {3 x \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}+\frac {14 x^5 \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}\right ) \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{7},\frac {6}{7},-2 x^7\right )}{x}+3 \int \frac {x \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx+14 \int \frac {x^5 \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 9.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\frac {3 \sqrt [3]{1+2 x^7}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+2 x^7}}\right )-\log \left (x+\sqrt [3]{1+2 x^7}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+2 x^7}+\left (1+2 x^7\right )^{2/3}\right ) \]

[In]

Integrate[((1 + 2*x^7)^(1/3)*(-3 + 8*x^7))/(x^2*(1 + x^3 + 2*x^7)),x]

[Out]

(3*(1 + 2*x^7)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + 2*x^7)^(1/3))] - Log[x + (1 + 2*x^7)^(1/3)]
+ Log[x^2 - x*(1 + 2*x^7)^(1/3) + (1 + 2*x^7)^(2/3)]/2

Maple [A] (verified)

Time = 20.93 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {1}{3}} \left (8 x^{7}+4\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x -2 \ln \left (\frac {2^{\frac {2}{3}} x +\left (8 x^{7}+4\right )^{\frac {1}{3}}}{x}\right ) x +\ln \left (\frac {-2^{\frac {2}{3}} \left (8 x^{7}+4\right )^{\frac {1}{3}} x +2 \,2^{\frac {1}{3}} x^{2}+\left (8 x^{7}+4\right )^{\frac {2}{3}}}{x^{2}}\right ) x +3 \,2^{\frac {1}{3}} \left (8 x^{7}+4\right )^{\frac {1}{3}}}{2 x}\)	\(115\)
risch	\(\frac {3 \left (2 x^{7}+1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{14}-4 x^{14}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{10}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{10}-6 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-4 x^{7}+3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (2 x^{7}+1\right ) \left (2 x^{7}+x^{3}+1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{14}-4 x^{14}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{10}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}+2 x^{10}-6 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-4 x^{7}+3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x +x^{3}-3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (2 x^{7}+1\right ) \left (2 x^{7}+x^{3}+1\right )}\right )\right ) {\left (\left (2 x^{7}+1\right )^{2}\right )}^{\frac {1}{3}}}{\left (2 x^{7}+1\right )^{\frac {2}{3}}}\)	\(498\)
trager	\(\text {Expression too large to display}\)	\(620\)

[In]

int((2*x^7+1)^(1/3)*(8*x^7-3)/x^2/(2*x^7+x^3+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)*(-2^(1/3)*(8*x^7+4)^(1/3)+x)/x)*x-2*ln((2^(2/3)*x+(8*x^7+4)^(1/3))/x)*x+ln((
-2^(2/3)*(8*x^7+4)^(1/3)*x+2*2^(1/3)*x^2+(8*x^7+4)^(2/3))/x^2)*x+3*2^(1/3)*(8*x^7+4)^(1/3))/x

Fricas [A] (verification not implemented)

none

Time = 8.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {8377128467638 \, \sqrt {3} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}} x^{2} + 15171948325814 \, \sqrt {3} {\left (2 \, x^{7} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2102123379894 \, x^{7} + 4448471619035 \, x^{3} + 1051061689947\right )}}{60468559237154 \, x^{7} - 5089335571601 \, x^{3} + 30234279618577}\right ) - x \log \left (\frac {2 \, x^{7} + x^{3} + 3 \, {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (2 \, x^{7} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{7} + x^{3} + 1}\right ) + 6 \, {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{2 \, x} \]

[In]

integrate((2*x^7+1)^(1/3)*(8*x^7-3)/x^2/(2*x^7+x^3+1),x, algorithm="fricas")

[Out]

1/2*(2*sqrt(3)*x*arctan((8377128467638*sqrt(3)*(2*x^7 + 1)^(1/3)*x^2 + 15171948325814*sqrt(3)*(2*x^7 + 1)^(2/3
)*x + sqrt(3)*(2102123379894*x^7 + 4448471619035*x^3 + 1051061689947))/(60468559237154*x^7 - 5089335571601*x^3
+ 30234279618577)) - x*log((2*x^7 + x^3 + 3*(2*x^7 + 1)^(1/3)*x^2 + 3*(2*x^7 + 1)^(2/3)*x + 1)/(2*x^7 + x^3 +
1)) + 6*(2*x^7 + 1)^(1/3))/x

Sympy [F]

\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int \frac {\sqrt [3]{2 x^{7} + 1} \cdot \left (8 x^{7} - 3\right )}{x^{2} \cdot \left (2 x^{7} + x^{3} + 1\right )}\, dx \]

[In]

integrate((2*x**7+1)**(1/3)*(8*x**7-3)/x**2/(2*x**7+x**3+1),x)

[Out]

Integral((2*x**7 + 1)**(1/3)*(8*x**7 - 3)/(x**2*(2*x**7 + x**3 + 1)), x)

Maxima [F]

\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int { \frac {{\left (8 \, x^{7} - 3\right )} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{7} + x^{3} + 1\right )} x^{2}} \,d x } \]

[In]

integrate((2*x^7+1)^(1/3)*(8*x^7-3)/x^2/(2*x^7+x^3+1),x, algorithm="maxima")

[Out]

integrate((8*x^7 - 3)*(2*x^7 + 1)^(1/3)/((2*x^7 + x^3 + 1)*x^2), x)

Giac [F]

\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int { \frac {{\left (8 \, x^{7} - 3\right )} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{7} + x^{3} + 1\right )} x^{2}} \,d x } \]

[In]

integrate((2*x^7+1)^(1/3)*(8*x^7-3)/x^2/(2*x^7+x^3+1),x, algorithm="giac")

[Out]

integrate((8*x^7 - 3)*(2*x^7 + 1)^(1/3)/((2*x^7 + x^3 + 1)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int \frac {{\left (2\,x^7+1\right )}^{1/3}\,\left (8\,x^7-3\right )}{x^2\,\left (2\,x^7+x^3+1\right )} \,d x \]

[In]

int(((2*x^7 + 1)^(1/3)*(8*x^7 - 3))/(x^2*(x^3 + 2*x^7 + 1)),x)

[Out]

int(((2*x^7 + 1)^(1/3)*(8*x^7 - 3))/(x^2*(x^3 + 2*x^7 + 1)), x)