\(\int \frac {x^{-1-n (1+p)} (a+b x^n)^p}{(c+d x^n)^3} \, dx\) [531]

Optimal result

Integrand size = 29, antiderivative size = 308 \[ \int \frac {x^{-1-n (1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=-\frac {d (2 b c-a d (3+p)) x^{-n p} \left (a+b x^n\right )^{1+p}}{2 a c^2 (b c-a d) n (1+p) \left (c+d x^n\right )^2}-\frac {x^{-n (1+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p) \left (c+d x^n\right )^2}-\frac {d \left (2 b^2 c^2-a b c d (9+5 p)+a^2 d^2 \left (6+5 p+p^2\right )\right ) x^{-n p} \left (a+b x^n\right )^{1+p}}{2 a c^3 (b c-a d)^2 n (1+p) \left (c+d x^n\right )}+\frac {d \left (6 b^2 c^2-6 a b c d (2+p)+a^2 d^2 \left (6+5 p+p^2\right )\right ) x^{-n p} \left (a+b x^n\right )^p \operatorname {Hypergeometric2F1}\left (1,-p,1-p,\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{2 c^4 (b c-a d)^2 n p} \] Output:

-1/2*d*(2*b*c-a*d*(3+p))*(a+b*x^n)^(p+1)/a/c^2/(-a*d+b*c)/n/(p+1)/(x^(n*p) 
)/(c+d*x^n)^2-(a+b*x^n)^(p+1)/a/c/n/(p+1)/(x^(n*(p+1)))/(c+d*x^n)^2-1/2*d* 
(2*b^2*c^2-a*b*c*d*(9+5*p)+a^2*d^2*(p^2+5*p+6))*(a+b*x^n)^(p+1)/a/c^3/(-a* 
d+b*c)^2/n/(p+1)/(x^(n*p))/(c+d*x^n)+1/2*d*(6*b^2*c^2-6*a*b*c*d*(2+p)+a^2* 
d^2*(p^2+5*p+6))*(a+b*x^n)^p*hypergeom([1, -p],[1-p],(-a*d+b*c)*x^n/c/(a+b 
*x^n))/c^4/(-a*d+b*c)^2/n/p/(x^(n*p))
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 1.84 (sec) , antiderivative size = 2698, normalized size of antiderivative = 8.76 \[ \int \frac {x^{-1-n (1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

((a + b*x^n)^(-1 + p)*(-2*c*(-6 + 11*p - 6*p^2 + p^3)*(a + b*x^n)*(c^3*(-1 
 + p)^3 + 3*c^2*d*(-1 + p)^3*x^n + 3*c*d^2*(-1 + 2*p - 4*p^2 + p^3)*x^(2*n 
) - d^3*(1 - p + 6*p^2)*x^(3*n))*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + 
 b*x^n)), 1, 1 - p] + c*(-6 + 11*p - 6*p^2 + p^3)*(a + b*x^n)*(c^3*(-2 + p 
)^3 + 3*c^2*d*(-2 + p)^3*x^n + 3*c*d^2*(-2 + p)^3*x^(2*n) - d^3*(8 - 13*p 
+ 6*p^2)*x^(3*n))*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 2 
- p] - 6*a*c^4*p^3*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, - 
p] + 11*a*c^4*p^4*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p 
] - 6*a*c^4*p^5*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] 
+ a*c^4*p^6*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] + 36 
*a*c^3*d*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 3 
0*a*c^3*d*p*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] 
- 30*a*c^3*d*p^2*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 
 -p] - 6*b*c^4*p^3*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 
1, -p] + 12*a*c^3*d*p^3*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^ 
n)), 1, -p] + 11*b*c^4*p^4*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b 
*x^n)), 1, -p] + 27*a*c^3*d*p^4*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*( 
a + b*x^n)), 1, -p] - 6*b*c^4*p^5*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c 
*(a + b*x^n)), 1, -p] - 18*a*c^3*d*p^5*x^n*HurwitzLerchPhi[((b*c - a*d)*x^ 
n)/(c*(a + b*x^n)), 1, -p] + b*c^4*p^6*x^n*HurwitzLerchPhi[((b*c - a*d)...
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 5.65 (sec) , antiderivative size = 2724, normalized size of antiderivative = 8.84, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1013, 1012}

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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle \frac {x^{-n (p+1)} \left (b x^n+a\right )^{p-1} \left (-b c^4 p^6 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n-3 a c^3 d p^6 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n+6 b c^4 p^5 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n+18 a c^3 d p^5 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n-11 b c^4 p^4 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n-27 a c^3 d p^4 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n+6 b c^4 p^3 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n-12 a c^3 d p^3 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n+30 a c^3 d p^2 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n-36 a c^3 d \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n+30 a c^3 d p \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^n-2 b c^4 \, _4F_3\left (2,2,2,1-p;1,1,4-p;\frac {(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n+2 a c^3 d \, _4F_3\left (2,2,2,1-p;1,1,4-p;\frac {(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^n-3 a c^2 d^2 p^6 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}-3 b c^3 d p^6 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}+24 a c^2 d^2 p^5 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}+18 b c^3 d p^5 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}-51 a c^2 d^2 p^4 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}-27 b c^3 d p^4 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}-12 a c^2 d^2 p^3 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}-12 b c^3 d p^3 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}-72 a c^2 d^2 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}+90 a c^2 d^2 p^2 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}+30 b c^3 d p^2 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}-36 b c^3 d \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}+24 a c^2 d^2 p \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}+30 b c^3 d p \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{2 n}+6 a c^2 d^2 \, _4F_3\left (2,2,2,1-p;1,1,4-p;\frac {(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^{2 n}-6 b c^3 d \, _4F_3\left (2,2,2,1-p;1,1,4-p;\frac {(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^{2 n}-3 b c^2 d^2 p^6 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}+6 a c d^3 p^5 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}+24 b c^2 d^2 p^5 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}-25 a c d^3 p^4 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}-51 b c^2 d^2 p^4 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}-36 a c d^3 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}+6 a c d^3 p^3 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}-12 b c^2 d^2 p^3 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}-72 b c^2 d^2 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}+49 a c d^3 p^2 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}+90 b c^2 d^2 p^2 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}+24 b c^2 d^2 p \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{3 n}+6 a c d^3 \, _4F_3\left (2,2,2,1-p;1,1,4-p;\frac {(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^{3 n}-6 b c^2 d^2 \, _4F_3\left (2,2,2,1-p;1,1,4-p;\frac {(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^{3 n}+6 b c d^3 p^5 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{4 n}-25 b c d^3 p^4 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{4 n}-36 b c d^3 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{4 n}+6 b c d^3 p^3 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{4 n}+49 b c d^3 p^2 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right ) x^{4 n}+2 a d^4 \, _4F_3\left (2,2,2,1-p;1,1,4-p;\frac {(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^{4 n}-2 b c d^3 \, _4F_3\left (2,2,2,1-p;1,1,4-p;\frac {(b c-a d) x^n}{c \left (b x^n+a\right )}\right ) x^{4 n}+2 c (1-p) (2-p) (3-p) \left (b x^n+a\right ) \left (3 c^2 d (1-p)^3 x^n+3 c d^2 \left (-p^3+4 p^2-2 p+1\right ) x^{2 n}+d^3 \left (6 p^2-p+1\right ) x^{3 n}+c^3 (1-p)^3\right ) \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,1-p\right )-c (1-p) (2-p) (3-p) \left (b x^n+a\right ) \left (3 c^2 d (2-p)^3 x^n+3 c d^2 (2-p)^3 x^{2 n}+d^3 \left (6 p^2-13 p+8\right ) x^{3 n}+c^3 (2-p)^3\right ) \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,2-p\right )-a c^4 p^6 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right )+6 a c^4 p^5 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right )-11 a c^4 p^4 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right )+6 a c^4 p^3 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right )\right )}{2 c^7 n (1-p) (2-p) (3-p) (p+1) \left (\frac {d x^n}{c}+1\right )^2}\)

Input:

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

Output:

((a + b*x^n)^(-1 + p)*(2*c*(1 - p)*(2 - p)*(3 - p)*(a + b*x^n)*(c^3*(1 - p 
)^3 + 3*c^2*d*(1 - p)^3*x^n + 3*c*d^2*(1 - 2*p + 4*p^2 - p^3)*x^(2*n) + d^ 
3*(1 - p + 6*p^2)*x^(3*n))*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n 
)), 1, 1 - p] - c*(1 - p)*(2 - p)*(3 - p)*(a + b*x^n)*(c^3*(2 - p)^3 + 3*c 
^2*d*(2 - p)^3*x^n + 3*c*d^2*(2 - p)^3*x^(2*n) + d^3*(8 - 13*p + 6*p^2)*x^ 
(3*n))*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 2 - p] + 6*a* 
c^4*p^3*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 11*a*c 
^4*p^4*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] + 6*a*c^4 
*p^5*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - a*c^4*p^6 
*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 36*a*c^3*d*x^ 
n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] + 30*a*c^3*d*p 
*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] + 30*a*c^3* 
d*p^2*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] + 6*b* 
c^4*p^3*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 12 
*a*c^3*d*p^3*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] 
 - 11*b*c^4*p^4*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 
-p] - 27*a*c^3*d*p^4*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)) 
, 1, -p] + 6*b*c^4*p^5*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n 
)), 1, -p] + 18*a*c^3*d*p^5*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + 
b*x^n)), 1, -p] - b*c^4*p^6*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a...
 

Defintions of rubi rules used

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

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

Maple [F]

\[\int \frac {x^{-1-n \left (p +1\right )} \left (a +b \,x^{n}\right )^{p}}{\left (c +d \,x^{n}\right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^{-1-n (1+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^3} \, dx=\text {too large to display} \] Input:

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

Output:

(6*x**(3*n)*(x**n*b + a)**p*a**4*b**3*d**6*p**4 + 48*x**(3*n)*(x**n*b + a) 
**p*a**4*b**3*d**6*p**3 + 138*x**(3*n)*(x**n*b + a)**p*a**4*b**3*d**6*p**2 
 + 168*x**(3*n)*(x**n*b + a)**p*a**4*b**3*d**6*p + 72*x**(3*n)*(x**n*b + a 
)**p*a**4*b**3*d**6 - 54*x**(3*n)*(x**n*b + a)**p*a**3*b**4*c*d**5*p**3 - 
282*x**(3*n)*(x**n*b + a)**p*a**3*b**4*c*d**5*p**2 - 480*x**(3*n)*(x**n*b 
+ a)**p*a**3*b**4*c*d**5*p - 264*x**(3*n)*(x**n*b + a)**p*a**3*b**4*c*d**5 
 + 120*x**(3*n)*(x**n*b + a)**p*a**2*b**5*c**2*d**4*p**2 + 408*x**(3*n)*(x 
**n*b + a)**p*a**2*b**5*c**2*d**4*p + 336*x**(3*n)*(x**n*b + a)**p*a**2*b* 
*5*c**2*d**4 - 72*x**(3*n)*(x**n*b + a)**p*a*b**6*c**3*d**3*p - 144*x**(3* 
n)*(x**n*b + a)**p*a*b**6*c**3*d**3 - 6*x**(2*n)*(x**n*b + a)**p*a**5*b**2 
*d**6*p**5 - 48*x**(2*n)*(x**n*b + a)**p*a**5*b**2*d**6*p**4 - 138*x**(2*n 
)*(x**n*b + a)**p*a**5*b**2*d**6*p**3 - 168*x**(2*n)*(x**n*b + a)**p*a**5* 
b**2*d**6*p**2 - 72*x**(2*n)*(x**n*b + a)**p*a**5*b**2*d**6*p + 78*x**(2*n 
)*(x**n*b + a)**p*a**4*b**3*c*d**5*p**4 + 426*x**(2*n)*(x**n*b + a)**p*a** 
4*b**3*c*d**5*p**3 + 768*x**(2*n)*(x**n*b + a)**p*a**4*b**3*c*d**5*p**2 + 
528*x**(2*n)*(x**n*b + a)**p*a**4*b**3*c*d**5*p + 144*x**(2*n)*(x**n*b + a 
)**p*a**4*b**3*c*d**5 - 12*x**(2*n)*(x**n*b + a)**p*a**3*b**4*c**2*d**4*p* 
*4 - 372*x**(2*n)*(x**n*b + a)**p*a**3*b**4*c**2*d**4*p**3 - 1152*x**(2*n) 
*(x**n*b + a)**p*a**3*b**4*c**2*d**4*p**2 - 1104*x**(2*n)*(x**n*b + a)**p* 
a**3*b**4*c**2*d**4*p - 384*x**(2*n)*(x**n*b + a)**p*a**3*b**4*c**2*d**...