\(\int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx\) [423]

Optimal result

Integrand size = 26, antiderivative size = 159 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx=\frac {d F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{h (d g-c h)}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}+\frac {(b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^2} \]

[Out]

d*F^(e+b*f/d-(-a*d+b*c)*f/d/(d*x+c))/h/(-c*h+d*g)-F^(e+f*(b*x+a)/(d*x+c))/h/(h*x+g)+(-a*d+b*c)*f*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))*ln(F)/(-c*h+d*g)^2

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2264, 6874, 2262, 2240, 2241, 2263, 2265, 2209} \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx=\frac {f \log (F) (b c-a d) F^{\frac {f (b g-a h)}{d g-c h}+e} \operatorname {ExpIntegralEi}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^2}+\frac {d F^{-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e}}{h (d g-c h)}-\frac {F^{\frac {f (a+b x)}{c+d x}+e}}{h (g+h x)} \]

[In]

Int[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^2,x]

[Out]

(d*F^(e + (b*f)/d - ((b*c - a*d)*f)/(d*(c + d*x))))/(h*(d*g - c*h)) - F^(e + (f*(a + b*x))/(c + d*x))/(h*(g +
h*x)) + ((b*c - a*d)*f*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)))]*Log[F])/(d*g - c*h)^2

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2241

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] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2262

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]

Rule 2263

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))/((g_.) + (h_.)*(x_)), x_Symbol] :> Dist[d
/h, Int[F^(e + f*((a + b*x)/(c + d*x)))/(c + d*x), x], x] - Dist[(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]

Rule 2264

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))*((g_.) + (h_.)*(x_))^(m_), x_Symbol] :> S
imp[(g + h*x)^(m + 1)*(F^(e + f*((a + b*x)/(c + d*x)))/(h*(m + 1))), x] - Dist[f*(b*c - a*d)*(Log[F]/(h*(m + 1
))), Int[(g + h*x)^(m + 1)*(F^(e + f*((a + b*x)/(c + d*x)))/(c + d*x)^2), 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] && ILtQ[m, -1]

Rule 2265

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))/(((g_.) + (h_.)*(x_))*((i_.) + (j_.)*(x_)
)), x_Symbol] :> Dist[-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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = -\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}+\frac {((b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2 (g+h x)} \, dx}{h} \\ & = -\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}+\frac {((b c-a d) f \log (F)) \int \left (\frac {d F^{e+\frac {f (a+b x)}{c+d x}}}{(d g-c h) (c+d x)^2}-\frac {d F^{e+\frac {f (a+b x)}{c+d x}} h}{(d g-c h)^2 (c+d x)}+\frac {F^{e+\frac {f (a+b x)}{c+d x}} h^2}{(d g-c h)^2 (g+h x)}\right ) \, dx}{h} \\ & = -\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}-\frac {(d (b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{(d g-c h)^2}+\frac {((b c-a d) f h \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{g+h x} \, dx}{(d g-c h)^2}+\frac {(d (b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x)^2} \, dx}{h (d g-c h)} \\ & = -\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}-\frac {(d (b c-a d) f \log (F)) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{(d g-c h)^2}+\frac {(d (b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{c+d x} \, dx}{(d g-c h)^2}-\frac {((b c-a d) f \log (F)) \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{d g-c h}+\frac {(d (b c-a d) f \log (F)) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{(c+d x)^2} \, dx}{h (d g-c h)} \\ & = \frac {d F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{h (d g-c h)}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}+\frac {(b c-a d) f F^{e+\frac {b f}{d}} \text {Ei}\left (-\frac {(b c-a d) f \log (F)}{d (c+d x)}\right ) \log (F)}{(d g-c h)^2}+\frac {((b c-a d) f \log (F)) \text {Subst}\left (\int \frac {F^{e+\frac {f (b g-a h)}{d g-c h}-\frac {(b c-a d) f x}{d g-c h}}}{x} \, dx,x,\frac {g+h x}{c+d x}\right )}{(d g-c h)^2}+\frac {(d (b c-a d) f \log (F)) \int \frac {F^{\frac {d e+b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{(d g-c h)^2} \\ & = \frac {d F^{e+\frac {b f}{d}-\frac {(b c-a d) f}{d (c+d x)}}}{h (d g-c h)}-\frac {F^{e+\frac {f (a+b x)}{c+d x}}}{h (g+h x)}+\frac {(b c-a d) f F^{e+\frac {f (b g-a h)}{d g-c h}} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log (F)}{(d g-c h)^2} \\ \end{align*}

Mathematica [F]

\[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx=\int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx \]

[In]

Integrate[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^2,x]

[Out]

Integrate[F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^2, x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(581\) vs. \(2(159)=318\).

Time = 0.53 (sec) , antiderivative size = 582, normalized size of antiderivative = 3.66

method	result	size
risch	\(\frac {\ln \left (F \right ) F^{\frac {f \left (a d -c b \right )}{d \left (d x +c \right )}} F^{\frac {b f +d e}{d}} a d f}{\left (c h -d g \right )^{2} \left (\frac {f \ln \left (F \right ) a}{d x +c}-\frac {f \ln \left (F \right ) c b}{d \left (d x +c \right )}+\frac {\ln \left (F \right ) b f}{d}+\ln \left (F \right ) e -\frac {\ln \left (F \right ) a f h}{c h -d g}+\frac {\ln \left (F \right ) b f g}{c h -d g}-\frac {\ln \left (F \right ) c e h}{c h -d g}+\frac {\ln \left (F \right ) d e g}{c h -d g}\right )}-\frac {\ln \left (F \right ) F^{\frac {f \left (a d -c b \right )}{d \left (d x +c \right )}} F^{\frac {b f +d e}{d}} b c f}{\left (c h -d g \right )^{2} \left (\frac {f \ln \left (F \right ) a}{d x +c}-\frac {f \ln \left (F \right ) c b}{d \left (d x +c \right )}+\frac {\ln \left (F \right ) b f}{d}+\ln \left (F \right ) e -\frac {\ln \left (F \right ) a f h}{c h -d g}+\frac {\ln \left (F \right ) b f g}{c h -d g}-\frac {\ln \left (F \right ) c e h}{c h -d g}+\frac {\ln \left (F \right ) d e g}{c h -d g}\right )}+\frac {\ln \left (F \right ) F^{\frac {a f h -b f g +c e h -d e g}{c h -d g}} \operatorname {Ei}_{1}\left (-\frac {\left (a d f -b c f \right ) \ln \left (F \right )}{d \left (d x +c \right )}-\frac {\left (b f +d e \right ) \ln \left (F \right )}{d}-\frac {-\ln \left (F \right ) a f h +\ln \left (F \right ) b f g -\ln \left (F \right ) c e h +\ln \left (F \right ) d e g}{c h -d g}\right ) a d f}{\left (c h -d g \right )^{2}}-\frac {\ln \left (F \right ) F^{\frac {a f h -b f g +c e h -d e g}{c h -d g}} \operatorname {Ei}_{1}\left (-\frac {\left (a d f -b c f \right ) \ln \left (F \right )}{d \left (d x +c \right )}-\frac {\left (b f +d e \right ) \ln \left (F \right )}{d}-\frac {-\ln \left (F \right ) a f h +\ln \left (F \right ) b f g -\ln \left (F \right ) c e h +\ln \left (F \right ) d e g}{c h -d g}\right ) b c f}{\left (c h -d g \right )^{2}}\)	\(582\)

[In]

int(F^(e+f*(b*x+a)/(d*x+c))/(h*x+g)^2,x,method=_RETURNVERBOSE)

[Out]

ln(F)/(c*h-d*g)^2*F^(f*(a*d-b*c)/d/(d*x+c))*F^((b*f+d*e)/d)/(f*ln(F)/(d*x+c)*a-f*ln(F)/d/(d*x+c)*c*b+ln(F)/d*b
*f+ln(F)*e-1/(c*h-d*g)*ln(F)*a*f*h+1/(c*h-d*g)*ln(F)*b*f*g-1/(c*h-d*g)*ln(F)*c*e*h+1/(c*h-d*g)*ln(F)*d*e*g)*a*
d*f-ln(F)/(c*h-d*g)^2*F^(f*(a*d-b*c)/d/(d*x+c))*F^((b*f+d*e)/d)/(f*ln(F)/(d*x+c)*a-f*ln(F)/d/(d*x+c)*c*b+ln(F)
/d*b*f+ln(F)*e-1/(c*h-d*g)*ln(F)*a*f*h+1/(c*h-d*g)*ln(F)*b*f*g-1/(c*h-d*g)*ln(F)*c*e*h+1/(c*h-d*g)*ln(F)*d*e*g
)*b*c*f+ln(F)/(c*h-d*g)^2*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*f-ln(F)/(c*h-d*g)^2*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*f

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.38 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx=\frac {{\left ({\left (b c - a d\right )} f h x + {\left (b c - a d\right )} f g\right )} F^{\frac {{\left (d e + b f\right )} g - {\left (c e + a f\right )} h}{d g - c h}} {\rm Ei}\left (-\frac {{\left ({\left (b c - a d\right )} f h x + {\left (b c - a d\right )} f g\right )} \log \left (F\right )}{c d g - c^{2} h + {\left (d^{2} g - c d h\right )} x}\right ) \log \left (F\right ) + {\left (c d g - c^{2} h + {\left (d^{2} g - c d h\right )} x\right )} F^{\frac {c e + a f + {\left (d e + b f\right )} x}{d x + c}}}{d^{2} g^{3} - 2 \, c d g^{2} h + c^{2} g h^{2} + {\left (d^{2} g^{2} h - 2 \, c d g h^{2} + c^{2} h^{3}\right )} x} \]

[In]

integrate(F^(e+f*(b*x+a)/(d*x+c))/(h*x+g)^2,x, algorithm="fricas")

[Out]

(((b*c - a*d)*f*h*x + (b*c - a*d)*f*g)*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))*log(F) + (c*d*g - c^2*h + (d^2*g - c*d*h)*x)*F^
((c*e + a*f + (d*e + b*f)*x)/(d*x + c)))/(d^2*g^3 - 2*c*d*g^2*h + c^2*g*h^2 + (d^2*g^2*h - 2*c*d*g*h^2 + c^2*h
^3)*x)

Sympy [F]

\[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx=\int \frac {F^{e + \frac {f \left (a + b x\right )}{c + d x}}}{\left (g + h x\right )^{2}}\, dx \]

[In]

integrate(F**(e+f*(b*x+a)/(d*x+c))/(h*x+g)**2,x)

[Out]

Integral(F**(e + f*(a + b*x)/(c + d*x))/(g + h*x)**2, x)

Maxima [F]

\[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx=\int { \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{2}} \,d x } \]

[In]

integrate(F^(e+f*(b*x+a)/(d*x+c))/(h*x+g)^2,x, algorithm="maxima")

[Out]

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^2, x)

Giac [F]

\[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx=\int { \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{2}} \,d x } \]

[In]

integrate(F^(e+f*(b*x+a)/(d*x+c))/(h*x+g)^2,x, algorithm="giac")

[Out]

integrate(F^(e + (b*x + a)*f/(d*x + c))/(h*x + g)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx=\int \frac {F^{e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}}}{{\left (g+h\,x\right )}^2} \,d x \]

[In]

int(F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^2,x)

[Out]

int(F^(e + (f*(a + b*x))/(c + d*x))/(g + h*x)^2, x)