\(\int \frac {-1+4 x^5}{(a-x+a x^5) \sqrt {x+x^6}} \, dx\) [249]

Optimal result

Integrand size = 29, antiderivative size = 24 \[ \int \frac {-1+4 x^5}{\left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {x+x^6}}\right )}{\sqrt {a}} \]

[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]*De
fer[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]*Sqrt[
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 {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 (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 = 1.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88

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

[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 = 0.34 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.38 \[ \int \frac {-1+4 x^5}{\left (a-x+a x^5\right ) \sqrt {x+x^6}} \, dx=\left [\frac {\log \left (-\frac {a^{2} x^{10} + 2 \, a^{2} x^{5} + 6 \, a x^{6} - 4 \, {\left (a x^{5} + a + x\right )} \sqrt {x^{6} + x} \sqrt {a} + a^{2} + 6 \, a x + x^{2}}{a^{2} x^{10} + 2 \, a^{2} x^{5} - 2 \, a x^{6} + a^{2} - 2 \, a x + x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a}}{a x^{5} + a + x}\right )}{a}\right ] \]

[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 + x^6)^(1/2)*(a - x + a*x^5)),x)

[Out]

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