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

3.7.16.1 Optimal result

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

output

-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)
 

3.7.16.2 Mathematica [F]

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

input

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

output

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

3.7.16.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

\(\displaystyle \frac {2 \sqrt {x} \sqrt {x^5+1} \int \frac {x \left (1-4 x^5\right )}{\sqrt {x^5+1} \left (x^{10}+2 x^5-a x^2+1\right )}d\sqrt {x}}{\sqrt {x^6+x}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {2 \sqrt {x} \sqrt {x^5+1} \int \left (\frac {x}{\sqrt {x^5+1} \left (x^{10}+2 x^5-a x^2+1\right )}-\frac {4 x^6}{\sqrt {x^5+1} \left (x^{10}+2 x^5-a x^2+1\right )}\right )d\sqrt {x}}{\sqrt {x^6+x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {x} \sqrt {x^5+1} \left (\int \frac {x}{\sqrt {x^5+1} \left (x^{10}+2 x^5-a x^2+1\right )}d\sqrt {x}-4 \int \frac {x^6}{\sqrt {x^5+1} \left (x^{10}+2 x^5-a x^2+1\right )}d\sqrt {x}\right )}{\sqrt {x^6+x}}\)

input

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

output

$Aborted
 

3.7.16.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.7.16.4 Maple [A] (verified)

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

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

input

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

output

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*ar 
ctan(1/a^(1/4)/x*(x^6+x)^(1/2)))
 

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

input

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

output

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))
 

3.7.16.6 Sympy [F]

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

input

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

output

-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) - Integral(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)
 

3.7.16.7 Maxima [F]

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

input

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

output

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

3.7.16.8 Giac [F]

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

input

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

output

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

3.7.16.9 Mupad [F(-1)]

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

input

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

output

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