\(\int \frac {1+4 x^5}{(-a-x+a x^5) \sqrt {-x+x^6}} \, dx\) [290]

Optimal result

Integrand size = 33, antiderivative size = 26 \[ \int \frac {1+4 x^5}{\left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {-x+x^6}}\right )}{\sqrt {a}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(8*x*Sqrt[1 - x^5]*Hypergeometric2F1[1/10, 1/2, 11/10, x^5])/(a*Sqrt[-x + x^6]) + (10*Sqrt[x]*Sqrt[-1 + x^5]*D
efer[Subst][Defer[Int][1/(Sqrt[-1 + x^10]*(-a - x^2 + a*x^10)), x], x, Sqrt[x]])/Sqrt[-x + x^6] + (8*Sqrt[x]*S
qrt[-1 + x^5]*Defer[Subst][Defer[Int][x^2/(Sqrt[-1 + x^10]*(-a - x^2 + a*x^10)), x], x, Sqrt[x]])/(a*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 (-a-x+a 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 (-a-x^2+a 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}{a \sqrt {-1+x^{10}}}+\frac {5 a+4 x^2}{a \sqrt {-1+x^{10}} \left (-a-x^2+a 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 a+4 x^2}{\sqrt {-1+x^{10}} \left (-a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \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 )}{a \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 )}{a \sqrt {-x+x^6}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \left (\frac {5 a}{\sqrt {-1+x^{10}} \left (-a-x^2+a x^{10}\right )}+\frac {4 x^2}{\sqrt {-1+x^{10}} \left (-a-x^2+a x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{a \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 )}{a \sqrt {-x+x^6}}+\frac {\left (10 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^{10}} \left (-a-x^2+a 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 {x^2}{\sqrt {-1+x^{10}} \left (-a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^6}} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88

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

[In]

int((4*x^5+1)/(a*x^5-a-x)/(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 = 0.37 (sec) , antiderivative size = 137, normalized size of antiderivative = 5.27 \[ \int \frac {1+4 x^5}{\left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\left [\frac {\log \left (-\frac {a^{2} x^{10} - 2 \, a^{2} x^{5} + 6 \, a x^{6} - 4 \, {\left (a x^{5} - a + x\right )} \sqrt {x^{6} - x} \sqrt {a} + a^{2} - 6 \, a x + x^{2}}{a^{2} x^{10} - 2 \, a^{2} x^{5} - 2 \, a x^{6} + a^{2} + 2 \, a x + x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} - x} \sqrt {-a}}{a x^{5} - a + x}\right )}{a}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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