Integrand size = 31, antiderivative size = 127 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\frac {(A b-a B) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a (b c-a d) e (1+m)}+\frac {(B c-A d) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )}{c (b c-a d) e (1+m)} \] Output:
(A*b-B*a)*(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)+(-A*d+B*c)*(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)
Time = 0.37 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.80 \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\frac {x (e x)^m \left ((-A b c+a B c) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )+a (-B c+A d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {d x^n}{c}\right )\right )}{a c (-b c+a d) (1+m)} \] Input:
Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)),x]
Output:
(x*(e*x)^m*((-(A*b*c) + a*B*c)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n) /n, -((b*x^n)/a)] + a*(-(B*c) + A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(a*c*(-(b*c) + a*d)*(1 + m))
Time = 0.52 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {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.
\(\displaystyle \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx\) |
\(\Big \downarrow \) 1067 |
\(\displaystyle \int \left (\frac {(e x)^m (A b-a B)}{(b c-a d) \left (a+b x^n\right )}+\frac {(e x)^m (B c-A d)}{(b c-a d) \left (c+d x^n\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(e x)^{m+1} (A b-a B) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{a e (m+1) (b c-a d)}+\frac {(e x)^{m+1} (B c-A d) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {d x^n}{c}\right )}{c e (m+1) (b c-a d)}\) |
Input:
Int[((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)),x]
Output:
((A*b - a*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)) + ((B*c - A*d)*(e*x)^(1 + m)*Hype rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*(b*c - a*d)*e *(1 + m))
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]
\[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right )}{\left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right )}d x\]
Input:
int((e*x)^m*(A+B*x^n)/(a+b*x^n)/(c+d*x^n),x)
Output:
int((e*x)^m*(A+B*x^n)/(a+b*x^n)/(c+d*x^n),x)
\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}} \,d x } \] Input:
integrate((e*x)^m*(A+B*x^n)/(a+b*x^n)/(c+d*x^n),x, algorithm="fricas")
Output:
integral((B*x^n + A)*(e*x)^m/(b*d*x^(2*n) + a*c + (b*c + a*d)*x^n), x)
Exception generated. \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((e*x)**m*(A+B*x**n)/(a+b*x**n)/(c+d*x**n),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}} \,d x } \] Input:
integrate((e*x)^m*(A+B*x^n)/(a+b*x^n)/(c+d*x^n),x, algorithm="maxima")
Output:
integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)*(d*x^n + c)), x)
\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int { \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}} \,d x } \] Input:
integrate((e*x)^m*(A+B*x^n)/(a+b*x^n)/(c+d*x^n),x, algorithm="giac")
Output:
integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)*(d*x^n + c)), x)
Timed out. \[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )}{\left (a+b\,x^n\right )\,\left (c+d\,x^n\right )} \,d x \] Input:
int(((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)),x)
Output:
int(((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)), x)
\[ \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx=e^{m} \left (\int \frac {x^{m}}{x^{n} d +c}d x \right ) \] Input:
int((e*x)^m*(A+B*x^n)/(a+b*x^n)/(c+d*x^n),x)
Output:
e**m*int(x**m/(x**n*d + c),x)