Optimal. Leaf size=155 \[ \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}-\frac {\log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}+\frac {b n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {b n \text {Li}_2\left (-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i} \]
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Rubi [A] time = 0.20, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2418, 2394, 2393, 2391} \[ \frac {b n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g h-f i}-\frac {b n \text {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{g h-f i}+\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i}-\frac {\log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g h-f i} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{(h+221 x) (f+g x)} \, dx &=\int \left (\frac {221 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(221 f-g h) (h+221 x)}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(221 f-g h) (f+g x)}\right ) \, dx\\ &=\frac {221 \int \frac {a+b \log \left (c (d+e x)^n\right )}{h+221 x} \, dx}{221 f-g h}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{221 f-g h}\\ &=\frac {\log \left (-\frac {e (h+221 x)}{221 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{221 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{221 f-g h}-\frac {(b e n) \int \frac {\log \left (\frac {e (h+221 x)}{-221 d+e h}\right )}{d+e x} \, dx}{221 f-g h}+\frac {(b e n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{221 f-g h}\\ &=\frac {\log \left (-\frac {e (h+221 x)}{221 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{221 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{221 f-g h}+\frac {(b n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{221 f-g h}-\frac {(b n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {221 x}{-221 d+e h}\right )}{x} \, dx,x,d+e x\right )}{221 f-g h}\\ &=\frac {\log \left (-\frac {e (h+221 x)}{221 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{221 f-g h}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{221 f-g h}-\frac {b n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{221 f-g h}+\frac {b n \text {Li}_2\left (\frac {221 (d+e x)}{221 d-e h}\right )}{221 f-g h}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 111, normalized size = 0.72 \[ \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (\log \left (\frac {e (f+g x)}{e f-d g}\right )-\log \left (\frac {e (h+i x)}{e h-d i}\right )\right )+b n \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )-b n \text {Li}_2\left (\frac {i (d+e x)}{d i-e h}\right )}{g h-f i} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{g i x^{2} + f h + {\left (g h + f i\right )} x}, x\right ) \]
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 \[ \int \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} {\left (i x + h\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 647, normalized size = 4.17 \[ \frac {i \pi b \,\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 f i -2 g h}-\frac {i \pi b \,\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 (i x +h \right )}{2 \left (f i -g h \right )}-\frac {i \pi b \,\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 \left (f i -g h \right )}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (i x +h \right )}{2 f i -2 g h}-\frac {i \pi b \,\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 \left (f i -g h \right )}+\frac {i \pi b \,\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 (i x +h \right )}{2 f i -2 g h}+\frac {i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right )}{2 f i -2 g h}-\frac {i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (i x +h \right )}{2 \left (f i -g h \right )}+\frac {b 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 )}{f i -g h}-\frac {b n \ln \left (\frac {d i -e h +\left (i x +h \right ) e}{d i -e h}\right ) \ln \left (i x +h \right )}{f i -g h}+\frac {b n \dilog \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right )}{f i -g h}-\frac {b n \dilog \left (\frac {d i -e h +\left (i x +h \right ) e}{d i -e h}\right )}{f i -g h}-\frac {b \ln \relax (c ) \ln \left (g x +f \right )}{f i -g h}+\frac {b \ln \relax (c ) \ln \left (i x +h \right )}{f i -g h}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{f i -g h}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (i x +h \right )}{f i -g h}-\frac {a \ln \left (g x +f \right )}{f i -g h}+\frac {a \ln \left (i x +h \right )}{f i -g h} \]
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 \[ a {\left (\frac {\log \left (g x + f\right )}{g h - f i} - \frac {\log \left (i x + h\right )}{g h - f i}\right )} + b \int \frac {\log \left ({\left (e x + d\right )}^{n}\right ) + \log \relax (c)}{g i x^{2} + f h + {\left (g h + f i\right )} x}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\left (f+g\,x\right )\,\left (h+i\,x\right )} \,d x \]
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 \[ \int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{\left (f + g x\right ) \left (h + i x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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