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

3.7.17.1 Optimal result

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

output

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)
 

3.7.17.2 Mathematica [F]

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

input

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

output

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

3.7.17.3 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.

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

\(\Big \downarrow \) 2027

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.7.17.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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] 
]
 

3.7.17.4 Maple [A] (verified)

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

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

input

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

output

-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)))
 

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

input

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

output

-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*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^1 
0 + 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)
 

3.7.17.6 Sympy [F]

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

input

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

output

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)
 

3.7.17.7 Maxima [F]

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

input

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

output

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

3.7.17.8 Giac [F]

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

input

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

output

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

3.7.17.9 Mupad [F(-1)]

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

input

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

output

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