Optimal. Leaf size=403 \[ \frac {\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g}-\frac {n \text {Li}_2\left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}-\frac {n \text {Li}_2\left (\frac {\sqrt {g} (a+b x)}{\sqrt {g} a+b \sqrt {f}}\right )}{2 g}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 g}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}+\frac {n \text {Li}_2\left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g}+\frac {n \text {Li}_2\left (\frac {\sqrt {g} (c+d x)}{\sqrt {g} c+d \sqrt {f}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g} \]
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Rubi [A] time = 0.35, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2513, 260, 2416, 2394, 2393, 2391} \[ -\frac {n \text {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}-\frac {n \text {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{a \sqrt {g}+b \sqrt {f}}\right )}{2 g}+\frac {n \text {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g}+\frac {n \text {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{c \sqrt {g}+d \sqrt {f}}\right )}{2 g}+\frac {\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 g}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2513
Rubi steps
\begin {align*} \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx &=n \int \frac {x \log (a+b x)}{f-g x^2} \, dx-n \int \frac {x \log (c+d x)}{f-g x^2} \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac {x}{f-g x^2} \, dx\\ &=\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g}+n \int \left (\frac {\log (a+b x)}{2 \sqrt {g} \left (\sqrt {f}-\sqrt {g} x\right )}-\frac {\log (a+b x)}{2 \sqrt {g} \left (\sqrt {f}+\sqrt {g} x\right )}\right ) \, dx-n \int \left (\frac {\log (c+d x)}{2 \sqrt {g} \left (\sqrt {f}-\sqrt {g} x\right )}-\frac {\log (c+d x)}{2 \sqrt {g} \left (\sqrt {f}+\sqrt {g} x\right )}\right ) \, dx\\ &=\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g}+\frac {n \int \frac {\log (a+b x)}{\sqrt {f}-\sqrt {g} x} \, dx}{2 \sqrt {g}}-\frac {n \int \frac {\log (a+b x)}{\sqrt {f}+\sqrt {g} x} \, dx}{2 \sqrt {g}}-\frac {n \int \frac {\log (c+d x)}{\sqrt {f}-\sqrt {g} x} \, dx}{2 \sqrt {g}}+\frac {n \int \frac {\log (c+d x)}{\sqrt {f}+\sqrt {g} x} \, dx}{2 \sqrt {g}}\\ &=-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g}+\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g}+\frac {(b n) \int \frac {\log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{a+b x} \, dx}{2 g}+\frac {(b n) \int \frac {\log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{a+b x} \, dx}{2 g}-\frac {(d n) \int \frac {\log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{c+d x} \, dx}{2 g}-\frac {(d n) \int \frac {\log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{c+d x} \, dx}{2 g}\\ &=-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g}+\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g}+\frac {n \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{b \sqrt {f}-a \sqrt {g}}\right )}{x} \, dx,x,a+b x\right )}{2 g}+\frac {n \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{b \sqrt {f}+a \sqrt {g}}\right )}{x} \, dx,x,a+b x\right )}{2 g}-\frac {n \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{d \sqrt {f}-c \sqrt {g}}\right )}{x} \, dx,x,c+d x\right )}{2 g}-\frac {n \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{d \sqrt {f}+c \sqrt {g}}\right )}{x} \, dx,x,c+d x\right )}{2 g}\\ &=-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g}+\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g}-\frac {n \text {Li}_2\left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}-\frac {n \text {Li}_2\left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g}+\frac {n \text {Li}_2\left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g}+\frac {n \text {Li}_2\left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 413, normalized size = 1.02 \[ -\frac {\log \left (\sqrt {f}-\sqrt {g} x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left (\sqrt {f}+\sqrt {g} x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \text {Li}_2\left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{\sqrt {g} a+b \sqrt {f}}\right )-n \text {Li}_2\left (\frac {b \left (\sqrt {g} x+\sqrt {f}\right )}{b \sqrt {f}-a \sqrt {g}}\right )-n \log \left (\sqrt {f}-\sqrt {g} x\right ) \log \left (\frac {\sqrt {g} (a+b x)}{a \sqrt {g}+b \sqrt {f}}\right )-n \log \left (\sqrt {f}+\sqrt {g} x\right ) \log \left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )+n \text {Li}_2\left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{\sqrt {g} c+d \sqrt {f}}\right )+n \text {Li}_2\left (\frac {d \left (\sqrt {g} x+\sqrt {f}\right )}{d \sqrt {f}-c \sqrt {g}}\right )+n \log \left (\sqrt {f}-\sqrt {g} x\right ) \log \left (\frac {\sqrt {g} (c+d x)}{c \sqrt {g}+d \sqrt {f}}\right )+n \log \left (\sqrt {f}+\sqrt {g} x\right ) \log \left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{-g \,x^{2}+f}\, dx \]
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 \[ -\int \frac {x \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{f-g\,x^2} \,d x \]
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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