\(\int \frac {(e x)^m (A+B x^n)}{(c+d x^n)^2} \, dx\) [50]

Optimal result

Integrand size = 22, antiderivative size = 107 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=-\frac {(B c-A d) (e x)^{1+m}}{c d e n \left (c+d x^n\right )}+\frac {(B c (1+m)-A d (1+m-n)) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c^2 d e (1+m) n} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.78 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\frac {x (e x)^m \left (B c \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )+(-B c+A d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )\right )}{c^2 d (1+m)} \] Input:

Integrate[((e*x)^m*(A + B*x^n))/(c + d*x^n)^2,x]
 

Output:

(x*(e*x)^m*(B*c*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c 
)] + (-(B*c) + A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^ 
n)/c)]))/(c^2*d*(1 + m))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {957, 888}

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.

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

\(\Big \downarrow \) 957

\(\displaystyle \frac {(B c (m+1)-A d (m-n+1)) \int \frac {(e x)^m}{d x^n+c}dx}{c d n}-\frac {(e x)^{m+1} (B c-A d)}{c d e n \left (c+d x^n\right )}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {(e x)^{m+1} (B c (m+1)-A d (m-n+1)) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {d x^n}{c}\right )}{c^2 d e (m+1) n}-\frac {(e x)^{m+1} (B c-A d)}{c d e n \left (c+d x^n\right )}\)

Input:

Int[((e*x)^m*(A + B*x^n))/(c + d*x^n)^2,x]
 

Output:

-(((B*c - A*d)*(e*x)^(1 + m))/(c*d*e*n*(c + d*x^n))) + ((B*c*(1 + m) - A*d 
*(1 + m - n))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, 
 -((d*x^n)/c)])/(c^2*d*e*(1 + m)*n)
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 957

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a 
*b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* 
(p + 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, 
 m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N 
eQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 
, m, (-n)*(p + 1)]))
 

Maple [F]

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

Input:

int((e*x)^m*(A+B*x^n)/(c+d*x^n)^2,x)
 

Output:

int((e*x)^m*(A+B*x^n)/(c+d*x^n)^2,x)
 

Fricas [F]

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

integrate((e*x)^m*(A+B*x^n)/(c+d*x^n)^2,x, algorithm="fricas")
 

Output:

integral((B*x^n + A)*(e*x)^m/(d^2*x^(2*n) + 2*c*d*x^n + c^2), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.94 (sec) , antiderivative size = 2382, normalized size of antiderivative = 22.26 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m*(A+B*x**n)/(c+d*x**n)**2,x)
 

Output:

A*(-c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*m**2*x**(m + 1)*lerchphi(d*x 
**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 
1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n)) + c*c**(m/n + 1/n)*c**(-m/n - 
 2 - 1/n)*e**m*m*n*x**(m + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 
1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n 
 + 1 + 1/n)) + c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*m*n*x**(m + 1)*ga 
mma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 
1/n)) - 2*c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*m*x**(m + 1)*lerchphi( 
d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n 
 + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n)) + c*c**(m/n + 1/n)*c**(-m/ 
n - 2 - 1/n)*e**m*n*x**(m + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 
 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/ 
n + 1 + 1/n)) + c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*n*x**(m + 1)*gam 
ma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1 
/n)) - c*c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*e**m*x**(m + 1)*lerchphi(d*x** 
n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 
+ 1/n) + d*n**3*x**n*gamma(m/n + 1 + 1/n)) - c**(m/n + 1/n)*c**(-m/n - 2 - 
 1/n)*d*e**m*m**2*x**n*x**(m + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/ 
n + 1/n)*gamma(m/n + 1/n)/(c*n**3*gamma(m/n + 1 + 1/n) + d*n**3*x**n*gamma 
(m/n + 1 + 1/n)) + c**(m/n + 1/n)*c**(-m/n - 2 - 1/n)*d*e**m*m*n*x**n*x...
 

Maxima [F]

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

integrate((e*x)^m*(A+B*x^n)/(c+d*x^n)^2,x, algorithm="maxima")
 

Output:

-(B*c*e^m - A*d*e^m)*x*x^m/(c*d^2*n*x^n + c^2*d*n) - (A*d*e^m*(m - n + 1) 
- B*c*e^m*(m + 1))*integrate(x^m/(c*d^2*n*x^n + c^2*d*n), x)
 

Giac [F]

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

integrate((e*x)^m*(A+B*x^n)/(c+d*x^n)^2,x, algorithm="giac")
 

Output:

integrate((B*x^n + A)*(e*x)^m/(d*x^n + c)^2, x)
 

Mupad [F(-1)]

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

int(((e*x)^m*(A + B*x^n))/(c + d*x^n)^2,x)
 

Output:

int(((e*x)^m*(A + B*x^n))/(c + d*x^n)^2, x)
 

Reduce [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\text {too large to display} \] Input:

int((e*x)^m*(A+B*x^n)/(c+d*x^n)^2,x)
 

Output:

(e**m*( - x**(m + n)*a*d*m*x + x**(m + n)*a*d*n*x - x**(m + n)*a*d*x + x** 
(m + n)*b*c*m*x + x**(m + n)*b*c*x + x**m*a*c*m*x + x**m*a*c*n*x + x**m*a* 
c*x + x**n*int(x**(m + 2*n)/(x**(2*n)*d**2*m + x**(2*n)*d**2*n + x**(2*n)* 
d**2 + 2*x**n*c*d*m + 2*x**n*c*d*n + 2*x**n*c*d + c**2*m + c**2*n + c**2), 
x)*a*d**3*m**3 + 3*x**n*int(x**(m + 2*n)/(x**(2*n)*d**2*m + x**(2*n)*d**2* 
n + x**(2*n)*d**2 + 2*x**n*c*d*m + 2*x**n*c*d*n + 2*x**n*c*d + c**2*m + c* 
*2*n + c**2),x)*a*d**3*m**2 - x**n*int(x**(m + 2*n)/(x**(2*n)*d**2*m + x** 
(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c*d*m + 2*x**n*c*d*n + 2*x**n*c*d + 
c**2*m + c**2*n + c**2),x)*a*d**3*m*n**2 + 3*x**n*int(x**(m + 2*n)/(x**(2* 
n)*d**2*m + x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c*d*m + 2*x**n*c*d*n 
+ 2*x**n*c*d + c**2*m + c**2*n + c**2),x)*a*d**3*m - x**n*int(x**(m + 2*n) 
/(x**(2*n)*d**2*m + x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c*d*m + 2*x** 
n*c*d*n + 2*x**n*c*d + c**2*m + c**2*n + c**2),x)*a*d**3*n**2 + x**n*int(x 
**(m + 2*n)/(x**(2*n)*d**2*m + x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x**n*c* 
d*m + 2*x**n*c*d*n + 2*x**n*c*d + c**2*m + c**2*n + c**2),x)*a*d**3 - x**n 
*int(x**(m + 2*n)/(x**(2*n)*d**2*m + x**(2*n)*d**2*n + x**(2*n)*d**2 + 2*x 
**n*c*d*m + 2*x**n*c*d*n + 2*x**n*c*d + c**2*m + c**2*n + c**2),x)*b*c*d** 
2*m**3 - x**n*int(x**(m + 2*n)/(x**(2*n)*d**2*m + x**(2*n)*d**2*n + x**(2* 
n)*d**2 + 2*x**n*c*d*m + 2*x**n*c*d*n + 2*x**n*c*d + c**2*m + c**2*n + c** 
2),x)*b*c*d**2*m**2*n - 3*x**n*int(x**(m + 2*n)/(x**(2*n)*d**2*m + x**(...