Optimal. Leaf size=148 \[ \frac {\log (g+h x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h}-\frac {B n \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {B n \log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{h}+\frac {B n \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {B n \log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{h} \]
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Rubi [A] time = 0.19, antiderivative size = 156, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6742, 2494, 2394, 2393, 2391} \[ -\frac {B n \text {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {B n \text {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {B \log (g+h x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h}-\frac {B n \log (g+h x) \log \left (-\frac {h (a+b x)}{b g-a h}\right )}{h}+\frac {A \log (g+h x)}{h}+\frac {B n \log (g+h x) \log \left (-\frac {h (c+d x)}{d g-c h}\right )}{h} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2494
Rule 6742
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx &=\int \left (\frac {A}{g+h x}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x}\right ) \, dx\\ &=\frac {A \log (g+h x)}{h}+B \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx\\ &=\frac {A \log (g+h x)}{h}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}-\frac {(b B n) \int \frac {\log (g+h x)}{a+b x} \, dx}{h}+\frac {(B d n) \int \frac {\log (g+h x)}{c+d x} \, dx}{h}\\ &=\frac {A \log (g+h x)}{h}-\frac {B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+(B n) \int \frac {\log \left (\frac {h (a+b x)}{-b g+a h}\right )}{g+h x} \, dx-(B n) \int \frac {\log \left (\frac {h (c+d x)}{-d g+c h}\right )}{g+h x} \, dx\\ &=\frac {A \log (g+h x)}{h}-\frac {B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac {(B n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h}-\frac {(B n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h}\\ &=\frac {A \log (g+h x)}{h}-\frac {B n \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac {B n \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}-\frac {B n \text {Li}_2\left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {B n \text {Li}_2\left (\frac {d (g+h x)}{d g-c h}\right )}{h}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 150, normalized size = 1.01 \[ \frac {\log (g+h x) \left (B \left (\log \left (e (a+b x)^n (c+d x)^{-n}\right )-n \log (a+b x)+n \log (c+d x)\right )+A\right )+B n \left (\text {Li}_2\left (\frac {h (a+b x)}{a h-b g}\right )+\log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )\right )-B n \left (\text {Li}_2\left (\frac {h (c+d x)}{c h-d g}\right )+\log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )\right )}{h} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{h x + g}, x\right ) \]
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 \[ \int \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{h x + g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.45, size = 597, normalized size = 4.03 \[ -\frac {i \pi B \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \ln \left (h x +g \right )}{2 h}+\frac {i \pi B \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} \ln \left (h x +g \right )}{2 h}-\frac {i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \ln \left (h x +g \right )}{2 h}+\frac {i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} \ln \left (h x +g \right )}{2 h}+\frac {i \pi B \,\mathrm {csgn}\left (i \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} \ln \left (h x +g \right )}{2 h}-\frac {i \pi B \mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3} \ln \left (h x +g \right )}{2 h}+\frac {i \pi B \,\mathrm {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} \ln \left (h x +g \right )}{2 h}-\frac {i \pi B \mathrm {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3} \ln \left (h x +g \right )}{2 h}-\frac {B n \ln \left (\frac {a h -b g +\left (h x +g \right ) b}{a h -b g}\right ) \ln \left (h x +g \right )}{h}+\frac {B n \ln \left (\frac {c h -d g +\left (h x +g \right ) d}{c h -d g}\right ) \ln \left (h x +g \right )}{h}-\frac {B n \dilog \left (\frac {a h -b g +\left (h x +g \right ) b}{a h -b g}\right )}{h}+\frac {B n \dilog \left (\frac {c h -d g +\left (h x +g \right ) d}{c h -d g}\right )}{h}+\frac {B \ln \relax (e ) \ln \left (h x +g \right )}{h}+\frac {B \ln \left (\left (b x +a \right )^{n}\right ) \ln \left (h x +g \right )}{h}-\frac {B \ln \left (\left (d x +c \right )^{n}\right ) \ln \left (h x +g \right )}{h}+\frac {A \ln \left (h x +g \right )}{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 \[ -B \int -\frac {\log \left ({\left (b x + a\right )}^{n}\right ) - \log \left ({\left (d x + c\right )}^{n}\right ) + \log \relax (e)}{h x + g}\,{d x} + \frac {A \log \left (h x + g\right )}{h} \]
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 \[ \int \frac {A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{g+h\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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