\(\int \frac {1}{x^{10} (a+b x^6)^2 \sqrt {c+d x^6}} \, dx\) [83]

Optimal result

Integrand size = 24, antiderivative size = 208 \[ \int \frac {1}{x^{10} \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=-\frac {(5 b c-2 a d) \sqrt {c+d x^6}}{18 a^2 c (b c-a d) x^9}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^6}}{18 a^3 c^2 (b c-a d) x^3}+\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^9 \left (a+b x^6\right )}+\frac {b^2 (5 b c-6 a d) \arctan \left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 a^{7/2} (b c-a d)^{3/2}} \] Output:

-1/18*(-2*a*d+5*b*c)*(d*x^6+c)^(1/2)/a^2/c/(-a*d+b*c)/x^9+1/18*(-4*a^2*d^2 
-8*a*b*c*d+15*b^2*c^2)*(d*x^6+c)^(1/2)/a^3/c^2/(-a*d+b*c)/x^3+1/6*b*(d*x^6 
+c)^(1/2)/a/(-a*d+b*c)/x^9/(b*x^6+a)+1/6*b^2*(-6*a*d+5*b*c)*arctan((-a*d+b 
*c)^(1/2)*x^3/a^(1/2)/(d*x^6+c)^(1/2))/a^(7/2)/(-a*d+b*c)^(3/2)
 

Mathematica [A] (verified)

Time = 2.17 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^{10} \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=-\frac {\sqrt {c+d x^6} \left (15 b^3 c^2 x^{12}+2 a b^2 c x^6 \left (5 c-4 d x^6\right )+2 a^3 d \left (c-2 d x^6\right )-2 a^2 b \left (c^2+3 c d x^6+2 d^2 x^{12}\right )\right )}{18 a^3 c^2 (-b c+a d) x^9 \left (a+b x^6\right )}+\frac {b^2 (5 b c-6 a d) \arctan \left (\frac {a \sqrt {d}+b x^3 \left (\sqrt {d} x^3+\sqrt {c+d x^6}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{6 a^{7/2} (b c-a d)^{3/2}} \] Input:

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

Output:

-1/18*(Sqrt[c + d*x^6]*(15*b^3*c^2*x^12 + 2*a*b^2*c*x^6*(5*c - 4*d*x^6) + 
2*a^3*d*(c - 2*d*x^6) - 2*a^2*b*(c^2 + 3*c*d*x^6 + 2*d^2*x^12)))/(a^3*c^2* 
(-(b*c) + a*d)*x^9*(a + b*x^6)) + (b^2*(5*b*c - 6*a*d)*ArcTan[(a*Sqrt[d] + 
 b*x^3*(Sqrt[d]*x^3 + Sqrt[c + d*x^6]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(6*a^( 
7/2)*(b*c - a*d)^(3/2))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {965, 374, 25, 445, 445, 27, 291, 218}

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.

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

\(\Big \downarrow \) 965

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

\(\Big \downarrow \) 374

\(\displaystyle \frac {1}{3} \left (\frac {b \sqrt {c+d x^6}}{2 a x^9 \left (a+b x^6\right ) (b c-a d)}-\frac {\int -\frac {4 b d x^6+5 b c-2 a d}{x^{12} \left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{2 a (b c-a d)}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \left (\frac {\int \frac {4 b d x^6+5 b c-2 a d}{x^{12} \left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^6}}{2 a x^9 \left (a+b x^6\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {\int \frac {2 b d (5 b c-2 a d) x^6+15 b^2 c^2-4 a^2 d^2-8 a b c d}{x^6 \left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{3 a c}-\frac {\sqrt {c+d x^6} (5 b c-2 a d)}{3 a c x^9}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^6}}{2 a x^9 \left (a+b x^6\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {-\frac {\int \frac {3 b^2 c^2 (5 b c-6 a d)}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{a c}-\frac {\sqrt {c+d x^6} \left (\frac {15 b^2 c}{a}-\frac {4 a d^2}{c}-8 b d\right )}{x^3}}{3 a c}-\frac {\sqrt {c+d x^6} (5 b c-2 a d)}{3 a c x^9}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^6}}{2 a x^9 \left (a+b x^6\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {-\frac {3 b^2 c (5 b c-6 a d) \int \frac {1}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{a}-\frac {\sqrt {c+d x^6} \left (\frac {15 b^2 c}{a}-\frac {4 a d^2}{c}-8 b d\right )}{x^3}}{3 a c}-\frac {\sqrt {c+d x^6} (5 b c-2 a d)}{3 a c x^9}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^6}}{2 a x^9 \left (a+b x^6\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {-\frac {3 b^2 c (5 b c-6 a d) \int \frac {1}{a-(a d-b c) x^6}d\frac {x^3}{\sqrt {d x^6+c}}}{a}-\frac {\sqrt {c+d x^6} \left (\frac {15 b^2 c}{a}-\frac {4 a d^2}{c}-8 b d\right )}{x^3}}{3 a c}-\frac {\sqrt {c+d x^6} (5 b c-2 a d)}{3 a c x^9}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^6}}{2 a x^9 \left (a+b x^6\right ) (b c-a d)}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{3} \left (\frac {-\frac {-\frac {3 b^2 c (5 b c-6 a d) \arctan \left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^6} \left (\frac {15 b^2 c}{a}-\frac {4 a d^2}{c}-8 b d\right )}{x^3}}{3 a c}-\frac {\sqrt {c+d x^6} (5 b c-2 a d)}{3 a c x^9}}{2 a (b c-a d)}+\frac {b \sqrt {c+d x^6}}{2 a x^9 \left (a+b x^6\right ) (b c-a d)}\right )\)

Input:

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

Output:

((b*Sqrt[c + d*x^6])/(2*a*(b*c - a*d)*x^9*(a + b*x^6)) + (-1/3*((5*b*c - 2 
*a*d)*Sqrt[c + d*x^6])/(a*c*x^9) - (-((((15*b^2*c)/a - 8*b*d - (4*a*d^2)/c 
)*Sqrt[c + d*x^6])/x^3) - (3*b^2*c*(5*b*c - 6*a*d)*ArcTan[(Sqrt[b*c - a*d] 
*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(a^(3/2)*Sqrt[b*c - a*d]))/(3*a*c))/(2*a 
*(b*c - a*d)))/3
 

Defintions of rubi rules used

rule 25

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

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 374

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
 + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - 
a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, 
c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, 
 c, d, e, m, 2, p, q, x]
 

rule 445

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

rule 965

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), 
 x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 
 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free 
Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 13.88 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {d \,x^{6}+c}\, \left (-2 a d \,x^{6}-6 b c \,x^{6}+a c \right )}{3 x^{9}}-\frac {b^{2} c^{2} \left (\frac {\sqrt {d \,x^{6}+c}\, b \,x^{3}}{b \,x^{6}+a}-\frac {\left (6 a d -5 c b \right ) \operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{6}+c}}{x^{3} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}\right )}{2 \left (a d -c b \right )}}{3 a^{3} c^{2}}\)	\(134\)

Input:

int(1/x^10/(b*x^6+a)^2/(d*x^6+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/3/a^3*(-1/3*(d*x^6+c)^(1/2)*(-2*a*d*x^6-6*b*c*x^6+a*c)/x^9-1/2*b^2*c^2/( 
a*d-b*c)*((d*x^6+c)^(1/2)*b*x^3/(b*x^6+a)-(6*a*d-5*b*c)/(a*(a*d-b*c))^(1/2 
)*arctanh(a*(d*x^6+c)^(1/2)/x^3/(a*(a*d-b*c))^(1/2))))/c^2
 

Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 760, normalized size of antiderivative = 3.65 \[ \int \frac {1}{x^{10} \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/72*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^15 + (5*a*b^3*c^3 - 6*a^2*b^2*c^2 
*d)*x^9)*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^12 
- 2*(3*a*b*c^2 - 4*a^2*c*d)*x^6 + a^2*c^2 - 4*((b*c - 2*a*d)*x^9 - a*c*x^3 
)*sqrt(d*x^6 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^12 + 2*a*b*x^6 + a^2)) - 4* 
((15*a*b^4*c^3 - 23*a^2*b^3*c^2*d + 4*a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^12 - 
2*a^3*b^2*c^3 + 4*a^4*b*c^2*d - 2*a^5*c*d^2 + 2*(5*a^2*b^3*c^3 - 8*a^3*b^2 
*c^2*d + a^4*b*c*d^2 + 2*a^5*d^3)*x^6)*sqrt(d*x^6 + c))/((a^4*b^3*c^4 - 2* 
a^5*b^2*c^3*d + a^6*b*c^2*d^2)*x^15 + (a^5*b^2*c^4 - 2*a^6*b*c^3*d + a^7*c 
^2*d^2)*x^9), 1/36*(3*((5*b^4*c^3 - 6*a*b^3*c^2*d)*x^15 + (5*a*b^3*c^3 - 6 
*a^2*b^2*c^2*d)*x^9)*sqrt(a*b*c - a^2*d)*arctan(1/2*((b*c - 2*a*d)*x^6 - a 
*c)*sqrt(d*x^6 + c)*sqrt(a*b*c - a^2*d)/((a*b*c*d - a^2*d^2)*x^9 + (a*b*c^ 
2 - a^2*c*d)*x^3)) + 2*((15*a*b^4*c^3 - 23*a^2*b^3*c^2*d + 4*a^3*b^2*c*d^2 
 + 4*a^4*b*d^3)*x^12 - 2*a^3*b^2*c^3 + 4*a^4*b*c^2*d - 2*a^5*c*d^2 + 2*(5* 
a^2*b^3*c^3 - 8*a^3*b^2*c^2*d + a^4*b*c*d^2 + 2*a^5*d^3)*x^6)*sqrt(d*x^6 + 
 c))/((a^4*b^3*c^4 - 2*a^5*b^2*c^3*d + a^6*b*c^2*d^2)*x^15 + (a^5*b^2*c^4 
- 2*a^6*b*c^3*d + a^7*c^2*d^2)*x^9)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^{10} \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {-2 \sqrt {d \,x^{6}+c}\, a^{2} c d +4 \sqrt {d \,x^{6}+c}\, a^{2} d^{2} x^{6}+\sqrt {d \,x^{6}+c}\, a b \,c^{2}+8 \sqrt {d \,x^{6}+c}\, a b c d \,x^{6}+4 \sqrt {d \,x^{6}+c}\, a b \,d^{2} x^{12}-5 \sqrt {d \,x^{6}+c}\, b^{2} c^{2} x^{6}+10 \sqrt {d \,x^{6}+c}\, b^{2} c d \,x^{12}+108 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{2}}{2 a \,b^{2} d^{2} x^{18}-b^{3} c d \,x^{18}+4 a^{2} b \,d^{2} x^{12}-b^{3} c^{2} x^{12}+2 a^{3} d^{2} x^{6}+3 a^{2} b c d \,x^{6}-2 a \,b^{2} c^{2} x^{6}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a^{3} b^{2} c^{2} d^{2} x^{9}-144 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{2}}{2 a \,b^{2} d^{2} x^{18}-b^{3} c d \,x^{18}+4 a^{2} b \,d^{2} x^{12}-b^{3} c^{2} x^{12}+2 a^{3} d^{2} x^{6}+3 a^{2} b c d \,x^{6}-2 a \,b^{2} c^{2} x^{6}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a^{2} b^{3} c^{3} d \,x^{9}+108 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{2}}{2 a \,b^{2} d^{2} x^{18}-b^{3} c d \,x^{18}+4 a^{2} b \,d^{2} x^{12}-b^{3} c^{2} x^{12}+2 a^{3} d^{2} x^{6}+3 a^{2} b c d \,x^{6}-2 a \,b^{2} c^{2} x^{6}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a^{2} b^{3} c^{2} d^{2} x^{15}+45 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{2}}{2 a \,b^{2} d^{2} x^{18}-b^{3} c d \,x^{18}+4 a^{2} b \,d^{2} x^{12}-b^{3} c^{2} x^{12}+2 a^{3} d^{2} x^{6}+3 a^{2} b c d \,x^{6}-2 a \,b^{2} c^{2} x^{6}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a \,b^{4} c^{4} x^{9}-144 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{2}}{2 a \,b^{2} d^{2} x^{18}-b^{3} c d \,x^{18}+4 a^{2} b \,d^{2} x^{12}-b^{3} c^{2} x^{12}+2 a^{3} d^{2} x^{6}+3 a^{2} b c d \,x^{6}-2 a \,b^{2} c^{2} x^{6}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) a \,b^{4} c^{3} d \,x^{15}+45 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{2}}{2 a \,b^{2} d^{2} x^{18}-b^{3} c d \,x^{18}+4 a^{2} b \,d^{2} x^{12}-b^{3} c^{2} x^{12}+2 a^{3} d^{2} x^{6}+3 a^{2} b c d \,x^{6}-2 a \,b^{2} c^{2} x^{6}+2 a^{3} c d -a^{2} b \,c^{2}}d x \right ) b^{5} c^{4} x^{15}}{9 a^{2} c^{2} x^{9} \left (2 a b d \,x^{6}-b^{2} c \,x^{6}+2 a^{2} d -a b c \right )} \] Input:

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

Output:

( - 2*sqrt(c + d*x**6)*a**2*c*d + 4*sqrt(c + d*x**6)*a**2*d**2*x**6 + sqrt 
(c + d*x**6)*a*b*c**2 + 8*sqrt(c + d*x**6)*a*b*c*d*x**6 + 4*sqrt(c + d*x** 
6)*a*b*d**2*x**12 - 5*sqrt(c + d*x**6)*b**2*c**2*x**6 + 10*sqrt(c + d*x**6 
)*b**2*c*d*x**12 + 108*int((sqrt(c + d*x**6)*x**2)/(2*a**3*c*d + 2*a**3*d* 
*2*x**6 - a**2*b*c**2 + 3*a**2*b*c*d*x**6 + 4*a**2*b*d**2*x**12 - 2*a*b**2 
*c**2*x**6 + 2*a*b**2*d**2*x**18 - b**3*c**2*x**12 - b**3*c*d*x**18),x)*a* 
*3*b**2*c**2*d**2*x**9 - 144*int((sqrt(c + d*x**6)*x**2)/(2*a**3*c*d + 2*a 
**3*d**2*x**6 - a**2*b*c**2 + 3*a**2*b*c*d*x**6 + 4*a**2*b*d**2*x**12 - 2* 
a*b**2*c**2*x**6 + 2*a*b**2*d**2*x**18 - b**3*c**2*x**12 - b**3*c*d*x**18) 
,x)*a**2*b**3*c**3*d*x**9 + 108*int((sqrt(c + d*x**6)*x**2)/(2*a**3*c*d + 
2*a**3*d**2*x**6 - a**2*b*c**2 + 3*a**2*b*c*d*x**6 + 4*a**2*b*d**2*x**12 - 
 2*a*b**2*c**2*x**6 + 2*a*b**2*d**2*x**18 - b**3*c**2*x**12 - b**3*c*d*x** 
18),x)*a**2*b**3*c**2*d**2*x**15 + 45*int((sqrt(c + d*x**6)*x**2)/(2*a**3* 
c*d + 2*a**3*d**2*x**6 - a**2*b*c**2 + 3*a**2*b*c*d*x**6 + 4*a**2*b*d**2*x 
**12 - 2*a*b**2*c**2*x**6 + 2*a*b**2*d**2*x**18 - b**3*c**2*x**12 - b**3*c 
*d*x**18),x)*a*b**4*c**4*x**9 - 144*int((sqrt(c + d*x**6)*x**2)/(2*a**3*c* 
d + 2*a**3*d**2*x**6 - a**2*b*c**2 + 3*a**2*b*c*d*x**6 + 4*a**2*b*d**2*x** 
12 - 2*a*b**2*c**2*x**6 + 2*a*b**2*d**2*x**18 - b**3*c**2*x**12 - b**3*c*d 
*x**18),x)*a*b**4*c**3*d*x**15 + 45*int((sqrt(c + d*x**6)*x**2)/(2*a**3*c* 
d + 2*a**3*d**2*x**6 - a**2*b*c**2 + 3*a**2*b*c*d*x**6 + 4*a**2*b*d**2*...