\(\int \frac {-x+4 x^6}{\sqrt {x+x^6} (a-x^2+2 a x^5+a x^{10})} \, dx\) [617]

Optimal result

Integrand size = 39, antiderivative size = 48 \[ \int \frac {-x+4 x^6}{\sqrt {x+x^6} \left (a-x^2+2 a x^5+a x^{10}\right )} \, dx=\frac {\arctan \left (\frac {x}{\sqrt [4]{a} \sqrt {x+x^6}}\right )}{\sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{a} \sqrt {x+x^6}}\right )}{\sqrt [4]{a}} \]

[Out]

arctan(x/a^(1/4)/(x^6+x)^(1/2))/a^(1/4)-arctanh(x/a^(1/4)/(x^6+x)^(1/2))/a^(1/4)

Rubi [F]

\[ \int \frac {-x+4 x^6}{\sqrt {x+x^6} \left (a-x^2+2 a x^5+a x^{10}\right )} \, dx=\int \frac {-x+4 x^6}{\sqrt {x+x^6} \left (a-x^2+2 a x^5+a x^{10}\right )} \, dx \]

[In]

Int[(-x + 4*x^6)/(Sqrt[x + x^6]*(a - x^2 + 2*a*x^5 + a*x^10)),x]

[Out]

(-2*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^2/(Sqrt[1 + x^10]*(a - x^4 + 2*a*x^10 + a*x^20)), x], x, S
qrt[x]])/Sqrt[x + x^6] + (8*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^12/(Sqrt[1 + x^10]*(a - x^4 + 2*a*
x^10 + a*x^20)), x], x, Sqrt[x]])/Sqrt[x + x^6]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (-1+4 x^5\right )}{\sqrt {x+x^6} \left (a-x^2+2 a x^5+a x^{10}\right )} \, dx \\ & = \frac {\left (\sqrt {x} \sqrt {1+x^5}\right ) \int \frac {\sqrt {x} \left (-1+4 x^5\right )}{\sqrt {1+x^5} \left (a-x^2+2 a x^5+a x^{10}\right )} \, dx}{\sqrt {x+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \left (-1+4 x^{10}\right )}{\sqrt {1+x^{10}} \left (a-x^4+2 a x^{10}+a x^{20}\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 {x^2}{\sqrt {1+x^{10}} \left (a-x^4+2 a x^{10}+a x^{20}\right )}+\frac {4 x^{12}}{\sqrt {1+x^{10}} \left (a-x^4+2 a x^{10}+a x^{20}\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 {x^2}{\sqrt {1+x^{10}} \left (a-x^4+2 a x^{10}+a x^{20}\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^{12}}{\sqrt {1+x^{10}} \left (a-x^4+2 a x^{10}+a x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {-x+4 x^6}{\sqrt {x+x^6} \left (a-x^2+2 a x^5+a x^{10}\right )} \, dx=\int \frac {-x+4 x^6}{\sqrt {x+x^6} \left (a-x^2+2 a x^5+a x^{10}\right )} \, dx \]

[In]

Integrate[(-x + 4*x^6)/(Sqrt[x + x^6]*(a - x^2 + 2*a*x^5 + a*x^10)),x]

[Out]

Integrate[(-x + 4*x^6)/(Sqrt[x + x^6]*(a - x^2 + 2*a*x^5 + a*x^10)), x]

Maple [A] (verified)

Time = 2.72 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.31

method	result	size
pseudoelliptic	\(-\frac {\left (\frac {1}{a}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\left (\frac {1}{a}\right )^{\frac {1}{4}} x +\sqrt {x^{6}+x}}{-\left (\frac {1}{a}\right )^{\frac {1}{4}} x +\sqrt {x^{6}+x}}\right )+2 \arctan \left (\frac {\sqrt {x^{6}+x}}{x \left (\frac {1}{a}\right )^{\frac {1}{4}}}\right )\right )}{2}\)	\(63\)

[In]

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

[Out]

-1/2*(1/a)^(1/4)*(ln(((1/a)^(1/4)*x+(x^6+x)^(1/2))/(-(1/a)^(1/4)*x+(x^6+x)^(1/2)))+2*arctan((x^6+x)^(1/2)/x/(1
/a)^(1/4)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.65 (sec) , antiderivative size = 334, normalized size of antiderivative = 6.96 \[ \int \frac {-x+4 x^6}{\sqrt {x+x^6} \left (a-x^2+2 a x^5+a x^{10}\right )} \, dx=-\frac {\log \left (-\frac {2 \, \sqrt {x^{6} + x} {\left (x^{5} + \frac {x}{\sqrt {a}} + 1\right )} + \frac {2 \, {\left (x^{6} + x\right )}}{a^{\frac {1}{4}}} + \frac {a x^{10} + 2 \, a x^{5} + x^{2} + a}{a^{\frac {3}{4}}}}{2 \, {\left (a x^{10} + 2 \, a x^{5} - x^{2} + a\right )}}\right )}{4 \, a^{\frac {1}{4}}} + \frac {\log \left (-\frac {2 \, \sqrt {x^{6} + x} {\left (x^{5} + \frac {x}{\sqrt {a}} + 1\right )} - \frac {2 \, {\left (x^{6} + x\right )}}{a^{\frac {1}{4}}} - \frac {a x^{10} + 2 \, a x^{5} + x^{2} + a}{a^{\frac {3}{4}}}}{2 \, {\left (a x^{10} + 2 \, a x^{5} - x^{2} + a\right )}}\right )}{4 \, a^{\frac {1}{4}}} - \frac {i \, \log \left (-\frac {2 \, \sqrt {x^{6} + x} {\left (x^{5} - \frac {x}{\sqrt {a}} + 1\right )} + \frac {2 \, {\left (i \, x^{6} + i \, x\right )}}{a^{\frac {1}{4}}} + \frac {-i \, a x^{10} - 2 i \, a x^{5} - i \, x^{2} - i \, a}{a^{\frac {3}{4}}}}{2 \, {\left (a x^{10} + 2 \, a x^{5} - x^{2} + a\right )}}\right )}{4 \, a^{\frac {1}{4}}} + \frac {i \, \log \left (-\frac {2 \, \sqrt {x^{6} + x} {\left (x^{5} - \frac {x}{\sqrt {a}} + 1\right )} + \frac {2 \, {\left (-i \, x^{6} - i \, x\right )}}{a^{\frac {1}{4}}} + \frac {i \, a x^{10} + 2 i \, a x^{5} + i \, x^{2} + i \, a}{a^{\frac {3}{4}}}}{2 \, {\left (a x^{10} + 2 \, a x^{5} - x^{2} + a\right )}}\right )}{4 \, a^{\frac {1}{4}}} \]

[In]

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

[Out]

-1/4*log(-1/2*(2*sqrt(x^6 + x)*(x^5 + x/sqrt(a) + 1) + 2*(x^6 + x)/a^(1/4) + (a*x^10 + 2*a*x^5 + x^2 + a)/a^(3
/4))/(a*x^10 + 2*a*x^5 - x^2 + a))/a^(1/4) + 1/4*log(-1/2*(2*sqrt(x^6 + x)*(x^5 + x/sqrt(a) + 1) - 2*(x^6 + x)
/a^(1/4) - (a*x^10 + 2*a*x^5 + x^2 + a)/a^(3/4))/(a*x^10 + 2*a*x^5 - x^2 + a))/a^(1/4) - 1/4*I*log(-1/2*(2*sqr
t(x^6 + x)*(x^5 - x/sqrt(a) + 1) + 2*(I*x^6 + I*x)/a^(1/4) + (-I*a*x^10 - 2*I*a*x^5 - I*x^2 - I*a)/a^(3/4))/(a
*x^10 + 2*a*x^5 - x^2 + a))/a^(1/4) + 1/4*I*log(-1/2*(2*sqrt(x^6 + x)*(x^5 - x/sqrt(a) + 1) + 2*(-I*x^6 - I*x)
/a^(1/4) + (I*a*x^10 + 2*I*a*x^5 + I*x^2 + I*a)/a^(3/4))/(a*x^10 + 2*a*x^5 - x^2 + a))/a^(1/4)

Sympy [F]

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

[In]

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

[Out]

Integral(x*(4*x**5 - 1)/(sqrt(x*(x + 1)*(x**4 - x**3 + x**2 - x + 1))*(a*x**10 + 2*a*x**5 + a - x**2)), x)

Maxima [F]

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

[In]

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

[Out]

integrate((4*x^6 - x)/((a*x^10 + 2*a*x^5 - x^2 + a)*sqrt(x^6 + x)), x)

Giac [F]

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

[In]

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

[Out]

integrate((4*x^6 - x)/((a*x^10 + 2*a*x^5 - x^2 + a)*sqrt(x^6 + x)), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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