\(\int \frac {1+4 x^5}{(-1-a x+x^5) \sqrt {-x+x^6}} \, dx\) [289]

Optimal result

Integrand size = 30, antiderivative size = 26 \[ \int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {-x+x^6}}\right )}{\sqrt {a}} \]

[Out]

-2*arctanh(a^(1/2)*x/(x^6-x)^(1/2))/a^(1/2)

Rubi [F]

\[ \int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx \]

[In]

Int[(1 + 4*x^5)/((-1 - a*x + x^5)*Sqrt[-x + x^6]),x]

[Out]

(8*x*Sqrt[1 - x^5]*Hypergeometric2F1[1/10, 1/2, 11/10, x^5])/Sqrt[-x + x^6] - (10*Sqrt[x]*Sqrt[-1 + x^5]*Defer
[Subst][Defer[Int][1/((1 + a*x^2 - x^10)*Sqrt[-1 + x^10]), x], x, Sqrt[x]])/Sqrt[-x + x^6] - (8*a*Sqrt[x]*Sqrt
[-1 + x^5]*Defer[Subst][Defer[Int][x^2/((1 + a*x^2 - x^10)*Sqrt[-1 + x^10]), x], x, Sqrt[x]])/Sqrt[-x + x^6]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-1+x^5}\right ) \int \frac {1+4 x^5}{\sqrt {x} \sqrt {-1+x^5} \left (-1-a x+x^5\right )} \, dx}{\sqrt {-x+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {1+4 x^{10}}{\sqrt {-1+x^{10}} \left (-1-a x^2+x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \left (\frac {4}{\sqrt {-1+x^{10}}}+\frac {5+4 a x^2}{\sqrt {-1+x^{10}} \left (-1-a x^2+x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {5+4 a x^2}{\sqrt {-1+x^{10}} \left (-1-a x^2+x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}} \\ & = \frac {\left (8 \sqrt {x} \sqrt {1-x^5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \left (-\frac {5}{\left (1+a x^2-x^{10}\right ) \sqrt {-1+x^{10}}}-\frac {4 a x^2}{\left (1+a x^2-x^{10}\right ) \sqrt {-1+x^{10}}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}} \\ & = \frac {8 x \sqrt {1-x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{10},\frac {1}{2},\frac {11}{10},x^5\right )}{\sqrt {-x+x^6}}-\frac {\left (10 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {1}{\left (1+a x^2-x^{10}\right ) \sqrt {-1+x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}-\frac {\left (8 a \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+a x^2-x^{10}\right ) \sqrt {-1+x^{10}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx \]

[In]

Integrate[(1 + 4*x^5)/((-1 - a*x + x^5)*Sqrt[-x + x^6]),x]

[Out]

Integrate[(1 + 4*x^5)/((-1 - a*x + x^5)*Sqrt[-x + x^6]), x]

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{6}-x}}{x \sqrt {a}}\right )}{\sqrt {a}}\)	\(23\)

[In]

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

[Out]

-2/a^(1/2)*arctanh((x^6-x)^(1/2)/x/a^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.73 \[ \int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\left [\frac {\log \left (-\frac {x^{10} + 6 \, a x^{6} - 2 \, x^{5} + a^{2} x^{2} - 4 \, \sqrt {x^{6} - x} {\left (x^{5} + a x - 1\right )} \sqrt {a} - 6 \, a x + 1}{x^{10} - 2 \, a x^{6} - 2 \, x^{5} + a^{2} x^{2} + 2 \, a x + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} - x} \sqrt {-a}}{x^{5} + a x - 1}\right )}{a}\right ] \]

[In]

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

[Out]

[1/2*log(-(x^10 + 6*a*x^6 - 2*x^5 + a^2*x^2 - 4*sqrt(x^6 - x)*(x^5 + a*x - 1)*sqrt(a) - 6*a*x + 1)/(x^10 - 2*a
*x^6 - 2*x^5 + a^2*x^2 + 2*a*x + 1))/sqrt(a), sqrt(-a)*arctan(2*sqrt(x^6 - x)*sqrt(-a)/(x^5 + a*x - 1))/a]

Sympy [F]

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

[In]

integrate((4*x**5+1)/(x**5-a*x-1)/(x**6-x)**(1/2),x)

[Out]

Integral((4*x**5 + 1)/(sqrt(x*(x - 1)*(x**4 + x**3 + x**2 + x + 1))*(-a*x + x**5 - 1)), x)

Maxima [F]

\[ \int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\int { \frac {4 \, x^{5} + 1}{\sqrt {x^{6} - x} {\left (x^{5} - a x - 1\right )}} \,d x } \]

[In]

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

[Out]

integrate((4*x^5 + 1)/(sqrt(x^6 - x)*(x^5 - a*x - 1)), x)

Giac [F]

\[ \int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\int { \frac {4 \, x^{5} + 1}{\sqrt {x^{6} - x} {\left (x^{5} - a x - 1\right )}} \,d x } \]

[In]

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

[Out]

integrate((4*x^5 + 1)/(sqrt(x^6 - x)*(x^5 - a*x - 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+4 x^5}{\left (-1-a x+x^5\right ) \sqrt {-x+x^6}} \, dx=\int -\frac {4\,x^5+1}{\sqrt {x^6-x}\,\left (-x^5+a\,x+1\right )} \,d x \]

[In]

int(-(4*x^5 + 1)/((x^6 - x)^(1/2)*(a*x - x^5 + 1)),x)

[Out]

int(-(4*x^5 + 1)/((x^6 - x)^(1/2)*(a*x - x^5 + 1)), x)