\(\int \frac {1}{x^{10} (a+b x^3+c x^6)^{3/2}} \, dx\) [227]

Optimal result

Integrand size = 20, antiderivative size = 256 \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {2 \left (b^2-2 a c+b c x^3\right )}{3 a \left (b^2-4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{9 a^2 \left (b^2-4 a c\right ) x^9}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{36 a^3 \left (b^2-4 a c\right ) x^6}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a+b x^3+c x^6}}{72 a^4 \left (b^2-4 a c\right ) x^3}+\frac {5 b \left (7 b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{48 a^{9/2}} \] Output:

2/3*(b*c*x^3-2*a*c+b^2)/a/(-4*a*c+b^2)/x^9/(c*x^6+b*x^3+a)^(1/2)-1/9*(-16* 
a*c+7*b^2)*(c*x^6+b*x^3+a)^(1/2)/a^2/(-4*a*c+b^2)/x^9+1/36*b*(-116*a*c+35* 
b^2)*(c*x^6+b*x^3+a)^(1/2)/a^3/(-4*a*c+b^2)/x^6-1/72*(256*a^2*c^2-460*a*b^ 
2*c+105*b^4)*(c*x^6+b*x^3+a)^(1/2)/a^4/(-4*a*c+b^2)/x^3+5/48*b*(-12*a*c+7* 
b^2)*arctanh(1/2*(b*x^3+2*a)/a^(1/2)/(c*x^6+b*x^3+a)^(1/2))/a^(9/2)
 

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {-32 a^4 c+105 b^4 x^9 \left (b+c x^3\right )+5 a b^2 x^6 \left (7 b^2-106 b c x^3-92 c^2 x^6\right )+8 a^3 \left (b^2+7 b c x^3+16 c^2 x^6\right )+2 a^2 x^3 \left (-7 b^3-86 b^2 c x^3+244 b c^2 x^6+128 c^3 x^9\right )}{72 a^4 \left (-b^2+4 a c\right ) x^9 \sqrt {a+b x^3+c x^6}}+\frac {5 b \left (-7 b^2+12 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^3-\sqrt {a+b x^3+c x^6}}{\sqrt {a}}\right )}{24 a^{9/2}} \] Input:

Integrate[1/(x^10*(a + b*x^3 + c*x^6)^(3/2)),x]
 

Output:

(-32*a^4*c + 105*b^4*x^9*(b + c*x^3) + 5*a*b^2*x^6*(7*b^2 - 106*b*c*x^3 - 
92*c^2*x^6) + 8*a^3*(b^2 + 7*b*c*x^3 + 16*c^2*x^6) + 2*a^2*x^3*(-7*b^3 - 8 
6*b^2*c*x^3 + 244*b*c^2*x^6 + 128*c^3*x^9))/(72*a^4*(-b^2 + 4*a*c)*x^9*Sqr 
t[a + b*x^3 + c*x^6]) + (5*b*(-7*b^2 + 12*a*c)*ArcTanh[(Sqrt[c]*x^3 - Sqrt 
[a + b*x^3 + c*x^6])/Sqrt[a]])/(24*a^(9/2))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1693, 1165, 27, 1237, 27, 1237, 27, 1228, 1154, 219}

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.

\(\displaystyle \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1693

\(\displaystyle \frac {1}{3} \int \frac {1}{x^{12} \left (c x^6+b x^3+a\right )^{3/2}}dx^3\)

\(\Big \downarrow \) 1165

\(\displaystyle \frac {1}{3} \left (\frac {2 \left (-2 a c+b^2+b c x^3\right )}{a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {2 \int -\frac {6 b c x^3+7 b^2-16 a c}{2 x^{12} \sqrt {c x^6+b x^3+a}}dx^3}{a \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {\int \frac {6 b c x^3+7 b^2-16 a c}{x^{12} \sqrt {c x^6+b x^3+a}}dx^3}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^3\right )}{a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {\int \frac {4 c \left (7 b^2-16 a c\right ) x^3+b \left (35 b^2-116 a c\right )}{2 x^9 \sqrt {c x^6+b x^3+a}}dx^3}{3 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{3 a x^9}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^3\right )}{a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {\int \frac {4 c \left (7 b^2-16 a c\right ) x^3+b \left (35 b^2-116 a c\right )}{x^9 \sqrt {c x^6+b x^3+a}}dx^3}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{3 a x^9}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^3\right )}{a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\right )\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {-\frac {\int \frac {105 b^4-460 a c b^2+2 c \left (35 b^2-116 a c\right ) x^3 b+256 a^2 c^2}{2 x^6 \sqrt {c x^6+b x^3+a}}dx^3}{2 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{2 a x^6}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{3 a x^9}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^3\right )}{a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {-\frac {\int \frac {105 b^4-460 a c b^2+2 c \left (35 b^2-116 a c\right ) x^3 b+256 a^2 c^2}{x^6 \sqrt {c x^6+b x^3+a}}dx^3}{4 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{2 a x^6}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{3 a x^9}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^3\right )}{a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\right )\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {-\frac {-\frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \int \frac {1}{x^3 \sqrt {c x^6+b x^3+a}}dx^3}{2 a}-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a+b x^3+c x^6}}{a x^3}}{4 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{2 a x^6}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{3 a x^9}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^3\right )}{a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\right )\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {-\frac {\frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \int \frac {1}{4 a-x^6}d\frac {b x^3+2 a}{\sqrt {c x^6+b x^3+a}}}{a}-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a+b x^3+c x^6}}{a x^3}}{4 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{2 a x^6}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{3 a x^9}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^3\right )}{a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {-\frac {\frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x^3}{2 \sqrt {a} \sqrt {a+b x^3+c x^6}}\right )}{2 a^{3/2}}-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a+b x^3+c x^6}}{a x^3}}{4 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a+b x^3+c x^6}}{2 a x^6}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a+b x^3+c x^6}}{3 a x^9}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x^3\right )}{a x^9 \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\right )\)

Input:

Int[1/(x^10*(a + b*x^3 + c*x^6)^(3/2)),x]
 

Output:

((2*(b^2 - 2*a*c + b*c*x^3))/(a*(b^2 - 4*a*c)*x^9*Sqrt[a + b*x^3 + c*x^6]) 
 + (-1/3*((7*b^2 - 16*a*c)*Sqrt[a + b*x^3 + c*x^6])/(a*x^9) - (-1/2*(b*(35 
*b^2 - 116*a*c)*Sqrt[a + b*x^3 + c*x^6])/(a*x^6) - (-(((105*b^4 - 460*a*b^ 
2*c + 256*a^2*c^2)*Sqrt[a + b*x^3 + c*x^6])/(a*x^3)) + (15*b*(7*b^2 - 12*a 
*c)*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6] 
)])/(2*a^(3/2)))/(4*a))/(6*a))/(a*(b^2 - 4*a*c)))/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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1693

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

Maple [F]

\[\int \frac {1}{x^{10} \left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}d x\]

Input:

int(1/x^10/(c*x^6+b*x^3+a)^(3/2),x)
 

Output:

int(1/x^10/(c*x^6+b*x^3+a)^(3/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 705, normalized size of antiderivative = 2.75 \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/x^10/(c*x^6+b*x^3+a)^(3/2),x, algorithm="fricas")
 

Output:

[-1/288*(15*((7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*x^15 + (7*b^6 - 40*a* 
b^4*c + 48*a^2*b^2*c^2)*x^12 + (7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*x^9 
)*sqrt(a)*log(-((b^2 + 4*a*c)*x^6 + 8*a*b*x^3 - 4*sqrt(c*x^6 + b*x^3 + a)* 
(b*x^3 + 2*a)*sqrt(a) + 8*a^2)/x^6) + 4*((105*a*b^4*c - 460*a^2*b^2*c^2 + 
256*a^3*c^3)*x^12 + (105*a*b^5 - 530*a^2*b^3*c + 488*a^3*b*c^2)*x^9 + (35* 
a^2*b^4 - 172*a^3*b^2*c + 128*a^4*c^2)*x^6 + 8*a^4*b^2 - 32*a^5*c - 14*(a^ 
3*b^3 - 4*a^4*b*c)*x^3)*sqrt(c*x^6 + b*x^3 + a))/((a^5*b^2*c - 4*a^6*c^2)* 
x^15 + (a^5*b^3 - 4*a^6*b*c)*x^12 + (a^6*b^2 - 4*a^7*c)*x^9), -1/144*(15*( 
(7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*x^15 + (7*b^6 - 40*a*b^4*c + 48*a^ 
2*b^2*c^2)*x^12 + (7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*x^9)*sqrt(-a)*ar 
ctan(1/2*sqrt(c*x^6 + b*x^3 + a)*(b*x^3 + 2*a)*sqrt(-a)/(a*c*x^6 + a*b*x^3 
 + a^2)) + 2*((105*a*b^4*c - 460*a^2*b^2*c^2 + 256*a^3*c^3)*x^12 + (105*a* 
b^5 - 530*a^2*b^3*c + 488*a^3*b*c^2)*x^9 + (35*a^2*b^4 - 172*a^3*b^2*c + 1 
28*a^4*c^2)*x^6 + 8*a^4*b^2 - 32*a^5*c - 14*(a^3*b^3 - 4*a^4*b*c)*x^3)*sqr 
t(c*x^6 + b*x^3 + a))/((a^5*b^2*c - 4*a^6*c^2)*x^15 + (a^5*b^3 - 4*a^6*b*c 
)*x^12 + (a^6*b^2 - 4*a^7*c)*x^9)]
 

Sympy [F]

\[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int \frac {1}{x^{10} \left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/x**10/(c*x**6+b*x**3+a)**(3/2),x)
 

Output:

Integral(1/(x**10*(a + b*x**3 + c*x**6)**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x^10/(c*x^6+b*x^3+a)^(3/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [F]

\[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}} x^{10}} \,d x } \] Input:

integrate(1/x^10/(c*x^6+b*x^3+a)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^10), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\int \frac {1}{x^{10}\,{\left (c\,x^6+b\,x^3+a\right )}^{3/2}} \,d x \] Input:

int(1/(x^10*(a + b*x^3 + c*x^6)^(3/2)),x)
 

Output:

int(1/(x^10*(a + b*x^3 + c*x^6)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\text {too large to display} \] Input:

int(1/x^10/(c*x^6+b*x^3+a)^(3/2),x)
 

Output:

( - 3072*sqrt(a + b*x**3 + c*x**6)*a**7*c**4 - 3072*sqrt(a + b*x**3 + c*x* 
*6)*a**6*b**2*c**3 - 6144*sqrt(a + b*x**3 + c*x**6)*a**6*b*c**4*x**3 + 353 
28*sqrt(a + b*x**3 + c*x**6)*a**6*c**5*x**6 + 5440*sqrt(a + b*x**3 + c*x** 
6)*a**5*b**4*c**2 + 11136*sqrt(a + b*x**3 + c*x**6)*a**5*b**3*c**3*x**3 + 
16128*sqrt(a + b*x**3 + c*x**6)*a**5*b**2*c**4*x**6 + 87168*sqrt(a + b*x** 
3 + c*x**6)*a**5*b*c**5*x**9 + 36096*sqrt(a + b*x**3 + c*x**6)*a**5*c**6*x 
**12 - 1120*sqrt(a + b*x**3 + c*x**6)*a**4*b**6*c + 12160*sqrt(a + b*x**3 
+ c*x**6)*a**4*b**5*c**2*x**3 - 92960*sqrt(a + b*x**3 + c*x**6)*a**4*b**4* 
c**3*x**6 + 63840*sqrt(a + b*x**3 + c*x**6)*a**4*b**3*c**4*x**9 + 70080*sq 
rt(a + b*x**3 + c*x**6)*a**4*b**2*c**5*x**12 - 15400*sqrt(a + b*x**3 + c*x 
**6)*a**3*b**7*c*x**3 + 8800*sqrt(a + b*x**3 + c*x**6)*a**3*b**6*c**2*x**6 
 - 217040*sqrt(a + b*x**3 + c*x**6)*a**3*b**5*c**3*x**9 - 108640*sqrt(a + 
b*x**3 + c*x**6)*a**3*b**4*c**4*x**12 + 2940*sqrt(a + b*x**3 + c*x**6)*a** 
2*b**9*x**3 + 32620*sqrt(a + b*x**3 + c*x**6)*a**2*b**8*c*x**6 + 27020*sqr 
t(a + b*x**3 + c*x**6)*a**2*b**7*c**2*x**9 - 139160*sqrt(a + b*x**3 + c*x* 
*6)*a**2*b**6*c**3*x**12 - 7350*sqrt(a + b*x**3 + c*x**6)*a*b**10*x**6 + 3 
6750*sqrt(a + b*x**3 + c*x**6)*a*b**9*c*x**9 + 102900*sqrt(a + b*x**3 + c* 
x**6)*a*b**8*c**2*x**12 + 34560*sqrt(a)*log(sqrt(a + b*x**3 + c*x**6) - sq 
rt(a))*a**5*b*c**5*x**9 + 14400*sqrt(a)*log(sqrt(a + b*x**3 + c*x**6) - sq 
rt(a))*a**4*b**3*c**4*x**9 + 34560*sqrt(a)*log(sqrt(a + b*x**3 + c*x**6...