\(\int \frac {-1+4 x^5}{(1-a x+x^5) \sqrt {x+x^6}} \, dx\) [248]

Optimal result

Integrand size = 28, antiderivative size = 24 \[ \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {x+x^6}}\right )}{\sqrt {a}} \]

[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 {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 (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 = 1.63 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{6}+x}}{x \sqrt {a}}\right )}{\sqrt {a}}\)	\(21\)

[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 = 119, normalized size of antiderivative = 4.96 \[ \int \frac {-1+4 x^5}{\left (1-a x+x^5\right ) \sqrt {x+x^6}} \, dx=\left [\frac {\log \left (-\frac {x^{10} + 6 \, a x^{6} + 2 \, x^{5} + a^{2} x^{2} - 4 \, \sqrt {x^{6} + x} {\left (x^{5} + a x + 1\right )} \sqrt {a} + 6 \, a x + 1}{x^{10} - 2 \, a x^{6} + 2 \, x^{5} + a^{2} x^{2} - 2 \, a x + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a}}{x^{5} + a x + 1}\right )}{a}\right ] \]

[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 + x^6)^(1/2)*(x^5 - a*x + 1)),x)

[Out]

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