\(\int \frac {(f x)^m (a+c x^{2 n})^p}{(d+e x^n)^2} \, dx\) [67]

Optimal result

Integrand size = 26, antiderivative size = 312 \[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\frac {(f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{2 n},-p,2,1+\frac {1+m}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 f (1+m)}-\frac {2 e x^n (f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m+n}{2 n},-p,2,\frac {1+m+3 n}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 f (1+m+n)}+\frac {e^2 x^{2 n} (f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m+2 n}{2 n},-p,2,\frac {1+m+4 n}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 f (1+m+2 n)} \] Output:

(f*x)^(1+m)*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m)/n,2,-p,1+1/2*(1+m)/n,e^2*x^ 
(2*n)/d^2,-c*x^(2*n)/a)/d^2/f/(1+m)/((1+c*x^(2*n)/a)^p)-2*e*x^n*(f*x)^(1+m 
)*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+n)/n,2,-p,1/2*(1+m+3*n)/n,e^2*x^(2*n)/ 
d^2,-c*x^(2*n)/a)/d^3/f/(1+m+n)/((1+c*x^(2*n)/a)^p)+e^2*x^(2*n)*(f*x)^(1+m 
)*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m+2*n)/n,2,-p,1/2*(1+m+4*n)/n,e^2*x^(2*n 
)/d^2,-c*x^(2*n)/a)/d^4/f/(1+m+2*n)/((1+c*x^(2*n)/a)^p)
 

Mathematica [F]

\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \] Input:

Integrate[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n)^2,x]
 

Output:

Integrate[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n)^2, x]
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1886, 2009}

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.

\(\displaystyle \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx\)

\(\Big \downarrow \) 1886

\(\displaystyle x^{-m} (f x)^m \int \left (\frac {d^2 \left (c x^{2 n}+a\right )^p x^m}{\left (d^2-e^2 x^{2 n}\right )^2}-\frac {2 d e \left (c x^{2 n}+a\right )^p x^{m+n}}{\left (d^2-e^2 x^{2 n}\right )^2}+\frac {e^2 \left (c x^{2 n}+a\right )^p x^{m+2 n}}{\left (d^2-e^2 x^{2 n}\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle x^{-m} (f x)^m \left (\frac {x^{m+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+1}{2 n},-p,2,\frac {m+1}{2 n}+1,-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (m+1)}+\frac {e^2 x^{m+2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+2 n+1}{2 n},-p,2,\frac {m+4 n+1}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (m+2 n+1)}-\frac {2 e x^{m+n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+n+1}{2 n},-p,2,\frac {m+3 n+1}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (m+n+1)}\right )\)

Input:

Int[((f*x)^m*(a + c*x^(2*n))^p)/(d + e*x^n)^2,x]
 

Output:

((f*x)^m*((x^(1 + m)*(a + c*x^(2*n))^p*AppellF1[(1 + m)/(2*n), -p, 2, 1 + 
(1 + m)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^2*(1 + m)*(1 + (c* 
x^(2*n))/a)^p) - (2*e*x^(1 + m + n)*(a + c*x^(2*n))^p*AppellF1[(1 + m + n) 
/(2*n), -p, 2, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/ 
(d^3*(1 + m + n)*(1 + (c*x^(2*n))/a)^p) + (e^2*x^(1 + m + 2*n)*(a + c*x^(2 
*n))^p*AppellF1[(1 + m + 2*n)/(2*n), -p, 2, (1 + m + 4*n)/(2*n), -((c*x^(2 
*n))/a), (e^2*x^(2*n))/d^2])/(d^4*(1 + m + 2*n)*(1 + (c*x^(2*n))/a)^p)))/x 
^m
 

Defintions of rubi rules used

rule 1886

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2 
_))^(p_), x_Symbol] :> Simp[(f*x)^m/x^m   Int[ExpandIntegrand[x^m*(a + c*x^ 
(2*n))^p, (d/(d^2 - e^2*x^(2*n)) - e*(x^n/(d^2 - e^2*x^(2*n))))^(-q), x], x 
], x] /; FreeQ[{a, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] &&  !RationalQ[ 
n] &&  !IntegerQ[p] && ILtQ[q, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \frac {\left (f x \right )^{m} \left (a +c \,x^{2 n}\right )^{p}}{\left (d +e \,x^{n}\right )^{2}}d x\]

Input:

int((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^2,x)
 

Output:

int((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^2,x)
 

Fricas [F]

\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \] Input:

integrate((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^2,x, algorithm="fricas")
 

Output:

integral((c*x^(2*n) + a)^p*(f*x)^m/(e^2*x^(2*n) + 2*d*e*x^n + d^2), x)
 

Sympy [F(-1)]

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

integrate((f*x)**m*(a+c*x**(2*n))**p/(d+e*x**n)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \] Input:

integrate((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^2,x, algorithm="maxima")
 

Output:

integrate((c*x^(2*n) + a)^p*(f*x)^m/(e*x^n + d)^2, x)
 

Giac [F]

\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \] Input:

integrate((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^2,x, algorithm="giac")
 

Output:

integrate((c*x^(2*n) + a)^p*(f*x)^m/(e*x^n + d)^2, x)
 

Mupad [F(-1)]

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

int(((a + c*x^(2*n))^p*(f*x)^m)/(d + e*x^n)^2,x)
 

Output:

int(((a + c*x^(2*n))^p*(f*x)^m)/(d + e*x^n)^2, x)
 

Reduce [F]

\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=f^{m} \left (\int \frac {x^{m} \left (x^{2 n} c +a \right )^{p}}{x^{2 n} e^{2}+2 x^{n} d e +d^{2}}d x \right ) \] Input:

int((f*x)^m*(a+c*x^(2*n))^p/(d+e*x^n)^2,x)
 

Output:

f**m*int((x**m*(x**(2*n)*c + a)**p)/(x**(2*n)*e**2 + 2*x**n*d*e + d**2),x)