Optimal. Leaf size=550 \[ -\frac {\sqrt {f} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{g^{3/2}}+\frac {x \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{g}+\frac {\sqrt {f} n \text {Li}_2\left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \text {Li}_2\left (\frac {\sqrt {g} (a+b x)}{\sqrt {g} a+b \sqrt {f}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {n (a+b x) \log (a+b x)}{b g}-\frac {\sqrt {f} n \text {Li}_2\left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} n \text {Li}_2\left (\frac {\sqrt {g} (c+d x)}{\sqrt {g} c+d \sqrt {f}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^{3/2}}+\frac {n (c+d x) \log (c+d x)}{d g} \]
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Rubi [A] time = 0.57, antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 10, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2513, 321, 208, 2416, 2389, 2295, 2409, 2394, 2393, 2391} \[ \frac {\sqrt {f} n \text {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \text {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{a \sqrt {g}+b \sqrt {f}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \text {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} n \text {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{c \sqrt {g}+d \sqrt {f}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{g^{3/2}}+\frac {x \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{g}-\frac {\sqrt {f} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {n (a+b x) \log (a+b x)}{b g}+\frac {\sqrt {f} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^{3/2}}+\frac {n (c+d x) \log (c+d x)}{d g} \]
Antiderivative was successfully verified.
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Rule 208
Rule 321
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 2416
Rule 2513
Rubi steps
\begin {align*} \int \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx &=n \int \frac {x^2 \log (a+b x)}{f-g x^2} \, dx-n \int \frac {x^2 \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^2}{f-g x^2} \, dx\\ &=\frac {x \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 )}{g}+n \int \left (-\frac {\log (a+b x)}{g}+\frac {f \log (a+b x)}{g \left (f-g x^2\right )}\right ) \, dx-n \int \left (-\frac {\log (c+d x)}{g}+\frac {f \log (c+d x)}{g \left (f-g x^2\right )}\right ) \, dx-\frac {\left (f \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 )\right ) \int \frac {1}{f-g x^2} \, dx}{g}\\ &=\frac {x \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 )}{g}-\frac {\sqrt {f} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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 )}{g^{3/2}}-\frac {n \int \log (a+b x) \, dx}{g}+\frac {n \int \log (c+d x) \, dx}{g}+\frac {(f n) \int \frac {\log (a+b x)}{f-g x^2} \, dx}{g}-\frac {(f n) \int \frac {\log (c+d x)}{f-g x^2} \, dx}{g}\\ &=\frac {x \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 )}{g}-\frac {\sqrt {f} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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 )}{g^{3/2}}-\frac {n \operatorname {Subst}(\int \log (x) \, dx,x,a+b x)}{b g}+\frac {n \operatorname {Subst}(\int \log (x) \, dx,x,c+d x)}{d g}+\frac {(f n) \int \left (\frac {\log (a+b x)}{2 \sqrt {f} \left (\sqrt {f}-\sqrt {g} x\right )}+\frac {\log (a+b x)}{2 \sqrt {f} \left (\sqrt {f}+\sqrt {g} x\right )}\right ) \, dx}{g}-\frac {(f n) \int \left (\frac {\log (c+d x)}{2 \sqrt {f} \left (\sqrt {f}-\sqrt {g} x\right )}+\frac {\log (c+d x)}{2 \sqrt {f} \left (\sqrt {f}+\sqrt {g} x\right )}\right ) \, dx}{g}\\ &=-\frac {n (a+b x) \log (a+b x)}{b g}+\frac {n (c+d x) \log (c+d x)}{d g}+\frac {x \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 )}{g}-\frac {\sqrt {f} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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 )}{g^{3/2}}+\frac {\left (\sqrt {f} n\right ) \int \frac {\log (a+b x)}{\sqrt {f}-\sqrt {g} x} \, dx}{2 g}+\frac {\left (\sqrt {f} n\right ) \int \frac {\log (a+b x)}{\sqrt {f}+\sqrt {g} x} \, dx}{2 g}-\frac {\left (\sqrt {f} n\right ) \int \frac {\log (c+d x)}{\sqrt {f}-\sqrt {g} x} \, dx}{2 g}-\frac {\left (\sqrt {f} n\right ) \int \frac {\log (c+d x)}{\sqrt {f}+\sqrt {g} x} \, dx}{2 g}\\ &=-\frac {n (a+b x) \log (a+b x)}{b g}+\frac {n (c+d x) \log (c+d x)}{d g}+\frac {x \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 )}{g}-\frac {\sqrt {f} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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 )}{g^{3/2}}-\frac {\sqrt {f} 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^{3/2}}+\frac {\sqrt {f} 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^{3/2}}+\frac {\sqrt {f} 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^{3/2}}-\frac {\sqrt {f} 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^{3/2}}+\frac {\left (b \sqrt {f} n\right ) \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^{3/2}}-\frac {\left (b \sqrt {f} n\right ) \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^{3/2}}-\frac {\left (d \sqrt {f} n\right ) \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^{3/2}}+\frac {\left (d \sqrt {f} n\right ) \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^{3/2}}\\ &=-\frac {n (a+b x) \log (a+b x)}{b g}+\frac {n (c+d x) \log (c+d x)}{d g}+\frac {x \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 )}{g}-\frac {\sqrt {f} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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 )}{g^{3/2}}-\frac {\sqrt {f} 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^{3/2}}+\frac {\sqrt {f} 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^{3/2}}+\frac {\sqrt {f} 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^{3/2}}-\frac {\sqrt {f} 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^{3/2}}-\frac {\left (\sqrt {f} n\right ) \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^{3/2}}+\frac {\left (\sqrt {f} n\right ) \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^{3/2}}+\frac {\left (\sqrt {f} n\right ) \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^{3/2}}-\frac {\left (\sqrt {f} n\right ) \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^{3/2}}\\ &=-\frac {n (a+b x) \log (a+b x)}{b g}+\frac {n (c+d x) \log (c+d x)}{d g}+\frac {x \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 )}{g}-\frac {\sqrt {f} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \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 )}{g^{3/2}}-\frac {\sqrt {f} 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^{3/2}}+\frac {\sqrt {f} 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^{3/2}}+\frac {\sqrt {f} 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^{3/2}}-\frac {\sqrt {f} 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^{3/2}}+\frac {\sqrt {f} n \text {Li}_2\left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \text {Li}_2\left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g^{3/2}}-\frac {\sqrt {f} n \text {Li}_2\left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^{3/2}}+\frac {\sqrt {f} n \text {Li}_2\left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 467, normalized size = 0.85 \[ \frac {-\sqrt {f} \log \left (\sqrt {f}-\sqrt {g} x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\sqrt {f} \log \left (\sqrt {f}+\sqrt {g} x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\frac {2 \sqrt {g} (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+\sqrt {f} n \left (\log \left (\sqrt {f}-\sqrt {g} x\right ) \left (\log \left (\frac {\sqrt {g} (a+b x)}{a \sqrt {g}+b \sqrt {f}}\right )-\log \left (\frac {\sqrt {g} (c+d x)}{c \sqrt {g}+d \sqrt {f}}\right )\right )+\text {Li}_2\left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{\sqrt {g} a+b \sqrt {f}}\right )-\text {Li}_2\left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{\sqrt {g} c+d \sqrt {f}}\right )\right )-\sqrt {f} n \left (\log \left (\sqrt {f}+\sqrt {g} x\right ) \left (\log \left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )-\log \left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )\right )+\text {Li}_2\left (\frac {b \left (\sqrt {g} x+\sqrt {f}\right )}{b \sqrt {f}-a \sqrt {g}}\right )-\text {Li}_2\left (\frac {d \left (\sqrt {g} x+\sqrt {f}\right )}{d \sqrt {f}-c \sqrt {g}}\right )\right )+\frac {2 \sqrt {g} n (b c-a d) \log (c+d x)}{b d}}{2 g^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f}, x\right ) \]
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 \[ \int \frac {x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{-g \,x^{2}+f}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.21, size = 1047, normalized size = 1.90 \[ \text {result too large to display} \]
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 \[ \int \frac {x^2\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{f-g\,x^2} \,d x \]
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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