\(\int \frac {(e x)^m (A+B x^n) (c+d x^n)}{(a+b x^n)^2} \, dx\) [24]

Optimal result

Integrand size = 29, antiderivative size = 175 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^2} \, dx=-\frac {d \left (A-\frac {a B (1+m+n)}{b (1+m)}\right ) (e x)^{1+m}}{a b e n}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )}{a b e n \left (a+b x^n\right )}+\frac {(b c (a B (1+m)-A b (1+m-n))+a d (A b (1+m)-a B (1+m+n))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a^2 b^2 e (1+m) n} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.63 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^2} \, dx=\frac {x (e x)^m \left (a^2 B d+a (b B c+A b d-2 a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )+(A b-a B) (b c-a d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )\right )}{a^2 b^2 (1+m)} \] Input:

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

Output:

(x*(e*x)^m*(a^2*B*d + a*(b*B*c + A*b*d - 2*a*B*d)*Hypergeometric2F1[1, (1 
+ m)/n, (1 + m + n)/n, -((b*x^n)/a)] + (A*b - a*B)*(b*c - a*d)*Hypergeomet 
ric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]))/(a^2*b^2*(1 + m))
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1064, 25, 959, 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.

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

\(\Big \downarrow \) 1064

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 1064

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^( 
m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*b*g*n*(p + 1))), x] + Simp[1/( 
a*b*n*(p + 1))   Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c 
*(b*e*n*(p + 1) + (b*e - a*f)*(m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m 
+ n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && LtQ 
[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 39.24 (sec) , antiderivative size = 5176, normalized size of antiderivative = 29.58 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{\left (a+b x^n\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

A*c*(-a*a**(m/n + 1/n)*a**(-m/n - 2 - 1/n)*e**m*m**2*x**(m + 1)*lerchphi(b 
*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*n**3*gamma(m/n 
+ 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n)) + a*a**(m/n + 1/n)*a**(-m/n 
 - 2 - 1/n)*e**m*m*n*x**(m + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n 
+ 1/n)*gamma(m/n + 1/n)/(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m 
/n + 1 + 1/n)) + a*a**(m/n + 1/n)*a**(-m/n - 2 - 1/n)*e**m*m*n*x**(m + 1)* 
gamma(m/n + 1/n)/(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 
+ 1/n)) - 2*a*a**(m/n + 1/n)*a**(-m/n - 2 - 1/n)*e**m*m*x**(m + 1)*lerchph 
i(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*n**3*gamma(m 
/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n)) + a*a**(m/n + 1/n)*a**(- 
m/n - 2 - 1/n)*e**m*n*x**(m + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n 
 + 1/n)*gamma(m/n + 1/n)/(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma( 
m/n + 1 + 1/n)) + a*a**(m/n + 1/n)*a**(-m/n - 2 - 1/n)*e**m*n*x**(m + 1)*g 
amma(m/n + 1/n)/(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 
 1/n)) - a*a**(m/n + 1/n)*a**(-m/n - 2 - 1/n)*e**m*x**(m + 1)*lerchphi(b*x 
**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*n**3*gamma(m/n + 
1 + 1/n) + b*n**3*x**n*gamma(m/n + 1 + 1/n)) - a**(m/n + 1/n)*a**(-m/n - 2 
 - 1/n)*b*e**m*m**2*x**n*x**(m + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 
m/n + 1/n)*gamma(m/n + 1/n)/(a*n**3*gamma(m/n + 1 + 1/n) + b*n**3*x**n*gam 
ma(m/n + 1 + 1/n)) + a**(m/n + 1/n)*a**(-m/n - 2 - 1/n)*b*e**m*m*n*x**n...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e**m*(x**m*d*x - int(x**m/(x**n*b + a),x)*a*d*m - int(x**m/(x**n*b + a),x 
)*a*d + int(x**m/(x**n*b + a),x)*b*c*m + int(x**m/(x**n*b + a),x)*b*c))/(b 
*(m + 1))