Integrand size = 26, antiderivative size = 104 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx=-\frac {F^{e+\frac {b f}{d}} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right )}{h}+\frac {F^{e+\frac {f (b g-a h)}{d g-c h}} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{h} \]
-F^(e+b*f/d)*Ei(-(-a*d+b*c)*f*ln(F)/d/(d*x+c))/h+F^(e+f*(-a*h+b*g)/(-c*h+d *g))*Ei(-(-a*d+b*c)*f*(h*x+g)*ln(F)/(-c*h+d*g)/(d*x+c))/h
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx=\frac {F^{e+\frac {b f}{d}} \left (-\operatorname {ExpIntegralEi}\left (\frac {(-b c+a d) f \log (F)}{d (c+d x)}\right )+F^{\frac {(b c-a d) f h}{d (d g-c h)}} \operatorname {ExpIntegralEi}\left (\frac {(b c-a d) f (g+h x) \log (F)}{(-d g+c h) (c+d x)}\right )\right )}{h} \]
(F^(e + (b*f)/d)*(-ExpIntegralEi[((-(b*c) + a*d)*f*Log[F])/(d*(c + d*x))] + F^(((b*c - a*d)*f*h)/(d*(d*g - c*h)))*ExpIntegralEi[((b*c - a*d)*f*(g + h*x)*Log[F])/((-(d*g) + c*h)*(c + d*x))]))/h
Time = 1.71 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2661, 2660, 2639, 2663, 2609}
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.
\(\displaystyle \int \frac {F^{\frac {f (a+b x)}{c+d x}+e}}{g+h x} \, dx\) |
\(\Big \downarrow \) 2661 |
\(\displaystyle \frac {d \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x}dx}{h}-\frac {(d g-c h) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x) (g+h x)}dx}{h}\) |
\(\Big \downarrow \) 2660 |
\(\displaystyle \frac {d \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x}dx}{h}-\frac {(d g-c h) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x) (g+h x)}dx}{h}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle -\frac {(d g-c h) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x) (g+h x)}dx}{h}-\frac {F^{\frac {b f}{d}+e} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right )}{h}\) |
\(\Big \downarrow \) 2663 |
\(\displaystyle \frac {\int \frac {F^{e-\frac {(b c-a d) f (g+h x)}{(d g-c h) (c+d x)}+\frac {f (b g-a h)}{d g-c h}} (c+d x)}{g+h x}d\frac {g+h x}{c+d x}}{h}-\frac {F^{\frac {b f}{d}+e} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right )}{h}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle \frac {F^{\frac {f (b g-a h)}{d g-c h}+e} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{h}-\frac {F^{\frac {b f}{d}+e} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right )}{h}\) |
-((F^(e + (b*f)/d)*ExpIntegralEi[-(((b*c - a*d)*f*Log[F])/(d*(c + d*x)))]) /h) + (F^(e + (f*(b*g - a*h))/(d*g - c*h))*ExpIntegralEi[-(((b*c - a*d)*f* (g + h*x)*Log[F])/((d*g - c*h)*(c + d*x)))])/h
3.5.22.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_ Symbol] :> Simp[F^a*(ExpIntegralEi[b*(c + d*x)^n*Log[F]]/(f*n)), x] /; Free Q[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Int[(g + h*x)^m*F^((d*e + b*f)/d - f*((b *c - a*d)/(d*(c + d*x)))), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m}, x] & & NeQ[b*c - a*d, 0] && EqQ[d*g - c*h, 0]
Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))/((g_.) + (h_.)*(x_)), x_Symbol] :> Simp[d/h Int[F^(e + f*((a + b*x)/(c + d*x))) /(c + d*x), x], x] - Simp[(d*g - c*h)/h Int[F^(e + f*((a + b*x)/(c + d*x) ))/((c + d*x)*(g + h*x)), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] & & NeQ[b*c - a*d, 0] && NeQ[d*g - c*h, 0]
Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))/(((g_. ) + (h_.)*(x_))*((i_.) + (j_.)*(x_))), x_Symbol] :> Simp[-d/(h*(d*i - c*j)) Subst[Int[F^(e + f*((b*i - a*j)/(d*i - c*j)) - (b*c - a*d)*f*(x/(d*i - c *j)))/x, x], x, (i + j*x)/(c + d*x)], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && EqQ[d*g - c*h, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(435\) vs. \(2(104)=208\).
Time = 0.38 (sec) , antiderivative size = 436, normalized size of antiderivative = 4.19
method | result | size |
risch | \(\frac {F^{\frac {b f +d e}{d}} \operatorname {Ei}_{1}\left (-\frac {\left (a d f -b c f \right ) \ln \left (F \right )}{d \left (d x +c \right )}-\frac {\left (b f +d e \right ) \ln \left (F \right )}{d}-\frac {-\ln \left (F \right ) b f -d e \ln \left (F \right )}{d}\right ) a d}{h \left (a d -c b \right )}-\frac {F^{\frac {b f +d e}{d}} \operatorname {Ei}_{1}\left (-\frac {\left (a d f -b c f \right ) \ln \left (F \right )}{d \left (d x +c \right )}-\frac {\left (b f +d e \right ) \ln \left (F \right )}{d}-\frac {-\ln \left (F \right ) b f -d e \ln \left (F \right )}{d}\right ) b c}{h \left (a d -c b \right )}-\frac {F^{\frac {a f h -b f g +c e h -d e g}{c h -d g}} \operatorname {Ei}_{1}\left (-\frac {\left (a d f -b c f \right ) \ln \left (F \right )}{d \left (d x +c \right )}-\frac {\left (b f +d e \right ) \ln \left (F \right )}{d}-\frac {-\ln \left (F \right ) a f h +\ln \left (F \right ) b f g -\ln \left (F \right ) c e h +\ln \left (F \right ) d e g}{c h -d g}\right ) a d}{h \left (a d -c b \right )}+\frac {F^{\frac {a f h -b f g +c e h -d e g}{c h -d g}} \operatorname {Ei}_{1}\left (-\frac {\left (a d f -b c f \right ) \ln \left (F \right )}{d \left (d x +c \right )}-\frac {\left (b f +d e \right ) \ln \left (F \right )}{d}-\frac {-\ln \left (F \right ) a f h +\ln \left (F \right ) b f g -\ln \left (F \right ) c e h +\ln \left (F \right ) d e g}{c h -d g}\right ) b c}{h \left (a d -c b \right )}\) | \(436\) |
1/h/(a*d-b*c)*F^((b*f+d*e)/d)*Ei(1,-(a*d*f-b*c*f)*ln(F)/d/(d*x+c)-(b*f+d*e )*ln(F)/d-(-ln(F)*b*f-d*e*ln(F))/d)*a*d-1/h/(a*d-b*c)*F^((b*f+d*e)/d)*Ei(1 ,-(a*d*f-b*c*f)*ln(F)/d/(d*x+c)-(b*f+d*e)*ln(F)/d-(-ln(F)*b*f-d*e*ln(F))/d )*b*c-1/h/(a*d-b*c)*F^((a*f*h-b*f*g+c*e*h-d*e*g)/(c*h-d*g))*Ei(1,-(a*d*f-b *c*f)*ln(F)/d/(d*x+c)-(b*f+d*e)*ln(F)/d-(-ln(F)*a*f*h+ln(F)*b*f*g-ln(F)*c* e*h+ln(F)*d*e*g)/(c*h-d*g))*a*d+1/h/(a*d-b*c)*F^((a*f*h-b*f*g+c*e*h-d*e*g) /(c*h-d*g))*Ei(1,-(a*d*f-b*c*f)*ln(F)/d/(d*x+c)-(b*f+d*e)*ln(F)/d-(-ln(F)* a*f*h+ln(F)*b*f*g-ln(F)*c*e*h+ln(F)*d*e*g)/(c*h-d*g))*b*c
Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.30 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx=-\frac {F^{\frac {d e + b f}{d}} {\rm Ei}\left (-\frac {{\left (b c - a d\right )} f \log \left (F\right )}{d^{2} x + c d}\right ) - F^{\frac {{\left (d e + b f\right )} g - {\left (c e + a f\right )} h}{d g - c h}} {\rm Ei}\left (-\frac {{\left ({\left (b c - a d\right )} f h x + {\left (b c - a d\right )} f g\right )} \log \left (F\right )}{c d g - c^{2} h + {\left (d^{2} g - c d h\right )} x}\right )}{h} \]
-(F^((d*e + b*f)/d)*Ei(-(b*c - a*d)*f*log(F)/(d^2*x + c*d)) - F^(((d*e + b *f)*g - (c*e + a*f)*h)/(d*g - c*h))*Ei(-((b*c - a*d)*f*h*x + (b*c - a*d)*f *g)*log(F)/(c*d*g - c^2*h + (d^2*g - c*d*h)*x)))/h
\[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx=\int \frac {F^{e + \frac {f \left (a + b x\right )}{c + d x}}}{g + h x}\, dx \]
\[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx=\int { \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{h x + g} \,d x } \]
\[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx=\int { \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{h x + g} \,d x } \]
Timed out. \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx=\int \frac {F^{e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}}}{g+h\,x} \,d x \]