\(\int \frac {x^6 (4+x^5)}{(-1+x^5)^{3/4} (1-2 x^5+x^8+x^{10})} \, dx\) [2699]

Optimal result

Integrand size = 33, antiderivative size = 245 \[ \int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx=-\frac {1}{2} \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^5}}{-x^2+\sqrt {-1+x^5}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{-1+x^5}}{-x^2+\sqrt {-1+x^5}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^5}}{x^2+\sqrt {-1+x^5}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^5}}{x^2+\sqrt {-1+x^5}}\right ) \]

[Out]

-1/2*(2+2^(1/2))^(1/2)*arctan((2+2^(1/2))^(1/2)*x*(x^5-1)^(1/4)/(-x^2+(x^5-1)^(1/2)))+1/2*(2-2^(1/2))^(1/2)*ar
ctan((2^(1/2)/(2-2^(1/2))^(1/2)-2/(2-2^(1/2))^(1/2))*x*(x^5-1)^(1/4)/(-x^2+(x^5-1)^(1/2)))+1/2*(2-2^(1/2))^(1/
2)*arctanh((2-2^(1/2))^(1/2)*x*(x^5-1)^(1/4)/(x^2+(x^5-1)^(1/2)))+1/2*(2+2^(1/2))^(1/2)*arctanh((2+2^(1/2))^(1
/2)*x*(x^5-1)^(1/4)/(x^2+(x^5-1)^(1/2)))

Rubi [F]

\[ \int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx=\int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx \]

[In]

Int[(x^6*(4 + x^5))/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)),x]

[Out]

(x^2*(1 - x^5)^(3/4)*Hypergeometric2F1[2/5, 3/4, 7/5, x^5])/(2*(-1 + x^5)^(3/4)) - Defer[Int][x/((-1 + x^5)^(3
/4)*(1 - 2*x^5 + x^8 + x^10)), x] + 6*Defer[Int][x^6/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)), x] - Defer[I
nt][x^9/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x}{\left (-1+x^5\right )^{3/4}}+\frac {x \left (-1+6 x^5-x^8\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )}\right ) \, dx \\ & = \int \frac {x}{\left (-1+x^5\right )^{3/4}} \, dx+\int \frac {x \left (-1+6 x^5-x^8\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx \\ & = \frac {\left (1-x^5\right )^{3/4} \int \frac {x}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\int \left (-\frac {x}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )}+\frac {6 x^6}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )}-\frac {x^9}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )}\right ) \, dx \\ & = \frac {x^2 \left (1-x^5\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {2}{5},\frac {3}{4},\frac {7}{5},x^5\right )}{2 \left (-1+x^5\right )^{3/4}}+6 \int \frac {x^6}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx-\int \frac {x}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx-\int \frac {x^9}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 7.05 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.89 \[ \int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx=\frac {1}{2} \left (\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^5}}{x^2-\sqrt {-1+x^5}}\right )+\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^5}}{x^2-\sqrt {-1+x^5}}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^5}}{x^2+\sqrt {-1+x^5}}\right )+\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^5}}{x^2+\sqrt {-1+x^5}}\right )\right ) \]

[In]

Integrate[(x^6*(4 + x^5))/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)),x]

[Out]

(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^5)^(1/4))/(x^2 - Sqrt[-1 + x^5])] + Sqrt[2 + Sqrt[2]]*A
rcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^5)^(1/4))/(x^2 - Sqrt[-1 + x^5])] + Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 - Sqr
t[2]]*x*(-1 + x^5)^(1/4))/(x^2 + Sqrt[-1 + x^5])] + Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^5)^
(1/4))/(x^2 + Sqrt[-1 + x^5])])/2

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 6.62 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.13

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{5}-1\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{7}}\right )}{2}\)	\(33\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} \sqrt {x^{5}-1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{2} \left (x^{5}-1\right )^{\frac {1}{4}} x^{3}-2 \left (x^{5}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3}}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{5}+1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} \sqrt {x^{5}-1}\, x^{2}-2 \left (x^{5}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{6} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) x^{5}+2 \left (x^{5}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{5}-1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{9} x^{4}+2 \left (x^{5}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{6} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5} x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{3} \sqrt {x^{5}-1}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{5}+2 \left (x^{5}-1\right )^{\frac {3}{4}} x}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{5}-1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{11} x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7} x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{7}-2 \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{2} \left (x^{5}-1\right )^{\frac {1}{4}} x^{3}-2 \sqrt {x^{5}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right ) x^{2}-2 \left (x^{5}-1\right )^{\frac {3}{4}} x}{\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{5}+1}\right )}{2}\)	\(462\)

[In]

int(x^6*(x^5+4)/(x^5-1)^(3/4)/(x^10+x^8-2*x^5+1),x,method=_RETURNVERBOSE)

[Out]

1/2*sum(ln((-_R*x+(x^5-1)^(1/4))/x)/_R^7,_R=RootOf(_Z^8+1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 62.70 (sec) , antiderivative size = 1214, normalized size of antiderivative = 4.96 \[ \int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx=\text {Too large to display} \]

[In]

integrate(x^6*(x^5+4)/(x^5-1)^(3/4)/(x^10+x^8-2*x^5+1),x, algorithm="fricas")

[Out]

-(1/8*I - 1/8)*sqrt(2)*(-1)^(1/8)*log(-(2*sqrt(2)*((I + 1)*(-1)^(3/8)*x^6 + (-1)^(7/8)*((I + 1)*x^7 - (I + 1)*
x^2))*sqrt(x^5 - 1) + 4*(I*(-1)^(1/4)*x^5 + (-1)^(3/4)*(I*x^6 - I*x))*(x^5 - 1)^(3/4) - sqrt(2)*((-1)^(5/8)*(-
(I - 1)*x^10 + (I - 1)*x^8 + (2*I - 2)*x^5 - I + 1) - 2*(-1)^(1/8)*((I - 1)*x^9 - (I - 1)*x^4)) - 4*(x^8 - I*x
^7 - x^3)*(x^5 - 1)^(1/4))/(x^10 + x^8 - 2*x^5 + 1)) + (1/8*I + 1/8)*sqrt(2)*(-1)^(1/8)*log(-(2*sqrt(2)*(-(I -
1)*(-1)^(3/8)*x^6 + (-1)^(7/8)*(-(I - 1)*x^7 + (I - 1)*x^2))*sqrt(x^5 - 1) + 4*(-I*(-1)^(1/4)*x^5 + (-1)^(3/4
)*(-I*x^6 + I*x))*(x^5 - 1)^(3/4) - sqrt(2)*((-1)^(5/8)*((I + 1)*x^10 - (I + 1)*x^8 - (2*I + 2)*x^5 + I + 1) -
2*(-1)^(1/8)*(-(I + 1)*x^9 + (I + 1)*x^4)) - 4*(x^8 - I*x^7 - x^3)*(x^5 - 1)^(1/4))/(x^10 + x^8 - 2*x^5 + 1))
- (1/8*I + 1/8)*sqrt(2)*(-1)^(1/8)*log(-(2*sqrt(2)*((I - 1)*(-1)^(3/8)*x^6 + (-1)^(7/8)*((I - 1)*x^7 - (I - 1
)*x^2))*sqrt(x^5 - 1) + 4*(-I*(-1)^(1/4)*x^5 + (-1)^(3/4)*(-I*x^6 + I*x))*(x^5 - 1)^(3/4) - sqrt(2)*((-1)^(5/8
)*(-(I + 1)*x^10 + (I + 1)*x^8 + (2*I + 2)*x^5 - I - 1) - 2*(-1)^(1/8)*((I + 1)*x^9 - (I + 1)*x^4)) - 4*(x^8 -
I*x^7 - x^3)*(x^5 - 1)^(1/4))/(x^10 + x^8 - 2*x^5 + 1)) + (1/8*I - 1/8)*sqrt(2)*(-1)^(1/8)*log(-(2*sqrt(2)*(-
(I + 1)*(-1)^(3/8)*x^6 + (-1)^(7/8)*(-(I + 1)*x^7 + (I + 1)*x^2))*sqrt(x^5 - 1) + 4*(I*(-1)^(1/4)*x^5 + (-1)^(
3/4)*(I*x^6 - I*x))*(x^5 - 1)^(3/4) - sqrt(2)*((-1)^(5/8)*((I - 1)*x^10 - (I - 1)*x^8 - (2*I - 2)*x^5 + I - 1)
- 2*(-1)^(1/8)*(-(I - 1)*x^9 + (I - 1)*x^4)) - 4*(x^8 - I*x^7 - x^3)*(x^5 - 1)^(1/4))/(x^10 + x^8 - 2*x^5 + 1
)) - 1/4*(-1)^(1/8)*log((2*((-1)^(1/4)*x^5 - (-1)^(3/4)*(x^6 - x))*(x^5 - 1)^(3/4) + (-1)^(5/8)*(x^10 - x^8 -
2*x^5 + 1) - 2*((-1)^(3/8)*x^6 - (-1)^(7/8)*(x^7 - x^2))*sqrt(x^5 - 1) + 2*(x^8 + I*x^7 - x^3)*(x^5 - 1)^(1/4)
- 2*(-1)^(1/8)*(x^9 - x^4))/(x^10 + x^8 - 2*x^5 + 1)) + 1/4*(-1)^(1/8)*log((2*((-1)^(1/4)*x^5 - (-1)^(3/4)*(x
^6 - x))*(x^5 - 1)^(3/4) - (-1)^(5/8)*(x^10 - x^8 - 2*x^5 + 1) + 2*((-1)^(3/8)*x^6 - (-1)^(7/8)*(x^7 - x^2))*s
qrt(x^5 - 1) + 2*(x^8 + I*x^7 - x^3)*(x^5 - 1)^(1/4) + 2*(-1)^(1/8)*(x^9 - x^4))/(x^10 + x^8 - 2*x^5 + 1)) - 1
/4*I*(-1)^(1/8)*log(-(2*((-1)^(1/4)*x^5 - (-1)^(3/4)*(x^6 - x))*(x^5 - 1)^(3/4) - (-1)^(5/8)*(I*x^10 - I*x^8 -
2*I*x^5 + I) + 2*(-I*(-1)^(3/8)*x^6 + (-1)^(7/8)*(I*x^7 - I*x^2))*sqrt(x^5 - 1) - 2*(x^8 + I*x^7 - x^3)*(x^5
- 1)^(1/4) + 2*(-1)^(1/8)*(I*x^9 - I*x^4))/(x^10 + x^8 - 2*x^5 + 1)) + 1/4*I*(-1)^(1/8)*log(-(2*((-1)^(1/4)*x^
5 - (-1)^(3/4)*(x^6 - x))*(x^5 - 1)^(3/4) - (-1)^(5/8)*(-I*x^10 + I*x^8 + 2*I*x^5 - I) + 2*(I*(-1)^(3/8)*x^6 +
(-1)^(7/8)*(-I*x^7 + I*x^2))*sqrt(x^5 - 1) - 2*(x^8 + I*x^7 - x^3)*(x^5 - 1)^(1/4) + 2*(-1)^(1/8)*(-I*x^9 + I
*x^4))/(x^10 + x^8 - 2*x^5 + 1))

Sympy [F]

\[ \int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx=\int \frac {x^{6} \left (x^{5} + 4\right )}{\left (\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )\right )^{\frac {3}{4}} \left (x^{10} + x^{8} - 2 x^{5} + 1\right )}\, dx \]

[In]

integrate(x**6*(x**5+4)/(x**5-1)**(3/4)/(x**10+x**8-2*x**5+1),x)

[Out]

Integral(x**6*(x**5 + 4)/(((x - 1)*(x**4 + x**3 + x**2 + x + 1))**(3/4)*(x**10 + x**8 - 2*x**5 + 1)), x)

Maxima [F]

\[ \int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx=\int { \frac {{\left (x^{5} + 4\right )} x^{6}}{{\left (x^{10} + x^{8} - 2 \, x^{5} + 1\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(x^6*(x^5+4)/(x^5-1)^(3/4)/(x^10+x^8-2*x^5+1),x, algorithm="maxima")

[Out]

integrate((x^5 + 4)*x^6/((x^10 + x^8 - 2*x^5 + 1)*(x^5 - 1)^(3/4)), x)

Giac [F]

\[ \int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx=\int { \frac {{\left (x^{5} + 4\right )} x^{6}}{{\left (x^{10} + x^{8} - 2 \, x^{5} + 1\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(x^6*(x^5+4)/(x^5-1)^(3/4)/(x^10+x^8-2*x^5+1),x, algorithm="giac")

[Out]

integrate((x^5 + 4)*x^6/((x^10 + x^8 - 2*x^5 + 1)*(x^5 - 1)^(3/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx=\int \frac {x^6\,\left (x^5+4\right )}{{\left (x^5-1\right )}^{3/4}\,\left (x^{10}+x^8-2\,x^5+1\right )} \,d x \]

[In]

int((x^6*(x^5 + 4))/((x^5 - 1)^(3/4)*(x^8 - 2*x^5 + x^10 + 1)),x)

[Out]

int((x^6*(x^5 + 4))/((x^5 - 1)^(3/4)*(x^8 - 2*x^5 + x^10 + 1)), x)