Optimal. Leaf size=634 \[ \frac{d^2 f \log (F) (b c-a d) F^{\frac{f (b g-a h)}{d g-c h}+e} \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^4}+\frac{d^3 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{3 h (d g-c h)^3}+\frac{5 d^2 f \log (F) (b c-a d) F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{6 (d g-c h)^4}+\frac{f^3 h^2 \log ^3(F) (b c-a d)^3 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{6 (d g-c h)^6}+\frac{d f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^5}+\frac{d f^2 h \log ^2(F) (b c-a d)^2 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{6 (d g-c h)^5}-\frac{f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (a+b x)}{c+d x}+e}}{6 (g+h x) (d g-c h)^4}-\frac{F^{\frac{f (a+b x)}{c+d x}+e}}{3 h (g+h x)^3}-\frac{2 d f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{3 (g+h x) (d g-c h)^3}-\frac{f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{6 (g+h x)^2 (d g-c h)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 9.40524, antiderivative size = 634, normalized size of antiderivative = 1., number of steps used = 48, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2232, 6742, 2230, 2209, 2210, 2231, 2233, 2178} \[ \frac{d^2 f \log (F) (b c-a d) F^{\frac{f (b g-a h)}{d g-c h}+e} \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^4}+\frac{d^3 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{3 h (d g-c h)^3}+\frac{5 d^2 f \log (F) (b c-a d) F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{6 (d g-c h)^4}+\frac{f^3 h^2 \log ^3(F) (b c-a d)^3 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{6 (d g-c h)^6}+\frac{d f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (b g-a h)}{d g-c h}+e} \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^5}+\frac{d f^2 h \log ^2(F) (b c-a d)^2 F^{-\frac{f (b c-a d)}{d (c+d x)}+\frac{b f}{d}+e}}{6 (d g-c h)^5}-\frac{f^2 h \log ^2(F) (b c-a d)^2 F^{\frac{f (a+b x)}{c+d x}+e}}{6 (g+h x) (d g-c h)^4}-\frac{F^{\frac{f (a+b x)}{c+d x}+e}}{3 h (g+h x)^3}-\frac{2 d f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{3 (g+h x) (d g-c h)^3}-\frac{f \log (F) (b c-a d) F^{\frac{f (a+b x)}{c+d x}+e}}{6 (g+h x)^2 (d g-c h)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2232
Rule 6742
Rule 2230
Rule 2209
Rule 2210
Rule 2231
Rule 2233
Rule 2178
Rubi steps
\begin{align*} \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^4} \, dx &=-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\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)^3} \, dx}{3 h}\\ &=-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac{((b c-a d) f \log (F)) \int \left (\frac{d^3 F^{e+\frac{f (a+b x)}{c+d x}}}{(d g-c h)^3 (c+d x)^2}-\frac{3 d^3 F^{e+\frac{f (a+b x)}{c+d x}} h}{(d g-c h)^4 (c+d x)}+\frac{F^{e+\frac{f (a+b x)}{c+d x}} h^2}{(d g-c h)^2 (g+h x)^3}+\frac{2 d F^{e+\frac{f (a+b x)}{c+d x}} h^2}{(d g-c h)^3 (g+h x)^2}+\frac{3 d^2 F^{e+\frac{f (a+b x)}{c+d x}} h^2}{(d g-c h)^4 (g+h x)}\right ) \, dx}{3 h}\\ &=-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}-\frac{\left (d^3 (b c-a d) f \log (F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{c+d x} \, dx}{(d g-c h)^4}+\frac{\left (d^2 (b c-a d) f h \log (F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{g+h x} \, dx}{(d g-c h)^4}+\frac{\left (d^3 (b c-a d) f \log (F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x)^2} \, dx}{3 h (d g-c h)^3}+\frac{(2 d (b c-a d) f h \log (F)) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx}{3 (d g-c h)^3}+\frac{((b c-a d) f h \log (F)) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^3} \, dx}{3 (d g-c h)^2}\\ &=-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}-\frac{(b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac{2 d (b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}-\frac{\left (d^3 (b c-a d) f \log (F)\right ) \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)^4}+\frac{\left (d^3 (b c-a d) f \log (F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{c+d x} \, dx}{(d g-c h)^4}-\frac{\left (d^2 (b c-a d) f \log (F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{(d g-c h)^3}+\frac{\left (d^3 (b c-a d) f \log (F)\right ) \int \frac{F^{\frac{d e+b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{(c+d x)^2} \, dx}{3 h (d g-c h)^3}+\frac{\left (2 d (b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x)^2 (g+h x)} \, dx}{3 (d g-c h)^3}+\frac{\left ((b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x)^2 (g+h x)^2} \, dx}{6 (d g-c h)^2}\\ &=\frac{d^3 F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}-\frac{(b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac{2 d (b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac{d^2 (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)^4}+\frac{\left (d^2 (b c-a d) f \log (F)\right ) \operatorname{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)^4}+\frac{\left (d^3 (b c-a d) f \log (F)\right ) \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)^4}+\frac{\left (2 d (b c-a d)^2 f^2 \log ^2(F)\right ) \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}{3 (d g-c h)^3}+\frac{\left ((b c-a d)^2 f^2 \log ^2(F)\right ) \int \left (\frac{d^2 F^{e+\frac{f (a+b x)}{c+d x}}}{(d g-c h)^2 (c+d x)^2}-\frac{2 d^2 F^{e+\frac{f (a+b x)}{c+d x}} h}{(d g-c h)^3 (c+d x)}+\frac{F^{e+\frac{f (a+b x)}{c+d x}} h^2}{(d g-c h)^2 (g+h x)^2}+\frac{2 d F^{e+\frac{f (a+b x)}{c+d x}} h^2}{(d g-c h)^3 (g+h x)}\right ) \, dx}{6 (d g-c h)^2}\\ &=\frac{d^3 F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}-\frac{(b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac{2 d (b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac{d^2 (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)^4}-\frac{\left (d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{c+d x} \, dx}{3 (d g-c h)^5}-\frac{\left (2 d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac{\left (d (b c-a d)^2 f^2 h^2 \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{g+h x} \, dx}{3 (d g-c h)^5}+\frac{\left (2 d (b c-a d)^2 f^2 h^2 \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{g+h x} \, dx}{3 (d g-c h)^5}+\frac{\left (d^2 (b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x)^2} \, dx}{6 (d g-c h)^4}+\frac{\left (2 d^2 (b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x)^2} \, dx}{3 (d g-c h)^4}+\frac{\left ((b c-a d)^2 f^2 h^2 \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^2} \, dx}{6 (d g-c h)^4}\\ &=\frac{d^3 F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}-\frac{(b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac{2 d (b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac{d^2 (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)^4}-\frac{(b c-a d)^2 f^2 F^{e+\frac{f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}-\frac{\left (d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac{F^{\frac{d e+b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac{\left (d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{c+d x} \, dx}{3 (d g-c h)^5}-\frac{\left (2 d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac{F^{\frac{d e+b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac{\left (2 d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac{\left (d^2 (b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac{F^{\frac{d e+b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{(c+d x)^2} \, dx}{6 (d g-c h)^4}+\frac{\left (2 d^2 (b c-a d)^2 f^2 \log ^2(F)\right ) \int \frac{F^{\frac{d e+b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{(c+d x)^2} \, dx}{3 (d g-c h)^4}-\frac{\left (d (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{3 (d g-c h)^4}-\frac{\left (2 d (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{3 (d g-c h)^4}+\frac{\left ((b c-a d)^3 f^3 h \log ^3(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x)^2 (g+h x)} \, dx}{6 (d g-c h)^4}\\ &=\frac{d^3 F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac{5 d^2 (b c-a d) f F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}} \log (F)}{6 (d g-c h)^4}-\frac{(b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac{2 d (b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac{d^2 (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)^4}-\frac{(b c-a d)^2 f^2 F^{e+\frac{f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}+\frac{d (b c-a d)^2 f^2 F^{e+\frac{b f}{d}} h \text{Ei}\left (-\frac{(b c-a d) f \log (F)}{d (c+d x)}\right ) \log ^2(F)}{(d g-c h)^5}+\frac{\left (d (b c-a d)^2 f^2 h \log ^2(F)\right ) \operatorname{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 )}{3 (d g-c h)^5}+\frac{\left (2 d (b c-a d)^2 f^2 h \log ^2(F)\right ) \operatorname{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 )}{3 (d g-c h)^5}+\frac{\left (d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac{F^{\frac{d e+b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac{\left (2 d^2 (b c-a d)^2 f^2 h \log ^2(F)\right ) \int \frac{F^{\frac{d e+b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{3 (d g-c h)^5}+\frac{\left ((b c-a d)^3 f^3 h \log ^3(F)\right ) \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}{6 (d g-c h)^4}\\ &=\frac{d^3 F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac{5 d^2 (b c-a d) f F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}} \log (F)}{6 (d g-c h)^4}-\frac{(b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac{2 d (b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac{d^2 (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)^4}-\frac{(b c-a d)^2 f^2 F^{e+\frac{f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}+\frac{d (b c-a d)^2 f^2 F^{e+\frac{f (b g-a h)}{d g-c h}} h \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^2(F)}{(d g-c h)^5}-\frac{\left (d (b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{c+d x} \, dx}{6 (d g-c h)^6}+\frac{\left ((b c-a d)^3 f^3 h^3 \log ^3(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{g+h x} \, dx}{6 (d g-c h)^6}+\frac{\left (d (b c-a d)^3 f^3 h \log ^3(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x)^2} \, dx}{6 (d g-c h)^5}\\ &=\frac{d^3 F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac{5 d^2 (b c-a d) f F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}} \log (F)}{6 (d g-c h)^4}-\frac{(b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac{2 d (b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac{d^2 (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)^4}-\frac{(b c-a d)^2 f^2 F^{e+\frac{f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}+\frac{d (b c-a d)^2 f^2 F^{e+\frac{f (b g-a h)}{d g-c h}} h \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^2(F)}{(d g-c h)^5}-\frac{\left (d (b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \int \frac{F^{\frac{d e+b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{6 (d g-c h)^6}+\frac{\left (d (b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{c+d x} \, dx}{6 (d g-c h)^6}+\frac{\left (d (b c-a d)^3 f^3 h \log ^3(F)\right ) \int \frac{F^{\frac{d e+b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{(c+d x)^2} \, dx}{6 (d g-c h)^5}-\frac{\left ((b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(c+d x) (g+h x)} \, dx}{6 (d g-c h)^5}\\ &=\frac{d^3 F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac{5 d^2 (b c-a d) f F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}} \log (F)}{6 (d g-c h)^4}-\frac{(b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac{2 d (b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac{d^2 (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)^4}+\frac{d (b c-a d)^2 f^2 F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}} h \log ^2(F)}{6 (d g-c h)^5}-\frac{(b c-a d)^2 f^2 F^{e+\frac{f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}+\frac{d (b c-a d)^2 f^2 F^{e+\frac{f (b g-a h)}{d g-c h}} h \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^2(F)}{(d g-c h)^5}+\frac{(b c-a d)^3 f^3 F^{e+\frac{b f}{d}} h^2 \text{Ei}\left (-\frac{(b c-a d) f \log (F)}{d (c+d x)}\right ) \log ^3(F)}{6 (d g-c h)^6}+\frac{\left ((b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \operatorname{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 )}{6 (d g-c h)^6}+\frac{\left (d (b c-a d)^3 f^3 h^2 \log ^3(F)\right ) \int \frac{F^{\frac{d e+b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{c+d x} \, dx}{6 (d g-c h)^6}\\ &=\frac{d^3 F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}}}{3 h (d g-c h)^3}-\frac{F^{e+\frac{f (a+b x)}{c+d x}}}{3 h (g+h x)^3}+\frac{5 d^2 (b c-a d) f F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}} \log (F)}{6 (d g-c h)^4}-\frac{(b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{6 (d g-c h)^2 (g+h x)^2}-\frac{2 d (b c-a d) f F^{e+\frac{f (a+b x)}{c+d x}} \log (F)}{3 (d g-c h)^3 (g+h x)}+\frac{d^2 (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)^4}+\frac{d (b c-a d)^2 f^2 F^{e+\frac{b f}{d}-\frac{(b c-a d) f}{d (c+d x)}} h \log ^2(F)}{6 (d g-c h)^5}-\frac{(b c-a d)^2 f^2 F^{e+\frac{f (a+b x)}{c+d x}} h \log ^2(F)}{6 (d g-c h)^4 (g+h x)}+\frac{d (b c-a d)^2 f^2 F^{e+\frac{f (b g-a h)}{d g-c h}} h \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^2(F)}{(d g-c h)^5}+\frac{(b c-a d)^3 f^3 F^{e+\frac{f (b g-a h)}{d g-c h}} h^2 \text{Ei}\left (-\frac{(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right ) \log ^3(F)}{6 (d g-c h)^6}\\ \end{align*}
Mathematica [F] time = 0.950908, size = 0, normalized size = 0. \[ \int \frac{F^{e+\frac{f (a+b x)}{c+d x}}}{(g+h x)^4} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.27, size = 4671, normalized size = 7.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{e + \frac{{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.9458, size = 4427, normalized size = 6.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{e + \frac{{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]