Optimal. Leaf size=119 \[ \frac {(g h-f i) \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {a i x}{g}+\frac {b i (d+e x) \log \left (c (d+e x)^n\right )}{e g}+\frac {b n (g h-f i) \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {b i n x}{g} \]
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Rubi [A] time = 0.14, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2418, 2389, 2295, 2394, 2393, 2391} \[ \frac {b n (g h-f i) \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {(g h-f i) \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {a i x}{g}+\frac {b i (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {b i n x}{g} \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rubi steps
\begin {align*} \int \frac {(h+219 x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx &=\int \left (\frac {219 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {(-219 f+g h) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (f+g x)}\right ) \, dx\\ &=\frac {219 \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g}+\frac {(-219 f+g h) \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g}\\ &=\frac {219 a x}{g}-\frac {(219 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {(219 b) \int \log \left (c (d+e x)^n\right ) \, dx}{g}+\frac {(b e (219 f-g h) n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^2}\\ &=\frac {219 a x}{g}-\frac {(219 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {(219 b) \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g}+\frac {(b (219 f-g h) n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}\\ &=\frac {219 a x}{g}-\frac {219 b n x}{g}+\frac {219 b (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {(219 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}-\frac {b (219 f-g h) n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 110, normalized size = 0.92 \[ \frac {(g h-f i) \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+a g i x+\frac {b g i (d+e x) \log \left (c (d+e x)^n\right )}{e}+b n (g h-f i) \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )-b g i n x}{g^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a i x + a h + {\left (b i x + b h\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}{g x + f}, x\right ) \]
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 \[ \int \frac {{\left (i x + h\right )} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 750, normalized size = 6.30 \[ \frac {i \pi b f i \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{2 g^{2}}-\frac {a f i \ln \left (g x +f \right )}{g^{2}}+\frac {a h \ln \left (g x +f \right )}{g}+\frac {b i x \ln \left (\left (e x +d \right )^{n}\right )}{g}+\frac {i \pi b f i \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right )}{2 g^{2}}+\frac {i \pi b h \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 g}+\frac {b d i n \ln \left (d g -e f +\left (g x +f \right ) e \right )}{e g}+\frac {b f i n \ln \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right ) \ln \left (g x +f \right )}{g^{2}}-\frac {b h n \dilog \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right )}{g}+\frac {b h \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g}-\frac {b f i \ln \relax (c ) \ln \left (g x +f \right )}{g^{2}}+\frac {i \pi b i x \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 g}+\frac {i \pi b h \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 g}+\frac {i \pi b i x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 g}+\frac {b f i n \dilog \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right )}{g^{2}}-\frac {b f i n}{g^{2}}+\frac {b h \ln \relax (c ) \ln \left (g x +f \right )}{g}+\frac {b i x \ln \relax (c )}{g}-\frac {i \pi b f i \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 g^{2}}-\frac {i \pi b f i \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 g^{2}}-\frac {i \pi b h \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{2 g}-\frac {i \pi b i x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{2 g}+\frac {a i x}{g}-\frac {i \pi b h \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right )}{2 g}-\frac {i \pi b i x \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2 g}-\frac {b f i \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g^{2}}-\frac {b h n \ln \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right ) \ln \left (g x +f \right )}{g}-\frac {b i n x}{g} \]
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 \[ a i {\left (\frac {x}{g} - \frac {f \log \left (g x + f\right )}{g^{2}}\right )} + \frac {a h \log \left (g x + f\right )}{g} + \int \frac {b i x \log \relax (c) + b h \log \relax (c) + {\left (b i x + b h\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (h+i\,x\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{f+g\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (h + i x\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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