\(\int \frac {x^{-1-n (-2+p)} (a+b x^n)^p}{(c+d x^n)^4} \, dx\) [552]

Optimal result

Integrand size = 29, antiderivative size = 140 \[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^4} \, dx=\frac {x^{n (2-p)} \left (a+b x^n\right )^{1+p}}{a c n (2-p) \left (c+d x^n\right )^3}-\frac {a^2 (3 b c-a d (1+p)) x^{n (3-p)} \left (a+b x^n\right )^{-3+p} \operatorname {Hypergeometric2F1}\left (4,3-p,4-p,\frac {(b c-a d) x^n}{c \left (a+b x^n\right )}\right )}{c^5 n (2-p) (3-p)} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.42 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.17 \[ \int \frac {x^{-1-n (-2+p)} \left (a+b x^n\right )^p}{\left (c+d x^n\right )^4} \, dx=-\frac {(-1+p) p x^{-n (-2+p)} \left (a+b x^n\right )^p \left (3 \left (2 c+d p x^n\right ) \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,1-p\right )-3 \left (c+d (-1+p) x^n\right ) \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,2-p\right )-2 d x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,3-p\right )+d p x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,3-p\right )-3 c \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,-p\right )-d x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,-p\right )-d p x^n \Phi \left (\frac {(b c-a d) x^n}{c \left (a+b x^n\right )},1,-p\right )\right )}{6 c^2 n \left (c+d x^n\right )^3} \] Input:

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

Output:

-1/6*((-1 + p)*p*(a + b*x^n)^p*(3*(2*c + d*p*x^n)*HurwitzLerchPhi[((b*c - 
a*d)*x^n)/(c*(a + b*x^n)), 1, 1 - p] - 3*(c + d*(-1 + p)*x^n)*HurwitzLerch 
Phi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 2 - p] - 2*d*x^n*HurwitzLerchPhi 
[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] + d*p*x^n*HurwitzLerchPhi[(( 
b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] - 3*c*HurwitzLerchPhi[((b*c - a 
*d)*x^n)/(c*(a + b*x^n)), 1, -p] - d*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n) 
/(c*(a + b*x^n)), 1, -p] - d*p*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n)/(c*(a 
 + b*x^n)), 1, -p]))/(c^2*n*x^(n*(-2 + p))*(c + d*x^n)^3)
 

Rubi [C] (verified)

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

Time = 0.99 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.24, 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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

((1 - p)*p*x^(n*(2 - p))*(a + b*x^n)^p*(3*(2*c + d*p*x^n)*HurwitzLerchPhi[ 
((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 1 - p] - 3*(c - d*(1 - p)*x^n)*Hurwi 
tzLerchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 2 - p] - 2*d*x^n*HurwitzL 
erchPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] + d*p*x^n*HurwitzLerc 
hPhi[((b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, 3 - p] - 3*c*HurwitzLerchPhi[(( 
b*c - a*d)*x^n)/(c*(a + b*x^n)), 1, -p] - d*x^n*HurwitzLerchPhi[((b*c - a* 
d)*x^n)/(c*(a + b*x^n)), 1, -p] - d*p*x^n*HurwitzLerchPhi[((b*c - a*d)*x^n 
)/(c*(a + b*x^n)), 1, -p]))/(6*c^5*n*(1 + (d*x^n)/c)^3)
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(x**(3*n)*(x**n*b + a)**p*a**2*b**2*d**3*p**2 + 5*x**(3*n)*(x**n*b + a)**p 
*a**2*b**2*d**3*p + 6*x**(3*n)*(x**n*b + a)**p*a**2*b**2*d**3 - 5*x**(3*n) 
*(x**n*b + a)**p*a*b**3*c*d**2*p - 11*x**(3*n)*(x**n*b + a)**p*a*b**3*c*d* 
*2 + 4*x**(3*n)*(x**n*b + a)**p*b**4*c**2*d - x**(2*n)*(x**n*b + a)**p*a** 
3*b*d**3*p**3 - 5*x**(2*n)*(x**n*b + a)**p*a**3*b*d**3*p**2 - 6*x**(2*n)*( 
x**n*b + a)**p*a**3*b*d**3*p + 8*x**(2*n)*(x**n*b + a)**p*a**2*b**2*c*d**2 
*p**2 + 26*x**(2*n)*(x**n*b + a)**p*a**2*b**2*c*d**2*p + 18*x**(2*n)*(x**n 
*b + a)**p*a**2*b**2*c*d**2 - 19*x**(2*n)*(x**n*b + a)**p*a*b**3*c**2*d*p 
- 33*x**(2*n)*(x**n*b + a)**p*a*b**3*c**2*d + 12*x**(2*n)*(x**n*b + a)**p* 
b**4*c**3 - x**n*(x**n*b + a)**p*a**4*d**3*p**3 - 6*x**n*(x**n*b + a)**p*a 
**4*d**3*p**2 - 11*x**n*(x**n*b + a)**p*a**4*d**3*p - 6*x**n*(x**n*b + a)* 
*p*a**4*d**3 + x**n*(x**n*b + a)**p*a**3*b*c*d**2*p**3 + 13*x**n*(x**n*b + 
 a)**p*a**3*b*c*d**2*p**2 + 30*x**n*(x**n*b + a)**p*a**3*b*c*d**2*p + 24*x 
**n*(x**n*b + a)**p*a**3*b*c*d**2 - 7*x**n*(x**n*b + a)**p*a**2*b**2*c**2* 
d*p**2 - 31*x**n*(x**n*b + a)**p*a**2*b**2*c**2*d*p - 18*x**n*(x**n*b + a) 
**p*a**2*b**2*c**2*d + 12*x**n*(x**n*b + a)**p*a*b**3*c**3*p + 6*x**(n*p + 
 3*n)*int((x**(3*n)*(x**n*b + a)**p)/(x**(n*p + 5*n)*a**8*b*d**12*p**8*x + 
 16*x**(n*p + 5*n)*a**8*b*d**12*p**7*x + 110*x**(n*p + 5*n)*a**8*b*d**12*p 
**6*x + 424*x**(n*p + 5*n)*a**8*b*d**12*p**5*x + 1001*x**(n*p + 5*n)*a**8* 
b*d**12*p**4*x + 1480*x**(n*p + 5*n)*a**8*b*d**12*p**3*x + 1336*x**(n*p...