\(\int (a+b x^n)^2 (c+d x^n)^{-4-\frac {1}{n}} \, dx\) [136]

Optimal result

Integrand size = 25, antiderivative size = 253 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=-\frac {d x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac {1}{n}}}{c (b c-a d) (1+3 n)}-\frac {(3 a d n-b (c+3 c n)) x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c^2 (b c-a d) (1+2 n) (1+3 n)}+\frac {2 a n (3 a d n-b (c+3 c n)) x \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^3 d (1+n) (1+2 n) (1+3 n)}+\frac {2 a n (b c+3 b c n-3 a d n) (b c+a d n) x \left (c+d x^n\right )^{-1/n}}{c^4 d (b c-a d) (1+n) (1+2 n) (1+3 n)} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.48 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.54 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-1/n} \left (1+\frac {d x^n}{c}\right )^{\frac {1}{n}} \left (b^2 c^2 \operatorname {Hypergeometric2F1}\left (2+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )-(b c-a d) \left (2 b c \operatorname {Hypergeometric2F1}\left (3+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )+(-b c+a d) \operatorname {Hypergeometric2F1}\left (4+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {d x^n}{c}\right )\right )\right )}{c^4 d^2} \] Input:

Integrate[(a + b*x^n)^2*(c + d*x^n)^(-4 - n^(-1)),x]
 

Output:

(x*(1 + (d*x^n)/c)^n^(-1)*(b^2*c^2*Hypergeometric2F1[2 + n^(-1), n^(-1), 1 
 + n^(-1), -((d*x^n)/c)] - (b*c - a*d)*(2*b*c*Hypergeometric2F1[3 + n^(-1) 
, n^(-1), 1 + n^(-1), -((d*x^n)/c)] + (-(b*c) + a*d)*Hypergeometric2F1[4 + 
 n^(-1), n^(-1), 1 + n^(-1), -((d*x^n)/c)])))/(c^4*d^2*(c + d*x^n)^n^(-1))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {907, 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.

\(\displaystyle \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-4} \, dx\)

\(\Big \downarrow \) 907

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

\(\Big \downarrow \) 903

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

\(\Big \downarrow \) 903

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

\(\Big \downarrow \) 903

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

\(\Big \downarrow \) 746

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

Input:

Int[(a + b*x^n)^2*(c + d*x^n)^(-4 - n^(-1)),x]
 

Output:

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

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]
 

rule 907

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1058\) vs. \(2(253)=506\).

Time = 1.40 (sec) , antiderivative size = 1059, normalized size of antiderivative = 4.19

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1059\)

Input:

int((a+b*x^n)^2*(c+d*x^n)^(-4-1/n),x,method=_RETURNVERBOSE)
 

Output:

(3*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^2*d^2*n+26*x*x^n*(c+d*x^n)^(-(1+ 
4*n)/n)*a^2*c^3*d*n^2+12*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^4*n^2+2*x*(x^n 
)^2*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^3*d+9*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^ 
3*d*n+10*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^4*n+4*x*(x^n)^4*(c+d*x^n)^(-(1 
+4*n)/n)*a*b*c*d^3*n^2+16*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^2*d^2*n^2 
+4*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^2*d^2*n+24*x*(x^n)^2*(c+d*x^n)^( 
-(1+4*n)/n)*a*b*c^3*d*n^2+14*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a*b*c^3*d*n+ 
24*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^2*c*d^3*n^3+x*(x^n)^4*(c+d*x^n)^(-(1 
+4*n)/n)*b^2*c^2*d^2*n+6*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^2*c*d^3*n^2+4* 
x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*b^2*c^3*d*n^2+36*x*(x^n)^2*(c+d*x^n)^(-(1 
+4*n)/n)*a^2*c^2*d^2*n^3+5*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*b^2*c^3*d*n+21 
*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^2*d^2*n^2+24*x*x^n*(c+d*x^n)^(-(1+ 
4*n)/n)*a^2*c^3*d*n^3+x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*b^2*c^2*d^2*n^2+x*( 
x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*b^2*c^3*d+4*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n) 
*b^2*c^4*n+x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^3*d+2*x*x^n*(c+d*x^n)^(-(1+4 
*n)/n)*a*b*c^4+x*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^4+6*x*(c+d*x^n)^(-(1+4*n)/n) 
*a^2*c^4*n^3+x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*b^2*c^4+11*x*(c+d*x^n)^(-(1+ 
4*n)/n)*a^2*c^4*n^2+6*x*(c+d*x^n)^(-(1+4*n)/n)*a^2*c^4*n+6*x*(x^n)^4*(c+d* 
x^n)^(-(1+4*n)/n)*a^2*d^4*n^3+3*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*b^2*c^4*n 
^2)/(2*n^2+3*n+1)/(1+3*n)/c^4
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.58 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {{\left (6 \, a^{2} d^{4} n^{3} + b^{2} c^{2} d^{2} n + {\left (b^{2} c^{2} d^{2} + 4 \, a b c d^{3}\right )} n^{2}\right )} x x^{4 \, n} + {\left (24 \, a^{2} c d^{3} n^{3} + b^{2} c^{3} d + 2 \, {\left (2 \, b^{2} c^{3} d + 8 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} n^{2} + {\left (5 \, b^{2} c^{3} d + 4 \, a b c^{2} d^{2}\right )} n\right )} x x^{3 \, n} + {\left (36 \, a^{2} c^{2} d^{2} n^{3} + b^{2} c^{4} + 2 \, a b c^{3} d + 3 \, {\left (b^{2} c^{4} + 8 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} n^{2} + {\left (4 \, b^{2} c^{4} + 14 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2}\right )} n\right )} x x^{2 \, n} + {\left (24 \, a^{2} c^{3} d n^{3} + 2 \, a b c^{4} + a^{2} c^{3} d + 2 \, {\left (6 \, a b c^{4} + 13 \, a^{2} c^{3} d\right )} n^{2} + {\left (10 \, a b c^{4} + 9 \, a^{2} c^{3} d\right )} n\right )} x x^{n} + {\left (6 \, a^{2} c^{4} n^{3} + 11 \, a^{2} c^{4} n^{2} + 6 \, a^{2} c^{4} n + a^{2} c^{4}\right )} x}{{\left (6 \, c^{4} n^{3} + 11 \, c^{4} n^{2} + 6 \, c^{4} n + c^{4}\right )} {\left (d x^{n} + c\right )}^{\frac {4 \, n + 1}{n}}} \] Input:

integrate((a+b*x^n)^2*(c+d*x^n)^(-4-1/n),x, algorithm="fricas")
 

Output:

((6*a^2*d^4*n^3 + b^2*c^2*d^2*n + (b^2*c^2*d^2 + 4*a*b*c*d^3)*n^2)*x*x^(4* 
n) + (24*a^2*c*d^3*n^3 + b^2*c^3*d + 2*(2*b^2*c^3*d + 8*a*b*c^2*d^2 + 3*a^ 
2*c*d^3)*n^2 + (5*b^2*c^3*d + 4*a*b*c^2*d^2)*n)*x*x^(3*n) + (36*a^2*c^2*d^ 
2*n^3 + b^2*c^4 + 2*a*b*c^3*d + 3*(b^2*c^4 + 8*a*b*c^3*d + 7*a^2*c^2*d^2)* 
n^2 + (4*b^2*c^4 + 14*a*b*c^3*d + 3*a^2*c^2*d^2)*n)*x*x^(2*n) + (24*a^2*c^ 
3*d*n^3 + 2*a*b*c^4 + a^2*c^3*d + 2*(6*a*b*c^4 + 13*a^2*c^3*d)*n^2 + (10*a 
*b*c^4 + 9*a^2*c^3*d)*n)*x*x^n + (6*a^2*c^4*n^3 + 11*a^2*c^4*n^2 + 6*a^2*c 
^4*n + a^2*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. 2746 vs. \(2 (219) = 438\).

Time = 15.89 (sec) , antiderivative size = 2746, normalized size of antiderivative = 10.85 \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\text {Too large to display} \] Input:

integrate((a+b*x**n)**2*(c+d*x**n)**(-4-1/n),x)
 

Output:

6*a**2*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**2*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**2*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**2*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**2*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]

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

integrate((a+b*x^n)^2*(c+d*x^n)^(-4-1/n),x, algorithm="maxima")
 

Output:

integrate((b*x^n + a)^2*(d*x^n + c)^(-1/n - 4), x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*x^n)^2*(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%%%{27,[1,0,4,3,1,3,2,0]%%%}+%%%{27,[1,0,4,2,1,3,2,0]%%%}+%%%{ 
9,[1,0,4,
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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)*b**2 + 2*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*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**2