\(\int \frac {(e x)^m (a+b x^n)^p (A+B x^n)}{c+d x^n} \, dx\) [60]

Optimal result

Integrand size = 31, antiderivative size = 164 \[ \int \frac {(e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right )}{c+d x^n} \, dx=-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{n},-p,1,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c d e (1+m)}+\frac {B (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{d e (1+m)} \] Output:

-(-A*d+B*c)*(e*x)^(1+m)*(a+b*x^n)^p*AppellF1((1+m)/n,-p,1,(1+m+n)/n,-b*x^n 
/a,-d*x^n/c)/c/d/e/(1+m)/((1+b*x^n/a)^p)+B*(e*x)^(1+m)*(a+b*x^n)^p*hyperge 
om([-p, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/d/e/(1+m)/((1+b*x^n/a)^p)
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.84 \[ \int \frac {(e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right )}{c+d x^n} \, dx=\frac {x (e x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (A (1+m+n) \operatorname {AppellF1}\left (\frac {1+m}{n},-p,1,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+B (1+m) x^n \operatorname {AppellF1}\left (\frac {1+m+n}{n},-p,1,\frac {1+m+2 n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{c (1+m) (1+m+n)} \] Input:

Integrate[((e*x)^m*(a + b*x^n)^p*(A + B*x^n))/(c + d*x^n),x]
 

Output:

(x*(e*x)^m*(a + b*x^n)^p*(A*(1 + m + n)*AppellF1[(1 + m)/n, -p, 1, (1 + m 
+ n)/n, -((b*x^n)/a), -((d*x^n)/c)] + B*(1 + m)*x^n*AppellF1[(1 + m + n)/n 
, -p, 1, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))/(c*(1 + m)*(1 + m 
+ n)*(1 + (b*x^n)/a)^p)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1067, 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.

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

\(\Big \downarrow \) 1067

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {B (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{n},-p,\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{d e (m+1)}-\frac {(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+1}{n},-p,1,\frac {m+n+1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c d e (m+1)}\)

Input:

Int[((e*x)^m*(a + b*x^n)^p*(A + B*x^n))/(c + d*x^n),x]
 

Output:

-(((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^n)^p*AppellF1[(1 + m)/n, -p, 1, (1 + 
 m + n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(c*d*e*(1 + m)*(1 + (b*x^n)/a)^p)) 
 + (B*(e*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(1 + m)/n, -p, (1 + m 
+ n)/n, -((b*x^n)/a)])/(d*e*(1 + m)*(1 + (b*x^n)/a)^p)
 

Defintions of rubi rules used

rule 1067

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n 
_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a 
+ b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, n, p}, x]
 

rule 2009

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

Maple [F]

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

Input:

int((e*x)^m*(a+b*x^n)^p*(A+B*x^n)/(c+d*x^n),x)
 

Output:

int((e*x)^m*(a+b*x^n)^p*(A+B*x^n)/(c+d*x^n),x)
 

Fricas [F]

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

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

Output:

integral((B*x^n + A)*(b*x^n + a)^p*(e*x)^m/(d*x^n + c), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right )}{c+d x^n} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((e*x)**m*(a+b*x**n)**p*(A+B*x**n)/(c+d*x**n),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

integrate((B*x^n + A)*(b*x^n + a)^p*(e*x)^m/(d*x^n + c), x)
 

Giac [F]

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

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

Output:

integrate((B*x^n + A)*(b*x^n + a)^p*(e*x)^m/(d*x^n + c), x)
 

Mupad [F(-1)]

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

int(((e*x)^m*(A + B*x^n)*(a + b*x^n)^p)/(c + d*x^n),x)
 

Output:

int(((e*x)^m*(A + B*x^n)*(a + b*x^n)^p)/(c + d*x^n), x)
 

Reduce [F]

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

int((e*x)^m*(a+b*x^n)^p*(A+B*x^n)/(c+d*x^n),x)
 

Output:

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