\(\int \frac {x-4 x^6}{\sqrt {x+x^6} (1-a x^2+2 x^5+x^{10})} \, dx\) [616]

Optimal result

Integrand size = 35, antiderivative size = 48 \[ \int \frac {x-4 x^6}{\sqrt {x+x^6} \left (1-a x^2+2 x^5+x^{10}\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt {x+x^6}}\right )}{a^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt {x+x^6}}\right )}{a^{3/4}} \]

[Out]

-arctan(a^(1/4)*x/(x^6+x)^(1/2))/a^(3/4)+arctanh(a^(1/4)*x/(x^6+x)^(1/2))/a^(3/4)

Rubi [F]

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

[In]

Int[(x - 4*x^6)/(Sqrt[x + x^6]*(1 - a*x^2 + 2*x^5 + x^10)),x]

[Out]

(2*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^2/(Sqrt[1 + x^10]*(1 - a*x^4 + 2*x^10 + x^20)), x], x, Sqrt
[x]])/Sqrt[x + x^6] - (8*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^12/(Sqrt[1 + x^10]*(1 - a*x^4 + 2*x^1
0 + 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 (1-a x^2+2 x^5+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 (1-a x^2+2 x^5+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 (1-a x^4+2 x^{10}+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 (1-a x^4+2 x^{10}+x^{20}\right )}-\frac {4 x^{12}}{\sqrt {1+x^{10}} \left (1-a x^4+2 x^{10}+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 (1-a x^4+2 x^{10}+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 (1-a x^4+2 x^{10}+x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}} \\ \end{align*}

Mathematica [F]

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

[In]

Integrate[(x - 4*x^6)/(Sqrt[x + x^6]*(1 - a*x^2 + 2*x^5 + x^10)),x]

[Out]

Integrate[(x - 4*x^6)/(Sqrt[x + x^6]*(1 - a*x^2 + 2*x^5 + x^10)), x]

Maple [A] (verified)

Time = 2.75 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.23

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {-a^{\frac {1}{4}} x -\sqrt {x^{6}+x}}{a^{\frac {1}{4}} x -\sqrt {x^{6}+x}}\right )+2 \arctan \left (\frac {\sqrt {x^{6}+x}}{a^{\frac {1}{4}} x}\right )}{2 a^{\frac {3}{4}}}\)	\(59\)

[In]

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

[Out]

1/2/a^(3/4)*(ln((-a^(1/4)*x-(x^6+x)^(1/2))/(a^(1/4)*x-(x^6+x)^(1/2)))+2*arctan(1/a^(1/4)/x*(x^6+x)^(1/2)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.65 (sec) , antiderivative size = 419, normalized size of antiderivative = 8.73 \[ \int \frac {x-4 x^6}{\sqrt {x+x^6} \left (1-a x^2+2 x^5+x^{10}\right )} \, dx=\frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (-\frac {{\left (a^{2} x^{10} + 2 \, a^{2} x^{5} + a^{3} x^{2} + a^{2}\right )} \frac {1}{a^{3}}^{\frac {3}{4}} + 2 \, \sqrt {x^{6} + x} {\left (x^{5} + a^{2} \sqrt {\frac {1}{a^{3}}} x + 1\right )} + 2 \, {\left (a x^{6} + a x\right )} \frac {1}{a^{3}}^{\frac {1}{4}}}{2 \, {\left (x^{10} + 2 \, x^{5} - a x^{2} + 1\right )}}\right ) - \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {{\left (a^{2} x^{10} + 2 \, a^{2} x^{5} + a^{3} x^{2} + a^{2}\right )} \frac {1}{a^{3}}^{\frac {3}{4}} - 2 \, \sqrt {x^{6} + x} {\left (x^{5} + a^{2} \sqrt {\frac {1}{a^{3}}} x + 1\right )} + 2 \, {\left (a x^{6} + a x\right )} \frac {1}{a^{3}}^{\frac {1}{4}}}{2 \, {\left (x^{10} + 2 \, x^{5} - a x^{2} + 1\right )}}\right ) - \frac {1}{4} i \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (-\frac {{\left (i \, a^{2} x^{10} + 2 i \, a^{2} x^{5} + i \, a^{3} x^{2} + i \, a^{2}\right )} \frac {1}{a^{3}}^{\frac {3}{4}} + 2 \, \sqrt {x^{6} + x} {\left (x^{5} - a^{2} \sqrt {\frac {1}{a^{3}}} x + 1\right )} + 2 \, {\left (-i \, a x^{6} - i \, a x\right )} \frac {1}{a^{3}}^{\frac {1}{4}}}{2 \, {\left (x^{10} + 2 \, x^{5} - a x^{2} + 1\right )}}\right ) + \frac {1}{4} i \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (-\frac {{\left (-i \, a^{2} x^{10} - 2 i \, a^{2} x^{5} - i \, a^{3} x^{2} - i \, a^{2}\right )} \frac {1}{a^{3}}^{\frac {3}{4}} + 2 \, \sqrt {x^{6} + x} {\left (x^{5} - a^{2} \sqrt {\frac {1}{a^{3}}} x + 1\right )} + 2 \, {\left (i \, a x^{6} + i \, a x\right )} \frac {1}{a^{3}}^{\frac {1}{4}}}{2 \, {\left (x^{10} + 2 \, x^{5} - a x^{2} + 1\right )}}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

-Integral(-x/(-a*x**2*sqrt(x**6 + x) + x**10*sqrt(x**6 + x) + 2*x**5*sqrt(x**6 + x) + sqrt(x**6 + x)), x) - In
tegral(4*x**6/(-a*x**2*sqrt(x**6 + x) + x**10*sqrt(x**6 + x) + 2*x**5*sqrt(x**6 + x) + sqrt(x**6 + x)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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