\(\int \frac {1+b x^3}{x^4 (-\sqrt {2} a^3 b+x^6)^2} \, dx\) [19]

Optimal result

Integrand size = 28, antiderivative size = 141 \[ \int \frac {1+b x^3}{x^4 \left (-\sqrt {2} a^3 b+x^6\right )^2} \, dx=\frac {-2 \sqrt {2} a^3 b+\sqrt {2} a^3 b^2 x^3+3 x^6}{12 a^6 b^2 x^3 \left (\sqrt {2} a^3 b-x^6\right )}+\frac {\text {arctanh}\left (\frac {x^3}{\sqrt [4]{2} a^{3/2} \sqrt {b}}\right )}{4 \sqrt [4]{2} a^{15/2} b^{5/2}}+\frac {\log (x)}{2 a^6 b}-\frac {\log \left (-\sqrt {2} a^3 b+x^6\right )}{12 a^6 b} \] Output:

1/12*(-2*2^(1/2)*a^3*b+2^(1/2)*a^3*b^2*x^3+3*x^6)/a^6/b^2/x^3/(2^(1/2)*a^3 
*b-x^6)+1/8*arctanh(1/2*x^3*2^(3/4)/a^(3/2)/b^(1/2))*2^(3/4)/a^(15/2)/b^(5 
/2)+1/2*ln(x)/a^6/b-1/12*ln(-2^(1/2)*a^3*b+x^6)/a^6/b
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.86 \[ \int \frac {1+b x^3}{x^4 \left (-\sqrt {2} a^3 b+x^6\right )^2} \, dx=\frac {-\frac {8 a^{3/2} \sqrt {b}}{x^3}+\frac {4 a^{3/2} \sqrt {b} \left (\sqrt {2} a^3 b^2+x^3\right )}{\sqrt {2} a^3 b-x^6}+24 a^{3/2} b^{3/2} \log (x)-3\ 2^{3/4} \log \left (2 \sqrt {a} \sqrt [6]{b}-2^{11/12} x\right )+3\ 2^{3/4} \log \left (2 \sqrt {a} \sqrt [6]{b}+2^{11/12} x\right )+3\ 2^{3/4} \log \left (2 a \sqrt [3]{b}-2^{11/12} \sqrt {a} \sqrt [6]{b} x+2^{5/6} x^2\right )-3\ 2^{3/4} \log \left (2 a \sqrt [3]{b}+2^{11/12} \sqrt {a} \sqrt [6]{b} x+2^{5/6} x^2\right )-4 a^{3/2} b^{3/2} \log \left (-\sqrt {2} a^3 b+x^6\right )}{48 a^{15/2} b^{5/2}} \] Input:

Integrate[(1 + b*x^3)/(x^4*(-(Sqrt[2]*a^3*b) + x^6)^2),x]
 

Output:

((-8*a^(3/2)*Sqrt[b])/x^3 + (4*a^(3/2)*Sqrt[b]*(Sqrt[2]*a^3*b^2 + x^3))/(S 
qrt[2]*a^3*b - x^6) + 24*a^(3/2)*b^(3/2)*Log[x] - 3*2^(3/4)*Log[2*Sqrt[a]* 
b^(1/6) - 2^(11/12)*x] + 3*2^(3/4)*Log[2*Sqrt[a]*b^(1/6) + 2^(11/12)*x] + 
3*2^(3/4)*Log[2*a*b^(1/3) - 2^(11/12)*Sqrt[a]*b^(1/6)*x + 2^(5/6)*x^2] - 3 
*2^(3/4)*Log[2*a*b^(1/3) + 2^(11/12)*Sqrt[a]*b^(1/6)*x + 2^(5/6)*x^2] - 4* 
a^(3/2)*b^(3/2)*Log[-(Sqrt[2]*a^3*b) + x^6])/(48*a^(15/2)*b^(5/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1803, 532, 27, 2333, 2009}

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.

\(\displaystyle \int \frac {b x^3+1}{x^4 \left (x^6-\sqrt {2} a^3 b\right )^2} \, dx\)

\(\Big \downarrow \) 1803

\(\displaystyle \frac {1}{3} \int \frac {b x^3+1}{x^6 \left (\sqrt {2} a^3 b-x^6\right )^2}dx^3\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {1}{3} \left (\frac {\sqrt {2} a^3 b^2+x^3}{4 a^6 b^2 \left (\sqrt {2} a^3 b-x^6\right )}-\frac {\int -\frac {\frac {\sqrt {2} x^6}{a^3 b}+4 b x^3+4}{2 x^6 \left (\sqrt {2} a^3 b-x^6\right )}dx^3}{2 \sqrt {2} a^3 b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {\int \frac {\frac {\sqrt {2} x^6}{a^3 b}+4 b x^3+4}{x^6 \left (\sqrt {2} a^3 b-x^6\right )}dx^3}{4 \sqrt {2} a^3 b}+\frac {\sqrt {2} a^3 b^2+x^3}{4 a^6 b^2 \left (\sqrt {2} a^3 b-x^6\right )}\right )\)

\(\Big \downarrow \) 2333

\(\displaystyle \frac {1}{3} \left (\frac {\int \left (\frac {\sqrt {2} \left (2 b x^3+3\right )}{a^3 b \left (\sqrt {2} a^3 b-x^6\right )}+\frac {2 \sqrt {2}}{a^3 x^3}+\frac {2 \sqrt {2}}{a^3 b x^6}\right )dx^3}{4 \sqrt {2} a^3 b}+\frac {\sqrt {2} a^3 b^2+x^3}{4 a^6 b^2 \left (\sqrt {2} a^3 b-x^6\right )}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (\frac {\frac {3 \sqrt [4]{2} \text {arctanh}\left (\frac {x^3}{\sqrt [4]{2} a^{3/2} \sqrt {b}}\right )}{a^{9/2} b^{3/2}}-\frac {\sqrt {2} \log \left (\sqrt {2} a^3 b-x^6\right )}{a^3}-\frac {2 \sqrt {2}}{a^3 b x^3}+\frac {2 \sqrt {2} \log \left (x^3\right )}{a^3}}{4 \sqrt {2} a^3 b}+\frac {\sqrt {2} a^3 b^2+x^3}{4 a^6 b^2 \left (\sqrt {2} a^3 b-x^6\right )}\right )\)

Input:

Int[(1 + b*x^3)/(x^4*(-(Sqrt[2]*a^3*b) + x^6)^2),x]
 

Output:

((Sqrt[2]*a^3*b^2 + x^3)/(4*a^6*b^2*(Sqrt[2]*a^3*b - x^6)) + ((-2*Sqrt[2]) 
/(a^3*b*x^3) + (3*2^(1/4)*ArcTanh[x^3/(2^(1/4)*a^(3/2)*Sqrt[b])])/(a^(9/2) 
*b^(3/2)) + (2*Sqrt[2]*Log[x^3])/a^3 - (Sqrt[2]*Log[Sqrt[2]*a^3*b - x^6])/ 
a^3)/(4*Sqrt[2]*a^3*b))/3
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

rule 1803

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x 
)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && 
 EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2009

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

rule 2333

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] 
&& PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.87

method	result	size
default	\(-\frac {1}{6 a^{6} b^{2} x^{3}}+\frac {\ln \left (x \right )}{2 a^{6} b}-\frac {\sqrt {2}\, \left (\frac {-\frac {x^{3}}{4}-\frac {\sqrt {2}\, a^{3} b^{2}}{4}}{-\frac {x^{6} \sqrt {2}}{2}+a^{3} b}+\frac {b \sqrt {2}\, \ln \left (x^{6} \sqrt {2}-2 a^{3} b \right )}{4}-\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {x^{3}}{a \sqrt {a b \sqrt {2}}}\right )}{4 a \sqrt {a b \sqrt {2}}}\right )}{6 a^{6} b^{2}}\)	\(122\)

Input:

int((b*x^3+1)/x^4/(-2^(1/2)*a^3*b+x^6)^2,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 432, normalized size of antiderivative = 3.06 \[ \int \frac {1+b x^3}{x^4 \left (-\sqrt {2} a^3 b+x^6\right )^2} \, dx=\left [-\frac {4 \, a^{7} b^{3} x^{3} + 6 \, a x^{12} - 8 \, a^{7} b^{2} - 3 \, \sqrt {\frac {1}{2}} {\left (x^{15} - 2 \, a^{6} b^{2} x^{3}\right )} \sqrt {\frac {\sqrt {2}}{a b}} \log \left (\frac {x^{12} + 2 \, \sqrt {2} a^{3} b x^{6} + 2 \, a^{6} b^{2} + 2 \, \sqrt {\frac {1}{2}} {\left (\sqrt {2} a^{2} b x^{9} + 2 \, a^{5} b^{2} x^{3}\right )} \sqrt {\frac {\sqrt {2}}{a b}}}{x^{12} - 2 \, a^{6} b^{2}}\right ) + 2 \, {\left (a b x^{15} - 2 \, a^{7} b^{3} x^{3}\right )} \log \left (x^{6} - \sqrt {2} a^{3} b\right ) - 12 \, {\left (a b x^{15} - 2 \, a^{7} b^{3} x^{3}\right )} \log \left (x\right ) + 2 \, \sqrt {2} {\left (a^{4} b^{2} x^{9} + a^{4} b x^{6}\right )}}{24 \, {\left (a^{7} b^{2} x^{15} - 2 \, a^{13} b^{4} x^{3}\right )}}, -\frac {2 \, a^{7} b^{3} x^{3} + 3 \, a x^{12} - 4 \, a^{7} b^{2} + 3 \, \sqrt {\frac {1}{2}} {\left (x^{15} - 2 \, a^{6} b^{2} x^{3}\right )} \sqrt {-\frac {\sqrt {2}}{a b}} \arctan \left (\frac {\sqrt {\frac {1}{2}} x^{3} \sqrt {-\frac {\sqrt {2}}{a b}}}{a}\right ) + {\left (a b x^{15} - 2 \, a^{7} b^{3} x^{3}\right )} \log \left (x^{6} - \sqrt {2} a^{3} b\right ) - 6 \, {\left (a b x^{15} - 2 \, a^{7} b^{3} x^{3}\right )} \log \left (x\right ) + \sqrt {2} {\left (a^{4} b^{2} x^{9} + a^{4} b x^{6}\right )}}{12 \, {\left (a^{7} b^{2} x^{15} - 2 \, a^{13} b^{4} x^{3}\right )}}\right ] \] Input:

integrate((b*x^3+1)/x^4/(-2^(1/2)*a^3*b+x^6)^2,x, algorithm="fricas")
 

Output:

[-1/24*(4*a^7*b^3*x^3 + 6*a*x^12 - 8*a^7*b^2 - 3*sqrt(1/2)*(x^15 - 2*a^6*b 
^2*x^3)*sqrt(sqrt(2)/(a*b))*log((x^12 + 2*sqrt(2)*a^3*b*x^6 + 2*a^6*b^2 + 
2*sqrt(1/2)*(sqrt(2)*a^2*b*x^9 + 2*a^5*b^2*x^3)*sqrt(sqrt(2)/(a*b)))/(x^12 
 - 2*a^6*b^2)) + 2*(a*b*x^15 - 2*a^7*b^3*x^3)*log(x^6 - sqrt(2)*a^3*b) - 1 
2*(a*b*x^15 - 2*a^7*b^3*x^3)*log(x) + 2*sqrt(2)*(a^4*b^2*x^9 + a^4*b*x^6)) 
/(a^7*b^2*x^15 - 2*a^13*b^4*x^3), -1/12*(2*a^7*b^3*x^3 + 3*a*x^12 - 4*a^7* 
b^2 + 3*sqrt(1/2)*(x^15 - 2*a^6*b^2*x^3)*sqrt(-sqrt(2)/(a*b))*arctan(sqrt( 
1/2)*x^3*sqrt(-sqrt(2)/(a*b))/a) + (a*b*x^15 - 2*a^7*b^3*x^3)*log(x^6 - sq 
rt(2)*a^3*b) - 6*(a*b*x^15 - 2*a^7*b^3*x^3)*log(x) + sqrt(2)*(a^4*b^2*x^9 
+ a^4*b*x^6))/(a^7*b^2*x^15 - 2*a^13*b^4*x^3)]
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {1+b x^3}{x^4 \left (-\sqrt {2} a^3 b+x^6\right )^2} \, dx=\text {Exception raised: PolynomialError} \] Input:

integrate((b*x**3+1)/x**4/(-2**(1/2)*a**3*b+x**6)**2,x)
 

Output:

Exception raised: PolynomialError >> 1/(606928896*sqrt(2)*_t**6*a**45*b**1 
5 + 151732224*sqrt(2)*_t**5*a**39*b**14 + 12644352*sqrt(2)*_t**4*a**33*b** 
13 - 24385536*_t**4*a**30*b**10 + 351232*sqrt(2)*_t**3*a**27*b**12 - 40642 
56*_t**3*a**24*
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00 \[ \int \frac {1+b x^3}{x^4 \left (-\sqrt {2} a^3 b+x^6\right )^2} \, dx=-\frac {\sqrt {2} a^{3} b^{2} x^{3} + 3 \, x^{6} - 2 \, \sqrt {2} a^{3} b}{12 \, {\left (a^{6} b^{2} x^{9} - \sqrt {2} a^{9} b^{3} x^{3}\right )}} - \frac {\log \left (x^{6} - \sqrt {2} a^{3} b\right )}{12 \, a^{6} b} + \frac {\log \left (x^{3}\right )}{6 \, a^{6} b} - \frac {2^{\frac {3}{4}} \log \left (\frac {x^{3} - \sqrt {\sqrt {2} a b} a}{x^{3} + \sqrt {\sqrt {2} a b} a}\right )}{16 \, \sqrt {a b} a^{7} b^{2}} \] Input:

integrate((b*x^3+1)/x^4/(-2^(1/2)*a^3*b+x^6)^2,x, algorithm="maxima")
 

Output:

-1/12*(sqrt(2)*a^3*b^2*x^3 + 3*x^6 - 2*sqrt(2)*a^3*b)/(a^6*b^2*x^9 - sqrt( 
2)*a^9*b^3*x^3) - 1/12*log(x^6 - sqrt(2)*a^3*b)/(a^6*b) + 1/6*log(x^3)/(a^ 
6*b) - 1/16*2^(3/4)*log((x^3 - sqrt(sqrt(2)*a*b)*a)/(x^3 + sqrt(sqrt(2)*a* 
b)*a))/(sqrt(a*b)*a^7*b^2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.94 \[ \int \frac {1+b x^3}{x^4 \left (-\sqrt {2} a^3 b+x^6\right )^2} \, dx=-\frac {\log \left (x^{6} - \sqrt {2} a^{3} b\right )}{12 \, a^{6} b} + \frac {\log \left ({\left | x \right |}\right )}{2 \, a^{6} b} - \frac {\sqrt {2} a^{3} b^{2} x^{3} + 3 \, x^{6} - 2 \, \sqrt {2} a^{3} b}{12 \, {\left (x^{9} - \sqrt {2} a^{3} b x^{3}\right )} a^{6} b^{2}} - \frac {\arctan \left (\frac {x^{3}}{\sqrt {a b} \sqrt {-\sqrt {2}} {\left | a \right |}}\right )}{4 \, \sqrt {a b} a^{6} b^{2} \sqrt {-\sqrt {2}} {\left | a \right |}} \] Input:

integrate((b*x^3+1)/x^4/(-2^(1/2)*a^3*b+x^6)^2,x, algorithm="giac")
 

Output:

-1/12*log(x^6 - sqrt(2)*a^3*b)/(a^6*b) + 1/2*log(abs(x))/(a^6*b) - 1/12*(s 
qrt(2)*a^3*b^2*x^3 + 3*x^6 - 2*sqrt(2)*a^3*b)/((x^9 - sqrt(2)*a^3*b*x^3)*a 
^6*b^2) - 1/4*arctan(x^3/(sqrt(a*b)*sqrt(-sqrt(2))*abs(a)))/(sqrt(a*b)*a^6 
*b^2*sqrt(-sqrt(2))*abs(a))
 

Mupad [B] (verification not implemented)

Time = 10.82 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.78 \[ \int \frac {1+b x^3}{x^4 \left (-\sqrt {2} a^3 b+x^6\right )^2} \, dx=\frac {\ln \left (x\right )}{2\,a^6\,b}-\frac {\frac {\sqrt {2}\,x^3}{12\,a^3}-\frac {\sqrt {2}}{6\,a^3\,b}+\frac {x^6}{4\,a^6\,b^2}}{x^9-\sqrt {2}\,a^3\,b\,x^3}+\frac {\sqrt {2}\,\ln \left (3\,2^{1/4}\,\sqrt {a^{15}\,b^5}-14\,\sqrt {2}\,a^9\,b^4+3\,a^6\,b^2\,x^3-14\,2^{1/4}\,b\,x^3\,\sqrt {a^{15}\,b^5}\right )\,\left (3\,\sqrt {\sqrt {2}\,a^{15}\,b^5}-2\,\sqrt {2}\,a^9\,b^4\right )}{48\,a^{15}\,b^5}-\frac {\sqrt {2}\,\ln \left (3\,2^{1/4}\,\sqrt {a^{15}\,b^5}+14\,\sqrt {2}\,a^9\,b^4-3\,a^6\,b^2\,x^3-14\,2^{1/4}\,b\,x^3\,\sqrt {a^{15}\,b^5}\right )\,\left (3\,\sqrt {\sqrt {2}\,a^{15}\,b^5}+2\,\sqrt {2}\,a^9\,b^4\right )}{48\,a^{15}\,b^5} \] Input:

int((b*x^3 + 1)/(x^4*(x^6 - 2^(1/2)*a^3*b)^2),x)
 

Output:

log(x)/(2*a^6*b) - ((2^(1/2)*x^3)/(12*a^3) - 2^(1/2)/(6*a^3*b) + x^6/(4*a^ 
6*b^2))/(x^9 - 2^(1/2)*a^3*b*x^3) + (2^(1/2)*log(3*2^(1/4)*(a^15*b^5)^(1/2 
) - 14*2^(1/2)*a^9*b^4 + 3*a^6*b^2*x^3 - 14*2^(1/4)*b*x^3*(a^15*b^5)^(1/2) 
)*(3*(2^(1/2)*a^15*b^5)^(1/2) - 2*2^(1/2)*a^9*b^4))/(48*a^15*b^5) - (2^(1/ 
2)*log(3*2^(1/4)*(a^15*b^5)^(1/2) + 14*2^(1/2)*a^9*b^4 - 3*a^6*b^2*x^3 - 1 
4*2^(1/4)*b*x^3*(a^15*b^5)^(1/2))*(3*(2^(1/2)*a^15*b^5)^(1/2) + 2*2^(1/2)* 
a^9*b^4))/(48*a^15*b^5)
 

Reduce [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 2054, normalized size of antiderivative = 14.57 \[ \int \frac {1+b x^3}{x^4 \left (-\sqrt {2} a^3 b+x^6\right )^2} \, dx =\text {Too large to display} \] Input:

int((b*x^3+1)/x^4/(-2^(1/2)*a^3*b+x^6)^2,x)
 

Output:

( - 12*sqrt(b)*sqrt(a)*sqrt(2)*2**(1/6)*2**(1/12)*atan((b**(1/6)*sqrt(a)*2 
**(1/12)*sqrt(3) - 2*x)/(b**(1/6)*sqrt(a)*2**(1/12)))*a**6*b**2*x**3 + 6*s 
qrt(b)*sqrt(a)*sqrt(2)*2**(1/6)*2**(1/12)*atan((b**(1/6)*sqrt(a)*2**(1/12) 
*sqrt(3) - 2*x)/(b**(1/6)*sqrt(a)*2**(1/12)))*x**15 + 12*sqrt(b)*sqrt(a)*2 
**(2/3)*2**(1/12)*atan((b**(1/6)*sqrt(a)*2**(1/12)*sqrt(3) - 2*x)/(b**(1/6 
)*sqrt(a)*2**(1/12)))*a**6*b**2*x**3 - 6*sqrt(b)*sqrt(a)*2**(2/3)*2**(1/12 
)*atan((b**(1/6)*sqrt(a)*2**(1/12)*sqrt(3) - 2*x)/(b**(1/6)*sqrt(a)*2**(1/ 
12)))*x**15 + 12*sqrt(b)*sqrt(a)*sqrt(2)*2**(1/6)*2**(1/12)*atan((b**(1/6) 
*sqrt(a)*2**(1/12)*sqrt(3) + 2*x)/(b**(1/6)*sqrt(a)*2**(1/12)))*a**6*b**2* 
x**3 - 6*sqrt(b)*sqrt(a)*sqrt(2)*2**(1/6)*2**(1/12)*atan((b**(1/6)*sqrt(a) 
*2**(1/12)*sqrt(3) + 2*x)/(b**(1/6)*sqrt(a)*2**(1/12)))*x**15 - 12*sqrt(b) 
*sqrt(a)*2**(2/3)*2**(1/12)*atan((b**(1/6)*sqrt(a)*2**(1/12)*sqrt(3) + 2*x 
)/(b**(1/6)*sqrt(a)*2**(1/12)))*a**6*b**2*x**3 + 6*sqrt(b)*sqrt(a)*2**(2/3 
)*2**(1/12)*atan((b**(1/6)*sqrt(a)*2**(1/12)*sqrt(3) + 2*x)/(b**(1/6)*sqrt 
(a)*2**(1/12)))*x**15 - 12*sqrt(b)*sqrt(a)*sqrt(2)*2**(1/6)*2**(1/12)*atan 
(x/(b**(1/6)*sqrt(a)*2**(1/12)))*a**6*b**2*x**3 + 6*sqrt(b)*sqrt(a)*sqrt(2 
)*2**(1/6)*2**(1/12)*atan(x/(b**(1/6)*sqrt(a)*2**(1/12)))*x**15 + 12*sqrt( 
b)*sqrt(a)*2**(2/3)*2**(1/12)*atan(x/(b**(1/6)*sqrt(a)*2**(1/12)))*a**6*b* 
*2*x**3 - 6*sqrt(b)*sqrt(a)*2**(2/3)*2**(1/12)*atan(x/(b**(1/6)*sqrt(a)*2* 
*(1/12)))*x**15 + 6*sqrt(b)*sqrt(a)*sqrt(2)*2**(1/6)*2**(1/12)*log( - b...