Optimal. Leaf size=634 \[ \frac {d^3 F^{-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e}}{3 h (d g-c h)^3}+\frac {d^2 f \log (F) (b c-a d) F^{\frac {f (b g-a h)}{d g-c h}+e} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^4}+\frac {5 d^2 f \log (F) (b c-a d) F^{-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e}}{6 (d g-c h)^4}+\frac {f^3 h^2 \log ^3(F) (b c-a d)^3 F^{\frac {f (b g-a h)}{d g-c h}+e} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{6 (d g-c h)^6}+\frac {d f^2 h \log ^2(F) (b c-a d)^2 F^{\frac {f (b g-a h)}{d g-c h}+e} \text {Ei}\left (-\frac {(b c-a d) f (g+h x) \log (F)}{(d g-c h) (c+d x)}\right )}{(d g-c h)^5}+\frac {d f^2 h \log ^2(F) (b c-a d)^2 F^{-\frac {f (b c-a d)}{d (c+d x)}+\frac {b f}{d}+e}}{6 (d g-c h)^5}-\frac {f^2 h \log ^2(F) (b c-a d)^2 F^{\frac {f (a+b x)}{c+d x}+e}}{6 (g+h x) (d g-c h)^4}-\frac {F^{\frac {f (a+b x)}{c+d x}+e}}{3 h (g+h x)^3}-\frac {2 d f \log (F) (b c-a d) F^{\frac {f (a+b x)}{c+d x}+e}}{3 (g+h x) (d g-c h)^3}-\frac {f \log (F) (b c-a d) F^{\frac {f (a+b x)}{c+d x}+e}}{6 (g+h x)^2 (d g-c h)^2} \]
[Out]
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Rubi [A] time = 9.41, antiderivative size = 634, normalized size of antiderivative = 1.00, 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 2178
Rule 2209
Rule 2210
Rule 2230
Rule 2231
Rule 2232
Rule 2233
Rule 6742
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*}
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Mathematica [F] time = 0.99, size = 0, normalized size = 0.00 \[ \int \frac {F^{e+\frac {f (a+b x)}{c+d x}}}{(g+h x)^4} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.51, size = 2250, normalized size = 3.55 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 4671, normalized size = 7.37 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{e + \frac {{\left (b x + a\right )} f}{d x + c}}}{{\left (h x + g\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {F^{e+\frac {f\,\left (a+b\,x\right )}{c+d\,x}}}{{\left (g+h\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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