\(\int \frac {x^{-1-n (-2+p)} (a+b x^n)^p}{(c+d x^n)^5} \, dx\) [553]

Optimal result

Integrand size = 29, antiderivative size = 245 \[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^5} \, dx=\frac {x^{n (2-p)} \left (a+b x^n\right )^{1+p}}{a c n (2-p) \left (c+d x^n\right )^4}+\frac {d (4 b c-a d (2+p)) x^{n (3-p)} \left (a+b x^n\right )^{1+p}}{4 a c^2 (b c-a d) n (2-p) \left (c+d x^n\right )^4}-\frac {a^2 \left (12 b^2 c^2-8 a b c d (1+p)+a^2 d^2 \left (2+3 p+p^2\right )\right ) x^{n (3-p)} \left (a+b x^n\right )^{-3+p} \operatorname {Hypergeometric2F1}\left (4,3-p,4-p,\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{4 c^6 (b c-a d) n (2-p) (3-p)} \] Output:

x^(n*(-p+2))*(a+b*x^n)^(p+1)/a/c/n/(-p+2)/(c+d*x^n)^4+1/4*d*(4*b*c-a*d*(2+ 
p))*x^(n*(3-p))*(a+b*x^n)^(p+1)/a/c^2/(-a*d+b*c)/n/(-p+2)/(c+d*x^n)^4-1/4* 
a^2*(12*b^2*c^2-8*a*b*c*d*(p+1)+a^2*d^2*(p^2+3*p+2))*x^(n*(3-p))*(a+b*x^n) 
^(-3+p)*hypergeom([4, 3-p],[4-p],(-a*d+b*c)*x^n/c/(a+b*x^n))/c^6/(-a*d+b*c 
)/n/(-p+2)/(3-p)
 

Mathematica [C] (verified)

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

Time = 0.98 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.95 \[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^5} \, dx=-\frac {(-1+p) p x^{-n (-2+p)} \left (a+b x^n\right )^p \left (4 \left (6 c^2+6 c d p x^n+d^2 p (1+p) x^{2 n}\right ) \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,1-p\right )-6 \left (2 c^2+4 c d (-1+p) x^n+d^2 (-1+p) p x^{2 n}\right ) \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,2-p\right )-16 c d x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,3-p\right )+8 c d p x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,3-p\right )+8 d^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,3-p\right )-12 d^2 p x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,3-p\right )+4 d^2 p^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,3-p\right )-6 d^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,4-p\right )+5 d^2 p x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,4-p\right )-d^2 p^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,4-p\right )-12 c^2 \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,-p\right )-8 c d x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,-p\right )-8 c d p x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,-p\right )-2 d^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,-p\right )-3 d^2 p x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,-p\right )-d^2 p^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,-p\right )\right )}{24 c^3 n \left (c+d x^n\right )^4} \] Input:

Integrate[(x^(-1 - n*(-2 + p))*(a + b*x^n)^p)/(c + d*x^n)^5,x]
 

Output:

-1/24*((-1 + p)*p*(a + b*x^n)^p*(4*(6*c^2 + 6*c*d*p*x^n + d^2*p*(1 + p)*x^ 
(2*n))*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 1 - p] - 6*(2 
*c^2 + 4*c*d*(-1 + p)*x^n + d^2*(-1 + p)*p*x^(2*n))*HurwitzLerchPhi[((b*c 
- a*d)*x^n)/(c*(a + b*x^n)), 1, 2 - p] - 16*c*d*x^n*HurwitzLerchPhi[((b*c 
- a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] + 8*c*d*p*x^n*HurwitzLerchPhi[((b*c 
 - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] + 8*d^2*x^(2*n)*HurwitzLerchPhi[(( 
b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] - 12*d^2*p*x^(2*n)*HurwitzLerch 
Phi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] + 4*d^2*p^2*x^(2*n)*Hurwi 
tzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] - 6*d^2*x^(2*n)*Hu 
rwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 4 - p] + 5*d^2*p*x^(2* 
n)*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 4 - p] - d^2*p^2* 
x^(2*n)*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 4 - p] - 12* 
c^2*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 8*c*d*x^n* 
HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 8*c*d*p*x^n*Hu 
rwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 2*d^2*x^(2*n)*Hu 
rwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 3*d^2*p*x^(2*n)* 
HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - d^2*p^2*x^(2*n 
)*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p]))/(c^3*n*x^(n* 
(-2 + p))*(c + d*x^n)^4)
 

Rubi [C] (verified)

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

Time = 2.21 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.99, 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.

\(\displaystyle \int \frac {x^{-n (p-2)-1} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^5} \, dx\)

\(\Big \downarrow \) 1013

\(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \int \frac {x^{n (2-p)-1} \left (\frac {b x^n}{a}+1\right )^p}{\left (d x^n+c\right )^5}dx\)

\(\Big \downarrow \) 1012

\(\displaystyle \frac {(1-p) p x^{n (2-p)} \left (a+b x^n\right )^p \left (4 \left (6 c^2+6 c d p x^n+d^2 p (p+1) x^{2 n}\right ) \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,1-p\right )-6 \left (2 c^2-4 c d (1-p) x^n-d^2 (1-p) p x^{2 n}\right ) \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,2-p\right )-12 c^2 \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right )+4 d^2 p^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,3-p\right )-d^2 p^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,4-p\right )-d^2 p^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right )+8 d^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,3-p\right )-12 d^2 p x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,3-p\right )-6 d^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,4-p\right )+5 d^2 p x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,4-p\right )-2 d^2 x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right )-3 d^2 p x^{2 n} \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right )-16 c d x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,3-p\right )+8 c d p x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,3-p\right )-8 c d x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right )-8 c d p x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (b x^n+a\right )},1,-p\right )\right )}{24 c^7 n \left (\frac {d x^n}{c}+1\right )^4}\)

Input:

Int[(x^(-1 - n*(-2 + p))*(a + b*x^n)^p)/(c + d*x^n)^5,x]
 

Output:

((1 - p)*p*x^(n*(2 - p))*(a + b*x^n)^p*(4*(6*c^2 + 6*c*d*p*x^n + d^2*p*(1 
+ p)*x^(2*n))*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 1 - p] 
 - 6*(2*c^2 - 4*c*d*(1 - p)*x^n - d^2*(1 - p)*p*x^(2*n))*HurwitzLerchPhi[( 
(b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 2 - p] - 16*c*d*x^n*HurwitzLerchPhi[( 
(b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] + 8*c*d*p*x^n*HurwitzLerchPhi[ 
((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] + 8*d^2*x^(2*n)*HurwitzLerchP 
hi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] - 12*d^2*p*x^(2*n)*Hurwitz 
LerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] + 4*d^2*p^2*x^(2*n)* 
HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] - 6*d^2*x^(2* 
n)*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 4 - p] + 5*d^2*p* 
x^(2*n)*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 4 - p] - d^2 
*p^2*x^(2*n)*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 4 - p] 
- 12*c^2*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 8*c*d 
*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 8*c*d*p*x 
^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 2*d^2*x^(2* 
n)*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - 3*d^2*p*x^( 
2*n)*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - d^2*p^2*x 
^(2*n)*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p]))/(24*c^7 
*n*(1 + (d*x^n)/c)^4)
 

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]

\[\int \frac {x^{-1-n \left (-2+p \right )} \left (a +b \,x^{n}\right )^{p}}{\left (c +d \,x^{n}\right )^{5}}d x\]

Input:

int(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^5,x)
 

Output:

int(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^5,x)
 

Fricas [F]

\[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^5} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 2\right )} - 1}}{{\left (d x^{n} + c\right )}^{5}} \,d x } \] Input:

integrate(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^5,x, algorithm="fricas")
 

Output:

integral((b*x^n + a)^p*x^(-n*p + 2*n - 1)/(d^5*x^(5*n) + 5*c*d^4*x^(4*n) + 
 10*c^2*d^3*x^(3*n) + 10*c^3*d^2*x^(2*n) + 5*c^4*d*x^n + c^5), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^5} \, dx=\text {Timed out} \] Input:

integrate(x**(-1-n*(-2+p))*(a+b*x**n)**p/(c+d*x**n)**5,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^5} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 2\right )} - 1}}{{\left (d x^{n} + c\right )}^{5}} \,d x } \] Input:

integrate(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^5,x, algorithm="maxima")
 

Output:

integrate((b*x^n + a)^p*x^(-n*(p - 2) - 1)/(d*x^n + c)^5, x)
 

Giac [F]

\[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^5} \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p} x^{-n {\left (p - 2\right )} - 1}}{{\left (d x^{n} + c\right )}^{5}} \,d x } \] Input:

integrate(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^5,x, algorithm="giac")
 

Output:

integrate((b*x^n + a)^p*x^(-n*(p - 2) - 1)/(d*x^n + c)^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^5} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p}{x^{n\,\left (p-2\right )+1}\,{\left (c+d\,x^n\right )}^5} \,d x \] Input:

int((a + b*x^n)^p/(x^(n*(p - 2) + 1)*(c + d*x^n)^5),x)
 

Output:

int((a + b*x^n)^p/(x^(n*(p - 2) + 1)*(c + d*x^n)^5), x)
 

Reduce [F]

\[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^5} \, dx=\text {too large to display} \] Input:

int(x^(-1-n*(-2+p))*(a+b*x^n)^p/(c+d*x^n)^5,x)
 

Output:

too large to display