Optimal. Leaf size=186 \[ -\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}-\frac {2 f \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {a x}{g^2}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {b e f^2 n \log (d+e x)}{g^3 (e f-d g)}-\frac {b e f^2 n \log (f+g x)}{g^3 (e f-d g)}-\frac {2 b f n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^3}-\frac {b n x}{g^2} \]
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Rubi [A] time = 0.20, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {43, 2416, 2389, 2295, 2395, 36, 31, 2394, 2393, 2391} \[ -\frac {2 b f n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^3}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}-\frac {2 f \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {a x}{g^2}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {b e f^2 n \log (d+e x)}{g^3 (e f-d g)}-\frac {b e f^2 n \log (f+g x)}{g^3 (e f-d g)}-\frac {b n x}{g^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 43
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(f+g x)^2} \, dx &=\int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{g^2}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)^2}-\frac {2 f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 (f+g x)}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g^2}-\frac {(2 f) \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g^2}+\frac {f^2 \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^2} \, dx}{g^2}\\ &=\frac {a x}{g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}-\frac {2 f \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^3}+\frac {b \int \log \left (c (d+e x)^n\right ) \, dx}{g^2}+\frac {(2 b e f n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^3}+\frac {\left (b e f^2 n\right ) \int \frac {1}{(d+e x) (f+g x)} \, dx}{g^3}\\ &=\frac {a x}{g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}-\frac {2 f \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^3}+\frac {b \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g^2}+\frac {(2 b f 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^3}+\frac {\left (b e^2 f^2 n\right ) \int \frac {1}{d+e x} \, dx}{g^3 (e f-d g)}-\frac {\left (b e f^2 n\right ) \int \frac {1}{f+g x} \, dx}{g^2 (e f-d g)}\\ &=\frac {a x}{g^2}-\frac {b n x}{g^2}+\frac {b e f^2 n \log (d+e x)}{g^3 (e f-d g)}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}-\frac {b e f^2 n \log (f+g x)}{g^3 (e f-d g)}-\frac {2 f \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^3}-\frac {2 b f n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{g^3}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 153, normalized size = 0.82 \[ \frac {-\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x}-2 f \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+a g x+\frac {b g (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {b e f^2 n (\log (d+e x)-\log (f+g x))}{e f-d g}-2 b f n \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )-b g n x}{g^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + a x^{2}}{g^{2} x^{2} + 2 \, f g x + f^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{2}}{{\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 791, normalized size = 4.25 \[ \frac {b x \ln \left (\left (e x +d \right )^{n}\right )}{g^{2}}-\frac {a \,f^{2}}{\left (g x +f \right ) g^{3}}-\frac {2 a f \ln \left (g x +f \right )}{g^{3}}+\frac {b x \ln \relax (c )}{g^{2}}+\frac {2 b f 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^{3}}-\frac {b e \,f^{2} n \ln \left (d g -e f +\left (g x +f \right ) e \right )}{\left (d g -e f \right ) g^{3}}+\frac {b e \,f^{2} n \ln \left (g x +f \right )}{\left (d g -e f \right ) g^{3}}+\frac {b \,d^{2} n \ln \left (d g -e f +\left (g x +f \right ) e \right )}{\left (d g -e f \right ) e g}-\frac {b d f n \ln \left (d g -e f +\left (g x +f \right ) e \right )}{\left (d g -e f \right ) g^{2}}-\frac {b f n}{g^{3}}+\frac {2 b f n \dilog \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right )}{g^{3}}-\frac {b \,f^{2} \ln \relax (c )}{\left (g x +f \right ) g^{3}}-\frac {2 b f \ln \relax (c ) \ln \left (g x +f \right )}{g^{3}}+\frac {i \pi b 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^{2}}-\frac {2 b f \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{g^{3}}-\frac {b \,f^{2} \ln \left (\left (e x +d \right )^{n}\right )}{\left (g x +f \right ) g^{3}}-\frac {i \pi b 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^{2}}-\frac {i \pi b \,f^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 \left (g x +f \right ) g^{3}}-\frac {i \pi b \,f^{2} \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 \left (g x +f \right ) g^{3}}-\frac {i \pi b f \,\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 )}{g^{3}}-\frac {i \pi b f \,\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 )}{g^{3}}-\frac {i \pi b x \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2 g^{2}}+\frac {i \pi b \,f^{2} \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 \left (g x +f \right ) g^{3}}+\frac {i \pi b f \,\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 )}{g^{3}}+\frac {i \pi b \,f^{2} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2 \left (g x +f \right ) g^{3}}+\frac {i \pi b f \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right )}{g^{3}}+\frac {i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2 g^{2}}+\frac {a x}{g^{2}}-\frac {b n x}{g^{2}} \]
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 \[ -a {\left (\frac {f^{2}}{g^{4} x + f g^{3}} - \frac {x}{g^{2}} + \frac {2 \, f \log \left (g x + f\right )}{g^{3}}\right )} + b \int \frac {x^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + x^{2} \log \relax (c)}{g^{2} x^{2} + 2 \, f g x + f^{2}}\,{d x} \]
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 \[ \int \frac {x^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{{\left (f+g\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{2} \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )}{\left (f + g x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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