\(\int \frac {(e x)^m (A+B x^n)}{(a+b x^n)^2 (c+d x^n)^2} \, dx\) [52]

Optimal result

Integrand size = 31, antiderivative size = 315 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\frac {d (A b c-2 a B c+a A d) (e x)^{1+m}}{a c (b c-a d)^2 e n \left (c+d x^n\right )}+\frac {(A b-a B) (e x)^{1+m}}{a (b c-a d) e n \left (a+b x^n\right ) \left (c+d x^n\right )}+\frac {b (a B (b c (1+m)-a d (1+m-2 n))+A b (a d (1+m-3 n)-b c (1+m-n))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^3 e (1+m) n}-\frac {d (b c (A d (1+m-3 n)-B c (1+m-2 n))+a d (B c (1+m)-A d (1+m-n))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c^2 (b c-a d)^3 e (1+m) n} \] Output:

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

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.66 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=\frac {x (e x)^m \left (\frac {b (b B c-2 A b d+a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a}-\frac {d (b B c-2 A b d+a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c}+\frac {b (-A b+a B) (-b c+a d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a^2}-\frac {d (b c-a d) (B c-A d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c^2}\right )}{(b c-a d)^3 (1+m)} \] Input:

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

Output:

(x*(e*x)^m*((b*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + m)/n, ( 
1 + m + n)/n, -((b*x^n)/a)])/a - (d*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometr 
ic2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/c + (b*(-(A*b) + a*B)*(- 
(b*c) + a*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]) 
/a^2 - (d*(b*c - a*d)*(B*c - A*d)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + 
 n)/n, -((d*x^n)/c)])/c^2))/((b*c - a*d)^3*(1 + m))
 

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1065, 25, 1065, 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.

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

\(\Big \downarrow \) 1065

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1065

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

\(\Big \downarrow \) 1067

\(\displaystyle \frac {\frac {\int \left (\frac {b c (a B (b c (m+1)-a d (m-2 n+1))+A b (a d (m-3 n+1)-b c (m-n+1))) n (e x)^m}{(b c-a d) \left (b x^n+a\right )}+\frac {a d (-b c (A d (m-3 n+1)-B c (m-2 n+1))-a d (B c (m+1)-A d (m-n+1))) n (e x)^m}{(b c-a d) \left (d x^n+c\right )}\right )dx}{c n (b c-a d)}+\frac {d (e x)^{m+1} (a A d-2 a B c+A b c)}{c e (b c-a d) \left (c+d x^n\right )}}{a n (b c-a d)}+\frac {(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 1065

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 + 1)/(a*g*n*(b*c - a*d)*(p + 1))) 
, x] + Simp[1/(a*n*(b*c - a*d)*(p + 1))   Int[(g*x)^m*(a + b*x^n)^(p + 1)*( 
c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e 
- a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, 
g, m, n, q}, x] && LtQ[p, -1]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((B*x^n + A)*(e*x)^m/(b^2*d^2*x^(4*n) + a^2*c^2 + 2*(b^2*c*d + a*b 
*d^2)*x^(3*n) + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(2*n) + 2*(a*b*c^2 + a^2 
*c*d)*x^n), x)
 

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

((b^3*c*e^m*(m - n + 1) - a*b^2*d*e^m*(m - 3*n + 1))*A + (a^2*b*d*e^m*(m - 
 2*n + 1) - a*b^2*c*e^m*(m + 1))*B)*integrate(-x^m/(a^2*b^3*c^3*n - 3*a^3* 
b^2*c^2*d*n + 3*a^4*b*c*d^2*n - a^5*d^3*n + (a*b^4*c^3*n - 3*a^2*b^3*c^2*d 
*n + 3*a^3*b^2*c*d^2*n - a^4*b*d^3*n)*x^n), x) - ((a*d^3*e^m*(m - n + 1) - 
 b*c*d^2*e^m*(m - 3*n + 1))*A + (b*c^2*d*e^m*(m - 2*n + 1) - a*c*d^2*e^m*( 
m + 1))*B)*integrate(-x^m/(b^3*c^5*n - 3*a*b^2*c^4*d*n + 3*a^2*b*c^3*d^2*n 
 - a^3*c^2*d^3*n + (b^3*c^4*d*n - 3*a*b^2*c^3*d^2*n + 3*a^2*b*c^2*d^3*n - 
a^3*c*d^4*n)*x^n), x) + (((b^2*c^2*e^m + a^2*d^2*e^m)*A - (a*b*c^2*e^m + a 
^2*c*d*e^m)*B)*x*x^m - (2*B*a*b*c*d*e^m - (b^2*c*d*e^m + a*b*d^2*e^m)*A)*x 
*e^(m*log(x) + n*log(x)))/(a^2*b^2*c^4*n - 2*a^3*b*c^3*d*n + a^4*c^2*d^2*n 
 + (a*b^3*c^3*d*n - 2*a^2*b^2*c^2*d^2*n + a^3*b*c*d^3*n)*x^(2*n) + (a*b^3* 
c^4*n - a^2*b^2*c^3*d*n - a^3*b*c^2*d^2*n + a^4*c*d^3*n)*x^n)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right )^2 \left (c+d x^n\right )^2} \, dx=e^{m} \left (\int \frac {x^{m}}{x^{3 n} b \,d^{2}+x^{2 n} a \,d^{2}+2 x^{2 n} b c d +2 x^{n} a c d +x^{n} b \,c^{2}+a \,c^{2}}d x \right ) \] Input:

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

Output:

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