Optimal. Leaf size=460 \[ -\frac{b^3 d^3 f^2 \log ^3(F) F^{a-\frac{b f}{d e-c f}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{6 (d e-c f)^6}+\frac{b^2 d^3 f \log ^2(F) F^{a-\frac{b f}{d e-c f}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^5}-\frac{b^2 d^2 f \log ^2(F) F^{a+\frac{b}{c+d x}}}{6 (e+f x) (d e-c f)^4}+\frac{b^2 d^3 f \log ^2(F) F^{a+\frac{b}{c+d x}}}{6 (d e-c f)^5}-\frac{b d^3 \log (F) F^{a-\frac{b f}{d e-c f}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^4}+\frac{d^3 F^{a+\frac{b}{c+d x}}}{3 f (d e-c f)^3}+\frac{2 b d^2 \log (F) F^{a+\frac{b}{c+d x}}}{3 (e+f x) (d e-c f)^3}-\frac{5 b d^3 \log (F) F^{a+\frac{b}{c+d x}}}{6 (d e-c f)^4}-\frac{F^{a+\frac{b}{c+d x}}}{3 f (e+f x)^3}+\frac{b d \log (F) F^{a+\frac{b}{c+d x}}}{6 (e+f x)^2 (d e-c f)^2} \]
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Rubi [A] time = 3.75106, antiderivative size = 460, normalized size of antiderivative = 1., 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 2223
Rule 6742
Rule 2209
Rule 2210
Rule 2222
Rule 2228
Rule 2178
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*}
Mathematica [F] time = 0.627395, size = 0, normalized size = 0. \[ \int \frac{F^{a+\frac{b}{c+d x}}}{(e+f x)^4} \, dx \]
Verification is Not applicable to the result.
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Maple [B] time = 0.2, size = 922, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (f x + e\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81402, size = 2736, normalized size = 5.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (f x + e\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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