Optimal. Leaf size=460 \[ -\frac {b^3 d^3 f^2 \log ^3(F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{6 (d e-c f)^6}+\frac {b^2 d^3 f \log ^2(F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^5}+\frac {b^2 d^3 f \log ^2(F) F^{a+\frac {b}{c+d x}}}{6 (d e-c f)^5}-\frac {b^2 d^2 f \log ^2(F) F^{a+\frac {b}{c+d x}}}{6 (e+f x) (d e-c f)^4}-\frac {b d^3 \log (F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^4}+\frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}-\frac {5 b d^3 \log (F) F^{a+\frac {b}{c+d x}}}{6 (d e-c f)^4}+\frac {2 b d^2 \log (F) F^{a+\frac {b}{c+d x}}}{3 (e+f x) (d e-c f)^3}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}+\frac {b d \log (F) F^{a+\frac {b}{c+d x}}}{6 (e+f x)^2 (d e-c f)^2} \]
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Rubi [A] time = 3.75, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2223, 6742, 2209, 2210, 2222, 2228, 2178} \[ -\frac {b^3 d^3 f^2 \log ^3(F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{6 (d e-c f)^6}+\frac {b^2 d^3 f \log ^2(F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^5}-\frac {b^2 d^2 f \log ^2(F) F^{a+\frac {b}{c+d x}}}{6 (e+f x) (d e-c f)^4}+\frac {b^2 d^3 f \log ^2(F) F^{a+\frac {b}{c+d x}}}{6 (d e-c f)^5}-\frac {b d^3 \log (F) F^{a-\frac {b f}{d e-c f}} \text {Ei}\left (\frac {b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^4}+\frac {d^3 F^{a+\frac {b}{c+d x}}}{3 f (d e-c f)^3}+\frac {2 b d^2 \log (F) F^{a+\frac {b}{c+d x}}}{3 (e+f x) (d e-c f)^3}-\frac {5 b d^3 \log (F) F^{a+\frac {b}{c+d x}}}{6 (d e-c f)^4}-\frac {F^{a+\frac {b}{c+d x}}}{3 f (e+f x)^3}+\frac {b d \log (F) F^{a+\frac {b}{c+d x}}}{6 (e+f x)^2 (d e-c f)^2} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2209
Rule 2210
Rule 2222
Rule 2223
Rule 2228
Rule 6742
Rubi steps
\begin {align*} \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx &=-\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
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Mathematica [F] time = 0.63, size = 0, normalized size = 0.00 \[ \int \frac {F^{a+\frac {b}{c+d x}}}{(e+f x)^4} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.49, size = 1376, normalized size = 2.99 \[ \text {result too large to display} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (f x + e\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 922, normalized size = 2.00 \[ \frac {b^{3} d^{3} f^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{3}}{3 \left (c f -d e \right )^{6} \left (-\frac {a c f \ln \relax (F )}{c f -d e}+\frac {a d e \ln \relax (F )}{c f -d e}-\frac {b f \ln \relax (F )}{c f -d e}+a \ln \relax (F )+\frac {b \ln \relax (F )}{d x +c}\right )^{3}}+\frac {b^{3} d^{3} f^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{3}}{6 \left (c f -d e \right )^{6} \left (-\frac {a c f \ln \relax (F )}{c f -d e}+\frac {a d e \ln \relax (F )}{c f -d e}-\frac {b f \ln \relax (F )}{c f -d e}+a \ln \relax (F )+\frac {b \ln \relax (F )}{d x +c}\right )^{2}}+\frac {b^{3} d^{3} f^{2} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{3}}{6 \left (c f -d e \right )^{6} \left (-\frac {a c f \ln \relax (F )}{c f -d e}+\frac {a d e \ln \relax (F )}{c f -d e}-\frac {b f \ln \relax (F )}{c f -d e}+a \ln \relax (F )+\frac {b \ln \relax (F )}{d x +c}\right )}+\frac {b^{3} d^{3} f^{2} F^{\frac {a c f -a d e +b f}{c f -d e}} \Ei \left (1, -a \ln \relax (F )-\frac {b \ln \relax (F )}{d x +c}-\frac {-a c f \ln \relax (F )+a d e \ln \relax (F )-b f \ln \relax (F )}{c f -d e}\right ) \ln \relax (F )^{3}}{6 \left (c f -d e \right )^{6}}+\frac {b^{2} d^{3} f \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{\left (c f -d e \right )^{5} \left (-\frac {a c f \ln \relax (F )}{c f -d e}+\frac {a d e \ln \relax (F )}{c f -d e}-\frac {b f \ln \relax (F )}{c f -d e}+a \ln \relax (F )+\frac {b \ln \relax (F )}{d x +c}\right )^{2}}+\frac {b^{2} d^{3} f \,F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )^{2}}{\left (c f -d e \right )^{5} \left (-\frac {a c f \ln \relax (F )}{c f -d e}+\frac {a d e \ln \relax (F )}{c f -d e}-\frac {b f \ln \relax (F )}{c f -d e}+a \ln \relax (F )+\frac {b \ln \relax (F )}{d x +c}\right )}+\frac {b^{2} d^{3} f \,F^{\frac {a c f -a d e +b f}{c f -d e}} \Ei \left (1, -a \ln \relax (F )-\frac {b \ln \relax (F )}{d x +c}-\frac {-a c f \ln \relax (F )+a d e \ln \relax (F )-b f \ln \relax (F )}{c f -d e}\right ) \ln \relax (F )^{2}}{\left (c f -d e \right )^{5}}+\frac {b \,d^{3} F^{a} F^{\frac {b}{d x +c}} \ln \relax (F )}{\left (c f -d e \right )^{4} \left (-\frac {a c f \ln \relax (F )}{c f -d e}+\frac {a d e \ln \relax (F )}{c f -d e}-\frac {b f \ln \relax (F )}{c f -d e}+a \ln \relax (F )+\frac {b \ln \relax (F )}{d x +c}\right )}+\frac {b \,d^{3} F^{\frac {a c f -a d e +b f}{c f -d e}} \Ei \left (1, -a \ln \relax (F )-\frac {b \ln \relax (F )}{d x +c}-\frac {-a c f \ln \relax (F )+a d e \ln \relax (F )-b f \ln \relax (F )}{c f -d e}\right ) \ln \relax (F )}{\left (c f -d e \right )^{4}} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{{\left (f x + e\right )}^{4}}\,{d x} \]
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 \[ \int \frac {F^{a+\frac {b}{c+d\,x}}}{{\left (e+f\,x\right )}^4} \,d x \]
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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