\(\int \frac {(e x)^m}{(a+b x^n) (c+d x^n)} \, dx\) [491]

Optimal result

Integrand size = 24, antiderivative size = 114 \[ \int \frac {(e x)^m}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\frac {b (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a (b c-a d) e (1+m)}-\frac {d (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c (b c-a d) e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77 \[ \int \frac {(e x)^m}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\frac {x (e x)^m \left (-b c \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )+a d \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )\right )}{a c (-b c+a d) (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1010, 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.

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

\(\Big \downarrow \) 1010

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

(b*(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)) - (d*(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))
 

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 1010

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((e*x)**m/((a + b*x**n)*(c + d*x**n)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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