Integrand size = 23, antiderivative size = 100 \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{(c+d x)^2} \, dx=-\frac {a}{d (c+d x)}-\frac {b \left (F^{e g+f g x}\right )^n}{d (c+d x)}+\frac {b f F^{\left (e-\frac {c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n g n \operatorname {ExpIntegralEi}\left (\frac {f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2} \] Output:
-a/d/(d*x+c)-b*(F^(f*g*x+e*g))^n/d/(d*x+c)+b*f*F^((e-c*f/d)*g*n-g*n*(f*x+e ))*(F^(f*g*x+e*g))^n*g*n*Ei(f*g*n*(d*x+c)*ln(F)/d)*ln(F)/d^2
Time = 0.59 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.78 \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{(c+d x)^2} \, dx=\frac {-\frac {d \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{c+d x}+b f F^{-\frac {f g n (c+d x)}{d}} \left (F^{g (e+f x)}\right )^n g n \operatorname {ExpIntegralEi}\left (\frac {f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2} \] Input:
Integrate[(a + b*(F^(g*(e + f*x)))^n)/(c + d*x)^2,x]
Output:
(-((d*(a + b*(F^(g*(e + f*x)))^n))/(c + d*x)) + (b*f*(F^(g*(e + f*x)))^n*g *n*ExpIntegralEi[(f*g*n*(c + d*x)*Log[F])/d]*Log[F])/F^((f*g*n*(c + d*x))/ d))/d^2
Time = 0.58 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2614, 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.
\(\displaystyle \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 2614 |
\(\displaystyle \int \left (\frac {a}{(c+d x)^2}+\frac {b \left (F^{e g+f g x}\right )^n}{(c+d x)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a}{d (c+d x)}+\frac {b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac {c f}{d}\right )-g n (e+f x)} \operatorname {ExpIntegralEi}\left (\frac {f g n (c+d x) \log (F)}{d}\right )}{d^2}-\frac {b \left (F^{e g+f g x}\right )^n}{d (c+d x)}\) |
Input:
Int[(a + b*(F^(g*(e + f*x)))^n)/(c + d*x)^2,x]
Output:
-(a/(d*(c + d*x))) - (b*(F^(e*g + f*g*x))^n)/(d*(c + d*x)) + (b*f*F^((e - (c*f)/d)*g*n - g*n*(e + f*x))*(F^(e*g + f*g*x))^n*g*n*ExpIntegralEi[(f*g*n *(c + d*x)*Log[F])/d]*Log[F])/d^2
Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*(F ^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
\[\int \frac {a +b \left (F^{g \left (f x +e \right )}\right )^{n}}{\left (d x +c \right )^{2}}d x\]
Input:
int((a+b*(F^(g*(f*x+e)))^n)/(d*x+c)^2,x)
Output:
int((a+b*(F^(g*(f*x+e)))^n)/(d*x+c)^2,x)
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{(c+d x)^2} \, dx=\frac {{\left (b d f g n x + b c f g n\right )} F^{\frac {{\left (d e - c f\right )} g n}{d}} {\rm Ei}\left (\frac {{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right ) - F^{f g n x + e g n} b d - a d}{d^{3} x + c d^{2}} \] Input:
integrate((a+b*(F^(g*(f*x+e)))^n)/(d*x+c)^2,x, algorithm="fricas")
Output:
((b*d*f*g*n*x + b*c*f*g*n)*F^((d*e - c*f)*g*n/d)*Ei((d*f*g*n*x + c*f*g*n)* log(F)/d)*log(F) - F^(f*g*n*x + e*g*n)*b*d - a*d)/(d^3*x + c*d^2)
\[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{(c+d x)^2} \, dx=\int \frac {a + b \left (F^{e g + f g x}\right )^{n}}{\left (c + d x\right )^{2}}\, dx \] Input:
integrate((a+b*(F**(g*(f*x+e)))**n)/(d*x+c)**2,x)
Output:
Integral((a + b*(F**(e*g + f*g*x))**n)/(c + d*x)**2, x)
\[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{(c+d x)^2} \, dx=\int { \frac {{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((a+b*(F^(g*(f*x+e)))^n)/(d*x+c)^2,x, algorithm="maxima")
Output:
F^(e*g*n)*b*integrate(F^(f*g*n*x)/(d^2*x^2 + 2*c*d*x + c^2), x) - a/(d^2*x + c*d)
\[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{(c+d x)^2} \, dx=\int { \frac {{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate((a+b*(F^(g*(f*x+e)))^n)/(d*x+c)^2,x, algorithm="giac")
Output:
integrate(((F^((f*x + e)*g))^n*b + a)/(d*x + c)^2, x)
Timed out. \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{(c+d x)^2} \, dx=\int \frac {a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int((a + b*(F^(g*(e + f*x)))^n)/(c + d*x)^2,x)
Output:
int((a + b*(F^(g*(e + f*x)))^n)/(c + d*x)^2, x)
\[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{(c+d x)^2} \, dx=\frac {f^{e g n} \left (\int \frac {f^{f g n x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) b \,c^{2}+f^{e g n} \left (\int \frac {f^{f g n x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) b c d x +a x}{c \left (d x +c \right )} \] Input:
int((a+b*(F^(g*(f*x+e)))^n)/(d*x+c)^2,x)
Output:
(f**(e*g*n)*int(f**(f*g*n*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*b*c**2 + f**( e*g*n)*int(f**(f*g*n*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*b*c*d*x + a*x)/(c* (c + d*x))