\(\int \frac {(e x)^m (A+B x^n)}{(c+d x^n)^3} \, dx\) [56]

Optimal result

Integrand size = 22, antiderivative size = 112 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=-\frac {(B c-A d) (e x)^{1+m}}{2 c d e n \left (c+d x^n\right )^2}+\frac {(B c (1+m)-A d (1+m-2 n)) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{2 c^3 d e (1+m) n} \] Output:

-1/2*(-A*d+B*c)*(e*x)^(1+m)/c/d/e/n/(c+d*x^n)^2+1/2*(B*c*(1+m)-A*d*(1+m-2* 
n))*(e*x)^(1+m)*hypergeom([2, (1+m)/n],[(1+m+n)/n],-d*x^n/c)/c^3/d/e/(1+m) 
/n
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.74 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\frac {x (e x)^m \left (B c \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )+(-B c+A d) \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )\right )}{c^3 d (1+m)} \] Input:

Integrate[((e*x)^m*(A + B*x^n))/(c + d*x^n)^3,x]
 

Output:

(x*(e*x)^m*(B*c*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c 
)] + (-(B*c) + A*d)*Hypergeometric2F1[3, (1 + m)/n, (1 + m + n)/n, -((d*x^ 
n)/c)]))/(c^3*d*(1 + m))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 112, 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.

\(\displaystyle \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx\)

\(\Big \downarrow \) 957

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

\(\Big \downarrow \) 888

\(\displaystyle \frac {(e x)^{m+1} (B c (m+1)-A d (m-2 n+1)) \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {d x^n}{c}\right )}{2 c^3 d e (m+1) n}-\frac {(e x)^{m+1} (B c-A d)}{2 c d e n \left (c+d x^n\right )^2}\)

Input:

Int[((e*x)^m*(A + B*x^n))/(c + d*x^n)^3,x]
 

Output:

-1/2*((B*c - A*d)*(e*x)^(1 + m))/(c*d*e*n*(c + d*x^n)^2) + ((B*c*(1 + m) - 
 A*d*(1 + m - 2*n))*(e*x)^(1 + m)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + 
 n)/n, -((d*x^n)/c)])/(2*c^3*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]

\[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right )}{\left (c +d \,x^{n}\right )^{3}}d x\]

Input:

int((e*x)^m*(A+B*x^n)/(c+d*x^n)^3,x)
 

Output:

int((e*x)^m*(A+B*x^n)/(c+d*x^n)^3,x)
 

Fricas [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:

integrate((e*x)^m*(A+B*x^n)/(c+d*x^n)^3,x, algorithm="fricas")
 

Output:

integral((B*x^n + A)*(e*x)^m/(d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 3*c^2*d*x^n 
+ c^3), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 128.35 (sec) , antiderivative size = 8303, normalized size of antiderivative = 74.13 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m*(A+B*x**n)/(c+d*x**n)**3,x)
 

Output:

A*(c**2*c**(m/n + 1/n)*c**(-m/n - 3 - 1/n)*e**m*m**3*x**(m + 1)*lerchphi(d 
*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*c**2*n**4*gamma 
(m/n + 1 + 1/n) + 4*c*d*n**4*x**n*gamma(m/n + 1 + 1/n) + 2*d**2*n**4*x**(2 
*n)*gamma(m/n + 1 + 1/n)) - 3*c**2*c**(m/n + 1/n)*c**(-m/n - 3 - 1/n)*e**m 
*m**2*n*x**(m + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma( 
m/n + 1/n)/(2*c**2*n**4*gamma(m/n + 1 + 1/n) + 4*c*d*n**4*x**n*gamma(m/n + 
 1 + 1/n) + 2*d**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n)) - c**2*c**(m/n + 1/ 
n)*c**(-m/n - 3 - 1/n)*e**m*m**2*n*x**(m + 1)*gamma(m/n + 1/n)/(2*c**2*n** 
4*gamma(m/n + 1 + 1/n) + 4*c*d*n**4*x**n*gamma(m/n + 1 + 1/n) + 2*d**2*n** 
4*x**(2*n)*gamma(m/n + 1 + 1/n)) + 3*c**2*c**(m/n + 1/n)*c**(-m/n - 3 - 1/ 
n)*e**m*m**2*x**(m + 1)*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*g 
amma(m/n + 1/n)/(2*c**2*n**4*gamma(m/n + 1 + 1/n) + 4*c*d*n**4*x**n*gamma( 
m/n + 1 + 1/n) + 2*d**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n)) + 2*c**2*c**(m 
/n + 1/n)*c**(-m/n - 3 - 1/n)*e**m*m*n**2*x**(m + 1)*lerchphi(d*x**n*exp_p 
olar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(2*c**2*n**4*gamma(m/n + 1 + 
1/n) + 4*c*d*n**4*x**n*gamma(m/n + 1 + 1/n) + 2*d**2*n**4*x**(2*n)*gamma(m 
/n + 1 + 1/n)) + 3*c**2*c**(m/n + 1/n)*c**(-m/n - 3 - 1/n)*e**m*m*n**2*x** 
(m + 1)*gamma(m/n + 1/n)/(2*c**2*n**4*gamma(m/n + 1 + 1/n) + 4*c*d*n**4*x* 
*n*gamma(m/n + 1 + 1/n) + 2*d**2*n**4*x**(2*n)*gamma(m/n + 1 + 1/n)) - 6*c 
**2*c**(m/n + 1/n)*c**(-m/n - 3 - 1/n)*e**m*m*n*x**(m + 1)*lerchphi(d*x...
 

Maxima [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:

integrate((e*x)^m*(A+B*x^n)/(c+d*x^n)^3,x, algorithm="maxima")
 

Output:

-((m^2 - m*(n - 2) - n + 1)*B*c*e^m - (m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1 
)*A*d*e^m)*integrate(1/2*x^m/(c^2*d^2*n^2*x^n + c^3*d*n^2), x) + 1/2*((B*c 
^2*e^m*(m - n + 1) - A*c*d*e^m*(m - 3*n + 1))*x*x^m - (A*d^2*e^m*(m - 2*n 
+ 1) - B*c*d*e^m*(m + 1))*x*e^(m*log(x) + n*log(x)))/(c^2*d^3*n^2*x^(2*n) 
+ 2*c^3*d^2*n^2*x^n + c^4*d*n^2)
 

Giac [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}} \,d x } \] Input:

integrate((e*x)^m*(A+B*x^n)/(c+d*x^n)^3,x, algorithm="giac")
 

Output:

integrate((B*x^n + A)*(e*x)^m/(d*x^n + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )}{{\left (c+d\,x^n\right )}^3} \,d x \] Input:

int(((e*x)^m*(A + B*x^n))/(c + d*x^n)^3,x)
 

Output:

int(((e*x)^m*(A + B*x^n))/(c + d*x^n)^3, x)
 

Reduce [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx=\text {too large to display} \] Input:

int((e*x)^m*(A+B*x^n)/(c+d*x^n)^3,x)
 

Output:

(e**m*( - x**(m + n)*a*d*m*x + 2*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**(2*n)*int(x**(m + 2*n)/(x**(3*n)*d**3*m**2 + x**(3*n)*d**3*m*n 
+ 2*x**(3*n)*d**3*m + x**(3*n)*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*c*d**2* 
m**2 + 3*x**(2*n)*c*d**2*m*n + 6*x**(2*n)*c*d**2*m + 3*x**(2*n)*c*d**2*n + 
 3*x**(2*n)*c*d**2 + 3*x**n*c**2*d*m**2 + 3*x**n*c**2*d*m*n + 6*x**n*c**2* 
d*m + 3*x**n*c**2*d*n + 3*x**n*c**2*d + c**3*m**2 + c**3*m*n + 2*c**3*m + 
c**3*n + c**3),x)*a*d**4*m**4 - 2*x**(2*n)*int(x**(m + 2*n)/(x**(3*n)*d**3 
*m**2 + x**(3*n)*d**3*m*n + 2*x**(3*n)*d**3*m + x**(3*n)*d**3*n + x**(3*n) 
*d**3 + 3*x**(2*n)*c*d**2*m**2 + 3*x**(2*n)*c*d**2*m*n + 6*x**(2*n)*c*d**2 
*m + 3*x**(2*n)*c*d**2*n + 3*x**(2*n)*c*d**2 + 3*x**n*c**2*d*m**2 + 3*x**n 
*c**2*d*m*n + 6*x**n*c**2*d*m + 3*x**n*c**2*d*n + 3*x**n*c**2*d + c**3*m** 
2 + c**3*m*n + 2*c**3*m + c**3*n + c**3),x)*a*d**4*m**3*n + 4*x**(2*n)*int 
(x**(m + 2*n)/(x**(3*n)*d**3*m**2 + x**(3*n)*d**3*m*n + 2*x**(3*n)*d**3*m 
+ x**(3*n)*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*c*d**2*m**2 + 3*x**(2*n)*c* 
d**2*m*n + 6*x**(2*n)*c*d**2*m + 3*x**(2*n)*c*d**2*n + 3*x**(2*n)*c*d**2 + 
 3*x**n*c**2*d*m**2 + 3*x**n*c**2*d*m*n + 6*x**n*c**2*d*m + 3*x**n*c**2*d* 
n + 3*x**n*c**2*d + c**3*m**2 + c**3*m*n + 2*c**3*m + c**3*n + c**3),x)*a* 
d**4*m**3 - x**(2*n)*int(x**(m + 2*n)/(x**(3*n)*d**3*m**2 + x**(3*n)*d**3* 
m*n + 2*x**(3*n)*d**3*m + x**(3*n)*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*...