\(\int (a+b x^n)^3 (c+d x^n)^{-4-\frac {1}{n}} \, dx\) [127]

Optimal result

Integrand size = 25, antiderivative size = 193 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac {1}{n}}}{c (1+3 n)}+\frac {3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}-\frac {6 a^2 (b c-a d) n^2 x \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^3 d (1+n) (1+2 n) (1+3 n)}+\frac {6 a^2 n^2 (b c+a d n) x \left (c+d x^n\right )^{-1/n}}{c^4 d (1+n) (1+2 n) (1+3 n)} \] Output:

x*(a+b*x^n)^3*(c+d*x^n)^(-3-1/n)/c/(1+3*n)+3*a*n*x*(a+b*x^n)^2*(c+d*x^n)^( 
-2-1/n)/c^2/(6*n^2+5*n+1)-6*a^2*(-a*d+b*c)*n^2*x*(c+d*x^n)^(-1-1/n)/c^3/d/ 
(1+n)/(1+2*n)/(1+3*n)+6*a^2*n^2*(a*d*n+b*c)*x/c^4/d/(1+n)/(1+2*n)/(1+3*n)/ 
((c+d*x^n)^(1/n))
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.13 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-3-\frac {1}{n}} \left (b^3 c^3 \left (1+3 n+2 n^2\right ) x^{3 n}+3 a b^2 c^2 (1+n) x^{2 n} \left (c+3 c n+d n x^n\right )+3 a^2 b c x^n \left (c^2 \left (1+5 n+6 n^2\right )+2 c d n (1+3 n) x^n+2 d^2 n^2 x^{2 n}\right )+a^3 \left (c^3 \left (1+6 n+11 n^2+6 n^3\right )+3 c^2 d n \left (1+5 n+6 n^2\right ) x^n+6 c d^2 n^2 (1+3 n) x^{2 n}+6 d^3 n^3 x^{3 n}\right )\right )}{c^4 (1+n) (1+2 n) (1+3 n)} \] Input:

Integrate[(a + b*x^n)^3*(c + d*x^n)^(-4 - n^(-1)),x]
 

Output:

(x*(c + d*x^n)^(-3 - n^(-1))*(b^3*c^3*(1 + 3*n + 2*n^2)*x^(3*n) + 3*a*b^2* 
c^2*(1 + n)*x^(2*n)*(c + 3*c*n + d*n*x^n) + 3*a^2*b*c*x^n*(c^2*(1 + 5*n + 
6*n^2) + 2*c*d*n*(1 + 3*n)*x^n + 2*d^2*n^2*x^(2*n)) + a^3*(c^3*(1 + 6*n + 
11*n^2 + 6*n^3) + 3*c^2*d*n*(1 + 5*n + 6*n^2)*x^n + 6*c*d^2*n^2*(1 + 3*n)* 
x^(2*n) + 6*d^3*n^3*x^(3*n))))/(c^4*(1 + n)*(1 + 2*n)*(1 + 3*n))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {903, 903, 903, 746}

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.

\(\displaystyle \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-4} \, dx\)

\(\Big \downarrow \) 903

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

\(\Big \downarrow \) 903

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

\(\Big \downarrow \) 903

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

\(\Big \downarrow \) 746

\(\displaystyle \frac {3 a n \left (\frac {2 a n \left (\frac {x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c (n+1)}+\frac {a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)}\right )}{c (2 n+1)}+\frac {x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c (2 n+1)}\right )}{c (3 n+1)}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)}\)

Input:

Int[(a + b*x^n)^3*(c + d*x^n)^(-4 - n^(-1)),x]
 

Output:

(x*(a + b*x^n)^3*(c + d*x^n)^(-3 - n^(-1)))/(c*(1 + 3*n)) + (3*a*n*((x*(a 
+ b*x^n)^2*(c + d*x^n)^(-2 - n^(-1)))/(c*(1 + 2*n)) + (2*a*n*((x*(a + b*x^ 
n)*(c + d*x^n)^(-1 - n^(-1)))/(c*(1 + n)) + (a*n*x)/(c^2*(1 + n)*(c + d*x^ 
n)^n^(-1))))/(c*(1 + 2*n))))/(c*(1 + 3*n))
 

Defintions of rubi rules used

rule 746

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1) 
/a), x] /; FreeQ[{a, b, n, p}, x] && EqQ[1/n + p + 1, 0]
 

rule 903

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] 
 :> Simp[(-x)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*n*(p + 1))), x] - Simp[ 
c*(q/(a*(p + 1)))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /; 
FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 
 0] && GtQ[q, 0] && NeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1289\) vs. \(2(193)=386\).

Time = 1.97 (sec) , antiderivative size = 1290, normalized size of antiderivative = 6.68

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1290\)

Input:

int((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x,method=_RETURNVERBOSE)
 

Output:

(2*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*b^3*c^3*d*n^2+24*x*(x^n)^3*(c+d*x^n)^( 
-(1+4*n)/n)*a^3*c*d^3*n^3+3*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*b^3*c^3*d*n+6 
*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^3*c*d^3*n^2+36*x*(x^n)^2*(c+d*x^n)^(-( 
1+4*n)/n)*a^3*c^2*d^2*n^3+21*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^2*d^2* 
n^2+9*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^4*n^2+24*x*x^n*(c+d*x^n)^(- 
(1+4*n)/n)*a^3*c^3*d*n^3+3*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^3*d+3* 
x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^2*d^2*n+12*x*(x^n)^2*(c+d*x^n)^(-(1 
+4*n)/n)*a*b^2*c^4*n+26*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^3*d*n^2+6*x*(x^ 
n)^4*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c*d^3*n^2+3*x*(x^n)^4*(c+d*x^n)^(-(1+4*n 
)/n)*a*b^2*c^2*d^2*n^2+3*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^2*d^2*n+ 
24*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^2*d^2*n^2+12*x*(x^n)^3*(c+d*x^ 
n)^(-(1+4*n)/n)*a*b^2*c^3*d*n^2+6*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c 
^2*d^2*n+15*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^3*d*n+6*x*(x^n)^4*(c+ 
d*x^n)^(-(1+4*n)/n)*a^3*d^4*n^3+2*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*b^3*c^4 
*n^2+36*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^3*d*n^2+21*x*(x^n)^2*(c+d 
*x^n)^(-(1+4*n)/n)*a^2*b*c^3*d*n+18*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^4 
*n^2+3*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^3*d+9*x*x^n*(c+d*x^n)^(-(1 
+4*n)/n)*a^3*c^3*d*n+15*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^4*n+x*(c+d*x^ 
n)^(-(1+4*n)/n)*a^3*c^4+x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*b^3*c^4+6*x*(c+d* 
x^n)^(-(1+4*n)/n)*a^3*c^4*n^3+11*x*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^4*n^2+6...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (193) = 386\).

Time = 0.13 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.48 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {{\left (6 \, a^{3} d^{4} n^{3} + b^{3} c^{3} d + {\left (2 \, b^{3} c^{3} d + 3 \, a b^{2} c^{2} d^{2} + 6 \, a^{2} b c d^{3}\right )} n^{2} + 3 \, {\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2}\right )} n\right )} x x^{4 \, n} + {\left (24 \, a^{3} c d^{3} n^{3} + b^{3} c^{4} + 3 \, a b^{2} c^{3} d + 2 \, {\left (b^{3} c^{4} + 6 \, a b^{2} c^{3} d + 12 \, a^{2} b c^{2} d^{2} + 3 \, a^{3} c d^{3}\right )} n^{2} + 3 \, {\left (b^{3} c^{4} + 5 \, a b^{2} c^{3} d + 2 \, a^{2} b c^{2} d^{2}\right )} n\right )} x x^{3 \, n} + 3 \, {\left (12 \, a^{3} c^{2} d^{2} n^{3} + a b^{2} c^{4} + a^{2} b c^{3} d + {\left (3 \, a b^{2} c^{4} + 12 \, a^{2} b c^{3} d + 7 \, a^{3} c^{2} d^{2}\right )} n^{2} + {\left (4 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2}\right )} n\right )} x x^{2 \, n} + {\left (24 \, a^{3} c^{3} d n^{3} + 3 \, a^{2} b c^{4} + a^{3} c^{3} d + 2 \, {\left (9 \, a^{2} b c^{4} + 13 \, a^{3} c^{3} d\right )} n^{2} + 3 \, {\left (5 \, a^{2} b c^{4} + 3 \, a^{3} c^{3} d\right )} n\right )} x x^{n} + {\left (6 \, a^{3} c^{4} n^{3} + 11 \, a^{3} c^{4} n^{2} + 6 \, a^{3} c^{4} n + a^{3} c^{4}\right )} x}{{\left (6 \, c^{4} n^{3} + 11 \, c^{4} n^{2} + 6 \, c^{4} n + c^{4}\right )} {\left (d x^{n} + c\right )}^{\frac {4 \, n + 1}{n}}} \] Input:

integrate((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x, algorithm="fricas")
 

Output:

((6*a^3*d^4*n^3 + b^3*c^3*d + (2*b^3*c^3*d + 3*a*b^2*c^2*d^2 + 6*a^2*b*c*d 
^3)*n^2 + 3*(b^3*c^3*d + a*b^2*c^2*d^2)*n)*x*x^(4*n) + (24*a^3*c*d^3*n^3 + 
 b^3*c^4 + 3*a*b^2*c^3*d + 2*(b^3*c^4 + 6*a*b^2*c^3*d + 12*a^2*b*c^2*d^2 + 
 3*a^3*c*d^3)*n^2 + 3*(b^3*c^4 + 5*a*b^2*c^3*d + 2*a^2*b*c^2*d^2)*n)*x*x^( 
3*n) + 3*(12*a^3*c^2*d^2*n^3 + a*b^2*c^4 + a^2*b*c^3*d + (3*a*b^2*c^4 + 12 
*a^2*b*c^3*d + 7*a^3*c^2*d^2)*n^2 + (4*a*b^2*c^4 + 7*a^2*b*c^3*d + a^3*c^2 
*d^2)*n)*x*x^(2*n) + (24*a^3*c^3*d*n^3 + 3*a^2*b*c^4 + a^3*c^3*d + 2*(9*a^ 
2*b*c^4 + 13*a^3*c^3*d)*n^2 + 3*(5*a^2*b*c^4 + 3*a^3*c^3*d)*n)*x*x^n + (6* 
a^3*c^4*n^3 + 11*a^3*c^4*n^2 + 6*a^3*c^4*n + a^3*c^4)*x)/((6*c^4*n^3 + 11* 
c^4*n^2 + 6*c^4*n + c^4)*(d*x^n + c)^((4*n + 1)/n))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2822 vs. \(2 (168) = 336\).

Time = 47.16 (sec) , antiderivative size = 2822, normalized size of antiderivative = 14.62 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\text {Too large to display} \] Input:

integrate((a+b*x**n)**3*(c+d*x**n)**(-4-1/n),x)
 

Output:

6*a**3*c**3*c**(1/n)*c**(-4 - 1/n)*n**3*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/ 
(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**4*x**n*(c/(d*x* 
*n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x** 
n) + 1)**(1/n)*gamma(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 
1)**(1/n)*gamma(4 + 1/n)) + 11*a**3*c**3*c**(1/n)*c**(-4 - 1/n)*n**2*gamma 
(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2* 
d*d**(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d** 
(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + d**3*d**(1/n) 
*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 6*a**3*c**3*c**(1 
/n)*c**(-4 - 1/n)*n*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n) 
*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamm 
a(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/n)*gamma 
(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 
1/n)) + a**3*c**3*c**(1/n)*c**(-4 - 1/n)*gamma(1/n)/(c**3*d**(1/n)*n**4*(c 
/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**4*x**n*(c/(d*x 
**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x* 
*n) + 1)**(1/n)*gamma(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 
 1)**(1/n)*gamma(4 + 1/n)) + 18*a**3*c**2*c**(1/n)*c**(-4 - 1/n)*d*n**3*x* 
*n*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 
 3*c**2*d*d**(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3...
 

Maxima [F]

\[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{3} {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 4} \,d x } \] Input:

integrate((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x, algorithm="maxima")
 

Output:

integrate((b*x^n + a)^3*(d*x^n + c)^(-1/n - 4), x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{81,[2,0,6,4,2,4,3,0]%%%}+%%%{108,[2,0,6,3,2,4,3,0]%%%}+%%% 
{54,[2,0,
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\int \frac {{\left (a+b\,x^n\right )}^3}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+4}} \,d x \] Input:

int((a + b*x^n)^3/(c + d*x^n)^(1/n + 4),x)
 

Output:

int((a + b*x^n)^3/(c + d*x^n)^(1/n + 4), x)
 

Reduce [F]

\[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\left (\int \frac {x^{3 n}}{x^{4 n} \left (x^{n} d +c \right )^{\frac {1}{n}} d^{4}+4 x^{3 n} \left (x^{n} d +c \right )^{\frac {1}{n}} c \,d^{3}+6 x^{2 n} \left (x^{n} d +c \right )^{\frac {1}{n}} c^{2} d^{2}+4 x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} c^{3} d +\left (x^{n} d +c \right )^{\frac {1}{n}} c^{4}}d x \right ) b^{3}+3 \left (\int \frac {x^{2 n}}{x^{4 n} \left (x^{n} d +c \right )^{\frac {1}{n}} d^{4}+4 x^{3 n} \left (x^{n} d +c \right )^{\frac {1}{n}} c \,d^{3}+6 x^{2 n} \left (x^{n} d +c \right )^{\frac {1}{n}} c^{2} d^{2}+4 x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} c^{3} d +\left (x^{n} d +c \right )^{\frac {1}{n}} c^{4}}d x \right ) a \,b^{2}+3 \left (\int \frac {x^{n}}{x^{4 n} \left (x^{n} d +c \right )^{\frac {1}{n}} d^{4}+4 x^{3 n} \left (x^{n} d +c \right )^{\frac {1}{n}} c \,d^{3}+6 x^{2 n} \left (x^{n} d +c \right )^{\frac {1}{n}} c^{2} d^{2}+4 x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} c^{3} d +\left (x^{n} d +c \right )^{\frac {1}{n}} c^{4}}d x \right ) a^{2} b +\left (\int \frac {1}{x^{4 n} \left (x^{n} d +c \right )^{\frac {1}{n}} d^{4}+4 x^{3 n} \left (x^{n} d +c \right )^{\frac {1}{n}} c \,d^{3}+6 x^{2 n} \left (x^{n} d +c \right )^{\frac {1}{n}} c^{2} d^{2}+4 x^{n} \left (x^{n} d +c \right )^{\frac {1}{n}} c^{3} d +\left (x^{n} d +c \right )^{\frac {1}{n}} c^{4}}d x \right ) a^{3} \] Input:

int((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x)
 

Output:

int(x**(3*n)/(x**(4*n)*(x**n*d + c)**(1/n)*d**4 + 4*x**(3*n)*(x**n*d + c)* 
*(1/n)*c*d**3 + 6*x**(2*n)*(x**n*d + c)**(1/n)*c**2*d**2 + 4*x**n*(x**n*d 
+ c)**(1/n)*c**3*d + (x**n*d + c)**(1/n)*c**4),x)*b**3 + 3*int(x**(2*n)/(x 
**(4*n)*(x**n*d + c)**(1/n)*d**4 + 4*x**(3*n)*(x**n*d + c)**(1/n)*c*d**3 + 
 6*x**(2*n)*(x**n*d + c)**(1/n)*c**2*d**2 + 4*x**n*(x**n*d + c)**(1/n)*c** 
3*d + (x**n*d + c)**(1/n)*c**4),x)*a*b**2 + 3*int(x**n/(x**(4*n)*(x**n*d + 
 c)**(1/n)*d**4 + 4*x**(3*n)*(x**n*d + c)**(1/n)*c*d**3 + 6*x**(2*n)*(x**n 
*d + c)**(1/n)*c**2*d**2 + 4*x**n*(x**n*d + c)**(1/n)*c**3*d + (x**n*d + c 
)**(1/n)*c**4),x)*a**2*b + int(1/(x**(4*n)*(x**n*d + c)**(1/n)*d**4 + 4*x* 
*(3*n)*(x**n*d + c)**(1/n)*c*d**3 + 6*x**(2*n)*(x**n*d + c)**(1/n)*c**2*d* 
*2 + 4*x**n*(x**n*d + c)**(1/n)*c**3*d + (x**n*d + c)**(1/n)*c**4),x)*a**3