\(\int \frac {(-1+x^7)^{2/3} (3+4 x^7)}{x^3 (-1+x^3+x^7)} \, dx\) [1249]

Optimal result

Integrand size = 30, antiderivative size = 90 \[ \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx=\frac {3 \left (-1+x^7\right )^{2/3}}{2 x^2}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^7}}\right )+\log \left (x+\sqrt [3]{-1+x^7}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^7}+\left (-1+x^7\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx=\int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx \]

[In]

Int[((-1 + x^7)^(2/3)*(3 + 4*x^7))/(x^3*(-1 + x^3 + x^7)),x]

[Out]

(3*(-1 + x^7)^(2/3)*Hypergeometric2F1[-2/3, -2/7, 5/7, x^7])/(2*x^2*(1 - x^7)^(2/3)) + 3*Defer[Int][(-1 + x^7)
^(2/3)/(-1 + x^3 + x^7), x] + 7*Defer[Int][(x^4*(-1 + x^7)^(2/3))/(-1 + x^3 + x^7), x]

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

Mathematica [A] (verified)

Time = 9.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx=\frac {3 \left (-1+x^7\right )^{2/3}}{2 x^2}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^7}}\right )+\log \left (x+\sqrt [3]{-1+x^7}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^7}+\left (-1+x^7\right )^{2/3}\right ) \]

[In]

Integrate[((-1 + x^7)^(2/3)*(3 + 4*x^7))/(x^3*(-1 + x^3 + x^7)),x]

[Out]

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

Maple [A] (verified)

Time = 74.72 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{7}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{2}+2 \ln \left (\frac {x +\left (x^{7}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-\ln \left (\frac {x^{2}-x \left (x^{7}-1\right )^{\frac {1}{3}}+\left (x^{7}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{7}-1\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(94\)
risch	\(\frac {3 \left (x^{7}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-x^{7}+\left (x^{7}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+2 \left (x^{7}-1\right )^{\frac {2}{3}} x -2 \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}+x^{3}+1}{x^{7}+x^{3}-1}\right )-\ln \left (\frac {x^{7}+\left (x^{7}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\left (x^{7}-1\right )^{\frac {2}{3}} x +\left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-1}{x^{7}+x^{3}-1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {x^{7}+\left (x^{7}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\left (x^{7}-1\right )^{\frac {2}{3}} x +\left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-1}{x^{7}+x^{3}-1}\right )\)	\(292\)
trager	\(\frac {3 \left (x^{7}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (-\frac {714529075649329276274990776617793122369024 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{7}+1010312028170994192094059614017027904963984 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{7}+86055206848169411129714472442189555841027 x^{7}-11343149075933102260865478578807465817608256 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+928417484526511344275296327342965046304892 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {2}{3}} x -995468846547627568662838493210705341689912 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-1918380815535014015605780986155414061362260 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+160323860922844909411511235046139198999567 \left (x^{7}-1\right )^{\frac {2}{3}} x +77368123710542612022941360611913753858741 \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-80634406416788660822330883626933520827419 x^{3}-714529075649329276274990776617793122369024 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-1010312028170994192094059614017027904963984 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-86055206848169411129714472442189555841027}{x^{7}+x^{3}-1}\right )+12 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (\frac {2341785786356484132789710368190607125878656 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{7}-2724206362092059470535164116415158469194828 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{7}-221429583339245270512301829212284847817484 x^{7}-37175849358409185608036652095025888123323664 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+2785252453579534032825888982028895138914676 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {2}{3}} x +5771658993222416738814404461661011163984412 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-3030936084512982576282178842051083714891952 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}-248867211636906892165709623302676335422478 \left (x^{7}-1\right )^{\frac {2}{3}} x +232104371131627836068824081835741261576223 \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-14886022409361026589062307846204023382688 x^{3}-2341785786356484132789710368190607125878656 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+2724206362092059470535164116415158469194828 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+221429583339245270512301829212284847817484}{x^{7}+x^{3}-1}\right )\)	\(400\)

[In]

int((x^7-1)^(2/3)*(4*x^7+3)/x^3/(x^7+x^3-1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 10.67 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {26962 \, \sqrt {3} {\left (x^{7} - 1\right )}^{\frac {1}{3}} x^{2} - 60268 \, \sqrt {3} {\left (x^{7} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (34656 \, x^{7} - 8959 \, x^{3} - 34656\right )}}{54872 \, x^{7} + 4913 \, x^{3} - 54872}\right ) + x^{2} \log \left (\frac {x^{7} + x^{3} + 3 \, {\left (x^{7} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{7} - 1\right )}^{\frac {2}{3}} x - 1}{x^{7} + x^{3} - 1}\right ) + 3 \, {\left (x^{7} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]

[In]

integrate((x^7-1)^(2/3)*(4*x^7+3)/x^3/(x^7+x^3-1),x, algorithm="fricas")

[Out]

1/2*(2*sqrt(3)*x^2*arctan(-(26962*sqrt(3)*(x^7 - 1)^(1/3)*x^2 - 60268*sqrt(3)*(x^7 - 1)^(2/3)*x + sqrt(3)*(346
56*x^7 - 8959*x^3 - 34656))/(54872*x^7 + 4913*x^3 - 54872)) + x^2*log((x^7 + x^3 + 3*(x^7 - 1)^(1/3)*x^2 + 3*(
x^7 - 1)^(2/3)*x - 1)/(x^7 + x^3 - 1)) + 3*(x^7 - 1)^(2/3))/x^2

Sympy [F]

\[ \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1\right )\right )^{\frac {2}{3}} \cdot \left (4 x^{7} + 3\right )}{x^{3} \left (x^{7} + x^{3} - 1\right )}\, dx \]

[In]

integrate((x**7-1)**(2/3)*(4*x**7+3)/x**3/(x**7+x**3-1),x)

[Out]

Integral(((x - 1)*(x**6 + x**5 + x**4 + x**3 + x**2 + x + 1))**(2/3)*(4*x**7 + 3)/(x**3*(x**7 + x**3 - 1)), x)

Maxima [F]

\[ \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx=\int { \frac {{\left (4 \, x^{7} + 3\right )} {\left (x^{7} - 1\right )}^{\frac {2}{3}}}{{\left (x^{7} + x^{3} - 1\right )} x^{3}} \,d x } \]

[In]

integrate((x^7-1)^(2/3)*(4*x^7+3)/x^3/(x^7+x^3-1),x, algorithm="maxima")

[Out]

integrate((4*x^7 + 3)*(x^7 - 1)^(2/3)/((x^7 + x^3 - 1)*x^3), x)

Giac [F]

\[ \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx=\int { \frac {{\left (4 \, x^{7} + 3\right )} {\left (x^{7} - 1\right )}^{\frac {2}{3}}}{{\left (x^{7} + x^{3} - 1\right )} x^{3}} \,d x } \]

[In]

integrate((x^7-1)^(2/3)*(4*x^7+3)/x^3/(x^7+x^3-1),x, algorithm="giac")

[Out]

integrate((4*x^7 + 3)*(x^7 - 1)^(2/3)/((x^7 + x^3 - 1)*x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx=\int \frac {{\left (x^7-1\right )}^{2/3}\,\left (4\,x^7+3\right )}{x^3\,\left (x^7+x^3-1\right )} \,d x \]

[In]

int(((x^7 - 1)^(2/3)*(4*x^7 + 3))/(x^3*(x^3 + x^7 - 1)),x)

[Out]

int(((x^7 - 1)^(2/3)*(4*x^7 + 3))/(x^3*(x^3 + x^7 - 1)), x)