Integrand size = 76, antiderivative size = 52 \[ \int \frac {e \left (\frac {a+b x}{c+d x}\right )^n+e^2 \left (\frac {a+b x}{c+d x}\right )^{2 n}}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx=\frac {e \left (\frac {a+b x}{c+d x}\right )^n}{(b c-a d) n \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^2} \]
Time = 0.72 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.98 \[ \int \frac {e \left (\frac {a+b x}{c+d x}\right )^n+e^2 \left (\frac {a+b x}{c+d x}\right )^{2 n}}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx=\frac {e \left (\frac {a+b x}{c+d x}\right )^n}{(b c-a d) n \left (-1+e \left (\frac {a+b x}{c+d x}\right )^n\right )^2} \]
Integrate[(e*((a + b*x)/(c + d*x))^n + e^2*((a + b*x)/(c + d*x))^(2*n))/(( a + b*x)*(c + d*x)*(1 - e*((a + b*x)/(c + d*x))^n)^3),x]
Time = 2.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {7292, 27, 7243, 38}
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.
\(\displaystyle \int \frac {e^2 \left (\frac {a+b x}{c+d x}\right )^{2 n}+e \left (\frac {a+b x}{c+d x}\right )^n}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e \left (\frac {a+b x}{c+d x}\right )^n \left (e \left (\frac {a+b x}{c+d x}\right )^n+1\right )}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e \int \frac {\left (\frac {a+b x}{c+d x}\right )^n \left (e \left (\frac {a+b x}{c+d x}\right )^n+1\right )}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3}dx\) |
\(\Big \downarrow \) 7243 |
\(\displaystyle \frac {\int \frac {e \left (\frac {a+b x}{c+d x}\right )^n+1}{\left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3}d\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)}\) |
\(\Big \downarrow \) 38 |
\(\displaystyle \frac {e \left (\frac {a+b x}{c+d x}\right )^n}{n (b c-a d) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^2}\) |
Int[(e*((a + b*x)/(c + d*x))^n + e^2*((a + b*x)/(c + d*x))^(2*n))/((a + b* x)*(c + d*x)*(1 - e*((a + b*x)/(c + d*x))^n)^3),x]
3.2.55.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_)), x_Symbol] :> Simp[d*x*(( a + b*x)^(m + 1)/(b*(m + 2))), x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[a*d - b*c*(m + 2), 0]
Int[(u_)*((c_.) + (d_.)*(v_))^(n_.)*((a_.) + (b_.)*(y_))^(m_.), x_Symbol] : > With[{q = DerivativeDivides[y, u, x]}, Simp[q Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, y], x] /; !FalseQ[q]] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[v, y]
Time = 1.77 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.02
method | result | size |
risch | \(-\frac {e \left (\frac {b x +a}{d x +c}\right )^{n}}{n \left (a d -b c \right ) \left (-1+e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2}}\) | \(53\) |
parallelrisch | \(-\frac {e \left (\frac {b x +a}{d x +c}\right )^{n}}{n \left (a d -b c \right ) \left (-1+e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2}}\) | \(53\) |
norman | \(-\frac {e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}}{n \left (a d -b c \right ) \left (-1+e \,{\mathrm e}^{n \ln \left (\frac {b x +a}{d x +c}\right )}\right )^{2}}\) | \(57\) |
derivativedivides | \(\frac {e \left (-\frac {1}{e \left (-1+e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2}}-\frac {1}{e \left (-1+e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\right )}{\left (a d -b c \right ) n}\) | \(69\) |
default | \(\frac {e \left (-\frac {1}{e \left (-1+e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2}}-\frac {1}{e \left (-1+e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\right )}{\left (a d -b c \right ) n}\) | \(69\) |
int(-(1+e*((b*x+a)/(d*x+c))^n)*e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/(d*x+c ))^n)^3/(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.67 \[ \int \frac {e \left (\frac {a+b x}{c+d x}\right )^n+e^2 \left (\frac {a+b x}{c+d x}\right )^{2 n}}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx=\frac {e \left (\frac {b x + a}{d x + c}\right )^{n}}{{\left (b c - a d\right )} e^{2} n \left (\frac {b x + a}{d x + c}\right )^{2 \, n} - 2 \, {\left (b c - a d\right )} e n \left (\frac {b x + a}{d x + c}\right )^{n} + {\left (b c - a d\right )} n} \]
integrate(-(1+e*((b*x+a)/(d*x+c))^n)*e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/ (d*x+c))^n)^3/(b*x+a)/(d*x+c),x, algorithm="fricas")
e*((b*x + a)/(d*x + c))^n/((b*c - a*d)*e^2*n*((b*x + a)/(d*x + c))^(2*n) - 2*(b*c - a*d)*e*n*((b*x + a)/(d*x + c))^n + (b*c - a*d)*n)
Timed out. \[ \int \frac {e \left (\frac {a+b x}{c+d x}\right )^n+e^2 \left (\frac {a+b x}{c+d x}\right )^{2 n}}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx=\text {Timed out} \]
integrate(-(1+e*((b*x+a)/(d*x+c))**n)*e*((b*x+a)/(d*x+c))**n/(-1+e*((b*x+a )/(d*x+c))**n)**3/(b*x+a)/(d*x+c),x)
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (51) = 102\).
Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 4.06 \[ \int \frac {e \left (\frac {a+b x}{c+d x}\right )^n+e^2 \left (\frac {a+b x}{c+d x}\right )^{2 n}}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx=\frac {1}{2} \, {\left (\frac {{\left (b x + a\right )}^{2 \, n} e}{{\left (b c e^{2} n - a d e^{2} n\right )} {\left (b x + a\right )}^{2 \, n} + {\left (b c n - a d n\right )} {\left (d x + c\right )}^{2 \, n} - 2 \, {\left (b c e n - a d e n\right )} e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}} - \frac {{\left (b x + a\right )}^{2 \, n} e - 2 \, e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}}{{\left (b c e^{2} n - a d e^{2} n\right )} {\left (b x + a\right )}^{2 \, n} + {\left (b c n - a d n\right )} {\left (d x + c\right )}^{2 \, n} - 2 \, {\left (b c e n - a d e n\right )} e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}}\right )} e \]
integrate(-(1+e*((b*x+a)/(d*x+c))^n)*e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/ (d*x+c))^n)^3/(b*x+a)/(d*x+c),x, algorithm="maxima")
1/2*((b*x + a)^(2*n)*e/((b*c*e^2*n - a*d*e^2*n)*(b*x + a)^(2*n) + (b*c*n - a*d*n)*(d*x + c)^(2*n) - 2*(b*c*e*n - a*d*e*n)*e^(n*log(b*x + a) + n*log( d*x + c))) - ((b*x + a)^(2*n)*e - 2*e^(n*log(b*x + a) + n*log(d*x + c)))/( (b*c*e^2*n - a*d*e^2*n)*(b*x + a)^(2*n) + (b*c*n - a*d*n)*(d*x + c)^(2*n) - 2*(b*c*e*n - a*d*e*n)*e^(n*log(b*x + a) + n*log(d*x + c))))*e
\[ \int \frac {e \left (\frac {a+b x}{c+d x}\right )^n+e^2 \left (\frac {a+b x}{c+d x}\right )^{2 n}}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx=\int { -\frac {{\left (e \left (\frac {b x + a}{d x + c}\right )^{n} + 1\right )} e \left (\frac {b x + a}{d x + c}\right )^{n}}{{\left (b x + a\right )} {\left (d x + c\right )} {\left (e \left (\frac {b x + a}{d x + c}\right )^{n} - 1\right )}^{3}} \,d x } \]
integrate(-(1+e*((b*x+a)/(d*x+c))^n)*e*((b*x+a)/(d*x+c))^n/(-1+e*((b*x+a)/ (d*x+c))^n)^3/(b*x+a)/(d*x+c),x, algorithm="giac")
integrate(-(e*((b*x + a)/(d*x + c))^n + 1)*e*((b*x + a)/(d*x + c))^n/((b*x + a)*(d*x + c)*(e*((b*x + a)/(d*x + c))^n - 1)^3), x)
Time = 4.78 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.56 \[ \int \frac {e \left (\frac {a+b x}{c+d x}\right )^n+e^2 \left (\frac {a+b x}{c+d x}\right )^{2 n}}{(a+b x) (c+d x) \left (1-e \left (\frac {a+b x}{c+d x}\right )^n\right )^3} \, dx=-\frac {e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n}{n\,\left (a\,d-b\,c\right )\,\left (e^2\,{\left (\frac {a}{c+d\,x}+\frac {b\,x}{c+d\,x}\right )}^{2\,n}-2\,e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n+1\right )} \]
int(-(e*(e*((a + b*x)/(c + d*x))^n + 1)*((a + b*x)/(c + d*x))^n)/((e*((a + b*x)/(c + d*x))^n - 1)^3*(a + b*x)*(c + d*x)),x)