\(\int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx\) [400]

Optimal result

Integrand size = 21, antiderivative size = 460 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx=\frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}-\frac {5 b d^3 F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^4}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^2 (e+f x)^2}+\frac {2 b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{3 (d e-c f)^3 (e+f x)}-\frac {b d^3 F^{a-\frac {b f}{d e-c f}} \operatorname {ExpIntegralEi}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^4}+\frac {b^2 d^3 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^5}-\frac {b^2 d^2 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^4 (e+f x)}+\frac {b^2 d^3 f F^{a-\frac {b f}{d e-c f}} \operatorname {ExpIntegralEi}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log ^2(F)}{(d e-c f)^5}-\frac {b^3 d^3 f^2 F^{a-\frac {b f}{d e-c f}} \operatorname {ExpIntegralEi}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log ^3(F)}{6 (d e-c f)^6} \]

[Out]

1/3*d^3*F^(a+b/(d*x+c))/f/(-c*f+d*e)^3-1/3*F^(a+b/(d*x+c))/f/(f*x+e)^3-5/6*b*d^3*F^(a+b/(d*x+c))*ln(F)/(-c*f+d
*e)^4+1/6*b*d*F^(a+b/(d*x+c))*ln(F)/(-c*f+d*e)^2/(f*x+e)^2+2/3*b*d^2*F^(a+b/(d*x+c))*ln(F)/(-c*f+d*e)^3/(f*x+e
)-b*d^3*F^(a-b*f/(-c*f+d*e))*Ei(b*d*(f*x+e)*ln(F)/(-c*f+d*e)/(d*x+c))*ln(F)/(-c*f+d*e)^4+1/6*b^2*d^3*f*F^(a+b/
(d*x+c))*ln(F)^2/(-c*f+d*e)^5-1/6*b^2*d^2*f*F^(a+b/(d*x+c))*ln(F)^2/(-c*f+d*e)^4/(f*x+e)+b^2*d^3*f*F^(a-b*f/(-
c*f+d*e))*Ei(b*d*(f*x+e)*ln(F)/(-c*f+d*e)/(d*x+c))*ln(F)^2/(-c*f+d*e)^5-1/6*b^3*d^3*f^2*F^(a-b*f/(-c*f+d*e))*E
i(b*d*(f*x+e)*ln(F)/(-c*f+d*e)/(d*x+c))*ln(F)^3/(-c*f+d*e)^6

Rubi [A] (verified)

Time = 2.32 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2255, 6874, 2240, 2241, 2254, 2260, 2209} \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx=-\frac {b^3 d^3 f^2 \log ^3(F) F^{a-\frac {b f}{d e-c f}} \operatorname {ExpIntegralEi}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{6 (d e-c f)^6}+\frac {b^2 d^3 f \log ^2(F) F^{a-\frac {b f}{d e-c f}} \operatorname {ExpIntegralEi}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^5}+\frac {b^2 d^3 f \log ^2(F) F^{a+\frac {b}{c+d x}}}{6 (d e-c f)^5}-\frac {b^2 d^2 f \log ^2(F) F^{a+\frac {b}{c+d x}}}{6 (e+f x) (d e-c f)^4}-\frac {b d^3 \log (F) F^{a-\frac {b f}{d e-c f}} \operatorname {ExpIntegralEi}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^4}+\frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {5 b d^3 \log (F) F^{a+\frac {b}{c+d x}}}{6 (d e-c f)^4}+\frac {2 b d^2 \log (F) F^{a+\frac {b}{c+d x}}}{3 (e+f x) (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}+\frac {b d \log (F) F^{a+\frac {b}{c+d x}}}{6 (e+f x)^2 (d e-c f)^2} \]

[In]

Int[F^(a + b/(c + d*x))/(e + f*x)^4,x]

[Out]

(d^3*F^(a + b/(c + d*x)))/(3*f*(d*e - c*f)^3) - F^(a + b/(c + d*x))/(3*f*(e + f*x)^3) - (5*b*d^3*F^(a + b/(c +
d*x))*Log[F])/(6*(d*e - c*f)^4) + (b*d*F^(a + b/(c + d*x))*Log[F])/(6*(d*e - c*f)^2*(e + f*x)^2) + (2*b*d^2*F
^(a + b/(c + d*x))*Log[F])/(3*(d*e - c*f)^3*(e + f*x)) - (b*d^3*F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(b*d*(
e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F])/(d*e - c*f)^4 + (b^2*d^3*f*F^(a + b/(c + d*x))*Log[F]^2)/(6*
(d*e - c*f)^5) - (b^2*d^2*f*F^(a + b/(c + d*x))*Log[F]^2)/(6*(d*e - c*f)^4*(e + f*x)) + (b^2*d^3*f*F^(a - (b*f
)/(d*e - c*f))*ExpIntegralEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F]^2)/(d*e - c*f)^5 - (b^3*d^
3*f^2*F^(a - (b*f)/(d*e - c*f))*ExpIntegralEi[(b*d*(e + f*x)*Log[F])/((d*e - c*f)*(c + d*x))]*Log[F]^3)/(6*(d*
e - c*f)^6)

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 2254

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[d/f, Int[F^(a + b/(c + d
*x))/(c + d*x), x], x] - Dist[(d*e - c*f)/f, Int[F^(a + b/(c + d*x))/((c + d*x)*(e + f*x)), x], x] /; FreeQ[{F
, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0]

Rule 2255

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(
F^(a + b/(c + d*x))/(f*(m + 1))), x] + Dist[b*d*(Log[F]/(f*(m + 1))), Int[(e + f*x)^(m + 1)*(F^(a + b/(c + d*x
))/(c + d*x)^2), x], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && ILtQ[m, -1]

Rule 2260

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/(((e_.) + (f_.)*(x_))*((g_.) + (h_.)*(x_))), x_Symbol] :> Dist[-
d/(f*(d*g - c*h)), Subst[Int[F^(a - b*(h/(d*g - c*h)) + d*b*(x/(d*g - c*h)))/x, x], x, (g + h*x)/(c + d*x)], x
] /; FreeQ[{F, a, b, c, d, e, f}, x] && EqQ[d*e - c*f, 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^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}-\frac {(b d \log (F)) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2 (e+f x)^3} \, dx}{3 f} \\ & = -\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}-\frac {(b d \log (F)) \int \left (\frac {d^3 F^{a+\frac {b}{c+d x}}}{(d e-c f)^3 (c+d x)^2}-\frac {3 d^3 f F^{a+\frac {b}{c+d x}}}{(d e-c f)^4 (c+d x)}+\frac {f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (e+f x)^3}+\frac {2 d f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^3 (e+f x)^2}+\frac {3 d^2 f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^4 (e+f x)}\right ) \, dx}{3 f} \\ & = -\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}+\frac {\left (b d^4 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{(d e-c f)^4}-\frac {\left (b d^3 f \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{e+f x} \, dx}{(d e-c f)^4}-\frac {\left (b d^4 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{3 f (d e-c f)^3}-\frac {\left (2 b d^2 f \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^2} \, dx}{3 (d e-c f)^3}-\frac {(b d f \log (F)) \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^3} \, dx}{3 (d e-c f)^2} \\ & = \frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^2 (e+f x)^2}+\frac {2 b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{3 (d e-c f)^3 (e+f x)}-\frac {b d^3 F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log (F)}{(d e-c f)^4}-\frac {\left (b d^4 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{(d e-c f)^4}+\frac {\left (b d^3 \log (F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x) (e+f x)} \, dx}{(d e-c f)^3}+\frac {\left (2 b^2 d^3 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2 (e+f x)} \, dx}{3 (d e-c f)^3}+\frac {\left (b^2 d^2 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2 (e+f x)^2} \, dx}{6 (d e-c f)^2} \\ & = \frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^2 (e+f x)^2}+\frac {2 b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{3 (d e-c f)^3 (e+f x)}-\frac {\left (b d^3 \log (F)\right ) \text {Subst}\left (\int \frac {F^{a-\frac {b f}{d e-c f}+\frac {b d x}{d e-c f}}}{x} \, dx,x,\frac {e+f x}{c+d x}\right )}{(d e-c f)^4}+\frac {\left (2 b^2 d^3 \log ^2(F)\right ) \int \left (\frac {d F^{a+\frac {b}{c+d x}}}{(d e-c f) (c+d x)^2}-\frac {d f F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (c+d x)}+\frac {f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (e+f x)}\right ) \, dx}{3 (d e-c f)^3}+\frac {\left (b^2 d^2 \log ^2(F)\right ) \int \left (\frac {d^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (c+d x)^2}-\frac {2 d^2 f F^{a+\frac {b}{c+d x}}}{(d e-c f)^3 (c+d x)}+\frac {f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (e+f x)^2}+\frac {2 d f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^3 (e+f x)}\right ) \, dx}{6 (d e-c f)^2} \\ & = \frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^2 (e+f x)^2}+\frac {2 b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{3 (d e-c f)^3 (e+f x)}-\frac {b d^3 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^4}-\frac {\left (b^2 d^4 f \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{3 (d e-c f)^5}-\frac {\left (2 b^2 d^4 f \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{3 (d e-c f)^5}+\frac {\left (b^2 d^3 f^2 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{e+f x} \, dx}{3 (d e-c f)^5}+\frac {\left (2 b^2 d^3 f^2 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{e+f x} \, dx}{3 (d e-c f)^5}+\frac {\left (b^2 d^4 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{6 (d e-c f)^4}+\frac {\left (2 b^2 d^4 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{3 (d e-c f)^4}+\frac {\left (b^2 d^2 f^2 \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^2} \, dx}{6 (d e-c f)^4} \\ & = \frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}-\frac {5 b d^3 F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^4}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^2 (e+f x)^2}+\frac {2 b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{3 (d e-c f)^3 (e+f x)}-\frac {b d^3 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^4}-\frac {b^2 d^2 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^4 (e+f x)}+\frac {b^2 d^3 f F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log ^2(F)}{(d e-c f)^5}+\frac {\left (b^2 d^4 f \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{3 (d e-c f)^5}+\frac {\left (2 b^2 d^4 f \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{3 (d e-c f)^5}-\frac {\left (b^2 d^3 f \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x) (e+f x)} \, dx}{3 (d e-c f)^4}-\frac {\left (2 b^2 d^3 f \log ^2(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x) (e+f x)} \, dx}{3 (d e-c f)^4}-\frac {\left (b^3 d^3 f \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2 (e+f x)} \, dx}{6 (d e-c f)^4} \\ & = \frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}-\frac {5 b d^3 F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^4}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^2 (e+f x)^2}+\frac {2 b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{3 (d e-c f)^3 (e+f x)}-\frac {b d^3 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^4}-\frac {b^2 d^2 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^4 (e+f x)}+\frac {\left (b^2 d^3 f \log ^2(F)\right ) \text {Subst}\left (\int \frac {F^{a-\frac {b f}{d e-c f}+\frac {b d x}{d e-c f}}}{x} \, dx,x,\frac {e+f x}{c+d x}\right )}{3 (d e-c f)^5}+\frac {\left (2 b^2 d^3 f \log ^2(F)\right ) \text {Subst}\left (\int \frac {F^{a-\frac {b f}{d e-c f}+\frac {b d x}{d e-c f}}}{x} \, dx,x,\frac {e+f x}{c+d x}\right )}{3 (d e-c f)^5}-\frac {\left (b^3 d^3 f \log ^3(F)\right ) \int \left (\frac {d F^{a+\frac {b}{c+d x}}}{(d e-c f) (c+d x)^2}-\frac {d f F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (c+d x)}+\frac {f^2 F^{a+\frac {b}{c+d x}}}{(d e-c f)^2 (e+f x)}\right ) \, dx}{6 (d e-c f)^4} \\ & = \frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}-\frac {5 b d^3 F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^4}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^2 (e+f x)^2}+\frac {2 b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{3 (d e-c f)^3 (e+f x)}-\frac {b d^3 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^4}-\frac {b^2 d^2 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^4 (e+f x)}+\frac {b^2 d^3 f F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log ^2(F)}{(d e-c f)^5}+\frac {\left (b^3 d^4 f^2 \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{6 (d e-c f)^6}-\frac {\left (b^3 d^3 f^3 \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{e+f x} \, dx}{6 (d e-c f)^6}-\frac {\left (b^3 d^4 f \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x)^2} \, dx}{6 (d e-c f)^5} \\ & = \frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}-\frac {5 b d^3 F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^4}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^2 (e+f x)^2}+\frac {2 b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{3 (d e-c f)^3 (e+f x)}-\frac {b d^3 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^4}+\frac {b^2 d^3 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^5}-\frac {b^2 d^2 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^4 (e+f x)}+\frac {b^2 d^3 f F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log ^2(F)}{(d e-c f)^5}-\frac {b^3 d^3 f^2 F^a \text {Ei}\left (\frac {b \log (F)}{c+d x}\right ) \log ^3(F)}{6 (d e-c f)^6}-\frac {\left (b^3 d^4 f^2 \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{c+d x} \, dx}{6 (d e-c f)^6}+\frac {\left (b^3 d^3 f^2 \log ^3(F)\right ) \int \frac {F^{a+\frac {b}{c+d x}}}{(c+d x) (e+f x)} \, dx}{6 (d e-c f)^5} \\ & = \frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}-\frac {5 b d^3 F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^4}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^2 (e+f x)^2}+\frac {2 b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{3 (d e-c f)^3 (e+f x)}-\frac {b d^3 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^4}+\frac {b^2 d^3 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^5}-\frac {b^2 d^2 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^4 (e+f x)}+\frac {b^2 d^3 f F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log ^2(F)}{(d e-c f)^5}-\frac {\left (b^3 d^3 f^2 \log ^3(F)\right ) \text {Subst}\left (\int \frac {F^{a-\frac {b f}{d e-c f}+\frac {b d x}{d e-c f}}}{x} \, dx,x,\frac {e+f x}{c+d x}\right )}{6 (d e-c f)^6} \\ & = \frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}-\frac {5 b d^3 F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^4}+\frac {b d F^{a+\frac {b}{c+d x}} \log (F)}{6 (d e-c f)^2 (e+f x)^2}+\frac {2 b d^2 F^{a+\frac {b}{c+d x}} \log (F)}{3 (d e-c f)^3 (e+f x)}-\frac {b d^3 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log (F)}{(d e-c f)^4}+\frac {b^2 d^3 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^5}-\frac {b^2 d^2 f F^{a+\frac {b}{c+d x}} \log ^2(F)}{6 (d e-c f)^4 (e+f x)}+\frac {b^2 d^3 f F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log ^2(F)}{(d e-c f)^5}-\frac {b^3 d^3 f^2 F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right ) \log ^3(F)}{6 (d e-c f)^6} \\ \end{align*}

Mathematica [F]

\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx=\int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx \]

[In]

Integrate[F^(a + b/(c + d*x))/(e + f*x)^4,x]

[Out]

Integrate[F^(a + b/(c + d*x))/(e + f*x)^4, x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(921\) vs. \(2(444)=888\).

Time = 0.82 (sec) , antiderivative size = 922, normalized size of antiderivative = 2.00

method	result	size
risch	\(\frac {b^{2} d^{3} \ln \left (F \right )^{2} f \,F^{a} F^{\frac {b}{d x +c}}}{\left (c f -d e \right )^{5} \left (\frac {b \ln \left (F \right )}{d x +c}+a \ln \left (F \right )-\frac {\ln \left (F \right ) a c f}{c f -d e}+\frac {\ln \left (F \right ) a d e}{c f -d e}-\frac {\ln \left (F \right ) b f}{c f -d e}\right )^{2}}+\frac {b^{2} d^{3} \ln \left (F \right )^{2} f \,F^{a} F^{\frac {b}{d x +c}}}{\left (c f -d e \right )^{5} \left (\frac {b \ln \left (F \right )}{d x +c}+a \ln \left (F \right )-\frac {\ln \left (F \right ) a c f}{c f -d e}+\frac {\ln \left (F \right ) a d e}{c f -d e}-\frac {\ln \left (F \right ) b f}{c f -d e}\right )}+\frac {b^{2} d^{3} \ln \left (F \right )^{2} f \,F^{\frac {a c f -a d e +b f}{c f -d e}} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (F \right )}{d x +c}-a \ln \left (F \right )-\frac {-\ln \left (F \right ) a c f +a e d \ln \left (F \right )-\ln \left (F \right ) b f}{c f -d e}\right )}{\left (c f -d e \right )^{5}}+\frac {b^{3} d^{3} \ln \left (F \right )^{3} f^{2} F^{a} F^{\frac {b}{d x +c}}}{3 \left (c f -d e \right )^{6} \left (\frac {b \ln \left (F \right )}{d x +c}+a \ln \left (F \right )-\frac {\ln \left (F \right ) a c f}{c f -d e}+\frac {\ln \left (F \right ) a d e}{c f -d e}-\frac {\ln \left (F \right ) b f}{c f -d e}\right )^{3}}+\frac {b^{3} d^{3} \ln \left (F \right )^{3} f^{2} F^{a} F^{\frac {b}{d x +c}}}{6 \left (c f -d e \right )^{6} \left (\frac {b \ln \left (F \right )}{d x +c}+a \ln \left (F \right )-\frac {\ln \left (F \right ) a c f}{c f -d e}+\frac {\ln \left (F \right ) a d e}{c f -d e}-\frac {\ln \left (F \right ) b f}{c f -d e}\right )^{2}}+\frac {b^{3} d^{3} \ln \left (F \right )^{3} f^{2} F^{a} F^{\frac {b}{d x +c}}}{6 \left (c f -d e \right )^{6} \left (\frac {b \ln \left (F \right )}{d x +c}+a \ln \left (F \right )-\frac {\ln \left (F \right ) a c f}{c f -d e}+\frac {\ln \left (F \right ) a d e}{c f -d e}-\frac {\ln \left (F \right ) b f}{c f -d e}\right )}+\frac {b^{3} d^{3} \ln \left (F \right )^{3} f^{2} F^{\frac {a c f -a d e +b f}{c f -d e}} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (F \right )}{d x +c}-a \ln \left (F \right )-\frac {-\ln \left (F \right ) a c f +a e d \ln \left (F \right )-\ln \left (F \right ) b f}{c f -d e}\right )}{6 \left (c f -d e \right )^{6}}+\frac {b \,d^{3} \ln \left (F \right ) F^{a} F^{\frac {b}{d x +c}}}{\left (c f -d e \right )^{4} \left (\frac {b \ln \left (F \right )}{d x +c}+a \ln \left (F \right )-\frac {\ln \left (F \right ) a c f}{c f -d e}+\frac {\ln \left (F \right ) a d e}{c f -d e}-\frac {\ln \left (F \right ) b f}{c f -d e}\right )}+\frac {b \,d^{3} \ln \left (F \right ) F^{\frac {a c f -a d e +b f}{c f -d e}} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (F \right )}{d x +c}-a \ln \left (F \right )-\frac {-\ln \left (F \right ) a c f +a e d \ln \left (F \right )-\ln \left (F \right ) b f}{c f -d e}\right )}{\left (c f -d e \right )^{4}}\)	\(922\)

[In]

int(F^(a+b/(d*x+c))/(f*x+e)^4,x,method=_RETURNVERBOSE)

[Out]

b^2*d^3*ln(F)^2*f/(c*f-d*e)^5*F^a*F^(b/(d*x+c))/(b*ln(F)/(d*x+c)+a*ln(F)-1/(c*f-d*e)*ln(F)*a*c*f+1/(c*f-d*e)*l
n(F)*a*d*e-1/(c*f-d*e)*ln(F)*b*f)^2+b^2*d^3*ln(F)^2*f/(c*f-d*e)^5*F^a*F^(b/(d*x+c))/(b*ln(F)/(d*x+c)+a*ln(F)-1
/(c*f-d*e)*ln(F)*a*c*f+1/(c*f-d*e)*ln(F)*a*d*e-1/(c*f-d*e)*ln(F)*b*f)+b^2*d^3*ln(F)^2*f/(c*f-d*e)^5*F^((a*c*f-
a*d*e+b*f)/(c*f-d*e))*Ei(1,-b*ln(F)/(d*x+c)-a*ln(F)-(-ln(F)*a*c*f+a*e*d*ln(F)-ln(F)*b*f)/(c*f-d*e))+1/3*b^3*d^
3*ln(F)^3*f^2/(c*f-d*e)^6*F^a*F^(b/(d*x+c))/(b*ln(F)/(d*x+c)+a*ln(F)-1/(c*f-d*e)*ln(F)*a*c*f+1/(c*f-d*e)*ln(F)
*a*d*e-1/(c*f-d*e)*ln(F)*b*f)^3+1/6*b^3*d^3*ln(F)^3*f^2/(c*f-d*e)^6*F^a*F^(b/(d*x+c))/(b*ln(F)/(d*x+c)+a*ln(F)
-1/(c*f-d*e)*ln(F)*a*c*f+1/(c*f-d*e)*ln(F)*a*d*e-1/(c*f-d*e)*ln(F)*b*f)^2+1/6*b^3*d^3*ln(F)^3*f^2/(c*f-d*e)^6*
F^a*F^(b/(d*x+c))/(b*ln(F)/(d*x+c)+a*ln(F)-1/(c*f-d*e)*ln(F)*a*c*f+1/(c*f-d*e)*ln(F)*a*d*e-1/(c*f-d*e)*ln(F)*b
*f)+1/6*b^3*d^3*ln(F)^3*f^2/(c*f-d*e)^6*F^((a*c*f-a*d*e+b*f)/(c*f-d*e))*Ei(1,-b*ln(F)/(d*x+c)-a*ln(F)-(-ln(F)*
a*c*f+a*e*d*ln(F)-ln(F)*b*f)/(c*f-d*e))+b*d^3*ln(F)/(c*f-d*e)^4*F^a*F^(b/(d*x+c))/(b*ln(F)/(d*x+c)+a*ln(F)-1/(
c*f-d*e)*ln(F)*a*c*f+1/(c*f-d*e)*ln(F)*a*d*e-1/(c*f-d*e)*ln(F)*b*f)+b*d^3*ln(F)/(c*f-d*e)^4*F^((a*c*f-a*d*e+b*
f)/(c*f-d*e))*Ei(1,-b*ln(F)/(d*x+c)-a*ln(F)-(-ln(F)*a*c*f+a*e*d*ln(F)-ln(F)*b*f)/(c*f-d*e))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1376 vs. \(2 (444) = 888\).

Time = 0.31 (sec) , antiderivative size = 1376, normalized size of antiderivative = 2.99 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx=\text {Too large to display} \]

[In]

integrate(F^(a+b/(d*x+c))/(f*x+e)^4,x, algorithm="fricas")

[Out]

-1/6*(((b^3*d^3*f^5*x^3 + 3*b^3*d^3*e*f^4*x^2 + 3*b^3*d^3*e^2*f^3*x + b^3*d^3*e^3*f^2)*log(F)^3 - 6*(b^2*d^4*e
^4*f - b^2*c*d^3*e^3*f^2 + (b^2*d^4*e*f^4 - b^2*c*d^3*f^5)*x^3 + 3*(b^2*d^4*e^2*f^3 - b^2*c*d^3*e*f^4)*x^2 + 3
*(b^2*d^4*e^3*f^2 - b^2*c*d^3*e^2*f^3)*x)*log(F)^2 + 6*(b*d^5*e^5 - 2*b*c*d^4*e^4*f + b*c^2*d^3*e^3*f^2 + (b*d
^5*e^2*f^3 - 2*b*c*d^4*e*f^4 + b*c^2*d^3*f^5)*x^3 + 3*(b*d^5*e^3*f^2 - 2*b*c*d^4*e^2*f^3 + b*c^2*d^3*e*f^4)*x^
2 + 3*(b*d^5*e^4*f - 2*b*c*d^4*e^3*f^2 + b*c^2*d^3*e^2*f^3)*x)*log(F))*F^((a*d*e - (a*c + b)*f)/(d*e - c*f))*E
i((b*d*f*x + b*d*e)*log(F)/(c*d*e - c^2*f + (d^2*e - c*d*f)*x)) - (6*c*d^5*e^5 - 24*c^2*d^4*e^4*f + 38*c^3*d^3
*e^3*f^2 - 30*c^4*d^2*e^2*f^3 + 12*c^5*d*e*f^4 - 2*c^6*f^5 + 2*(d^6*e^3*f^2 - 3*c*d^5*e^2*f^3 + 3*c^2*d^4*e*f^
4 - c^3*d^3*f^5)*x^3 + 6*(d^6*e^4*f - 3*c*d^5*e^3*f^2 + 3*c^2*d^4*e^2*f^3 - c^3*d^3*e*f^4)*x^2 + (b^2*c*d^3*e^
3*f^2 - b^2*c^2*d^2*e^2*f^3 + (b^2*d^4*e*f^4 - b^2*c*d^3*f^5)*x^3 + (2*b^2*d^4*e^2*f^3 - b^2*c*d^3*e*f^4 - b^2
*c^2*d^2*f^5)*x^2 + (b^2*d^4*e^3*f^2 + b^2*c*d^3*e^2*f^3 - 2*b^2*c^2*d^2*e*f^4)*x)*log(F)^2 + 6*(d^6*e^5 - 3*c
*d^5*e^4*f + 3*c^2*d^4*e^3*f^2 - c^3*d^3*e^2*f^3)*x - (6*b*c*d^4*e^4*f - 13*b*c^2*d^3*e^3*f^2 + 8*b*c^3*d^2*e^
2*f^3 - b*c^4*d*e*f^4 + 5*(b*d^5*e^2*f^3 - 2*b*c*d^4*e*f^4 + b*c^2*d^3*f^5)*x^3 + (11*b*d^5*e^3*f^2 - 18*b*c*d
^4*e^2*f^3 + 3*b*c^2*d^3*e*f^4 + 4*b*c^3*d^2*f^5)*x^2 + (6*b*d^5*e^4*f - 2*b*c*d^4*e^3*f^2 - 15*b*c^2*d^3*e^2*
f^3 + 12*b*c^3*d^2*e*f^4 - b*c^4*d*f^5)*x)*log(F))*F^((a*d*x + a*c + b)/(d*x + c)))/(d^6*e^9 - 6*c*d^5*e^8*f +
15*c^2*d^4*e^7*f^2 - 20*c^3*d^3*e^6*f^3 + 15*c^4*d^2*e^5*f^4 - 6*c^5*d*e^4*f^5 + c^6*e^3*f^6 + (d^6*e^6*f^3 -
6*c*d^5*e^5*f^4 + 15*c^2*d^4*e^4*f^5 - 20*c^3*d^3*e^3*f^6 + 15*c^4*d^2*e^2*f^7 - 6*c^5*d*e*f^8 + c^6*f^9)*x^3
+ 3*(d^6*e^7*f^2 - 6*c*d^5*e^6*f^3 + 15*c^2*d^4*e^5*f^4 - 20*c^3*d^3*e^4*f^5 + 15*c^4*d^2*e^3*f^6 - 6*c^5*d*e
^2*f^7 + c^6*e*f^8)*x^2 + 3*(d^6*e^8*f - 6*c*d^5*e^7*f^2 + 15*c^2*d^4*e^6*f^3 - 20*c^3*d^3*e^5*f^4 + 15*c^4*d^
2*e^4*f^5 - 6*c^5*d*e^3*f^6 + c^6*e^2*f^7)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx=\text {Timed out} \]

[In]

integrate(F**(a+b/(d*x+c))/(f*x+e)**4,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx=\int { \frac {F^{a + \frac {b}{d x + c}}}{{\left (f x + e\right )}^{4}} \,d x } \]

[In]

integrate(F^(a+b/(d*x+c))/(f*x+e)^4,x, algorithm="maxima")

[Out]

integrate(F^(a + b/(d*x + c))/(f*x + e)^4, x)

Giac [F]

\[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx=\int { \frac {F^{a + \frac {b}{d x + c}}}{{\left (f x + e\right )}^{4}} \,d x } \]

[In]

integrate(F^(a+b/(d*x+c))/(f*x+e)^4,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c))/(f*x + e)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx=\int \frac {F^{a+\frac {b}{c+d\,x}}}{{\left (e+f\,x\right )}^4} \,d x \]

[In]

int(F^(a + b/(c + d*x))/(e + f*x)^4,x)

[Out]

int(F^(a + b/(c + d*x))/(e + f*x)^4, x)