Optimal. Leaf size=112 \[ -\frac {a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac {b e^2 n \log (d+e x)}{2 g (e f-d g)^2}-\frac {b e^2 n \log (f+g x)}{2 g (e f-d g)^2}+\frac {b e n}{2 g (f+g x) (e f-d g)} \]
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Rubi [A] time = 0.06, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2395, 44} \[ -\frac {a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac {b e^2 n \log (d+e x)}{2 g (e f-d g)^2}-\frac {b e^2 n \log (f+g x)}{2 g (e f-d g)^2}+\frac {b e n}{2 g (f+g x) (e f-d g)} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2395
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^3} \, dx &=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac {(b e n) \int \frac {1}{(d+e x) (f+g x)^2} \, dx}{2 g}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac {(b e n) \int \left (\frac {e^2}{(e f-d g)^2 (d+e x)}-\frac {g}{(e f-d g) (f+g x)^2}-\frac {e g}{(e f-d g)^2 (f+g x)}\right ) \, dx}{2 g}\\ &=\frac {b e n}{2 g (e f-d g) (f+g x)}+\frac {b e^2 n \log (d+e x)}{2 g (e f-d g)^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}-\frac {b e^2 n \log (f+g x)}{2 g (e f-d g)^2}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 83, normalized size = 0.74 \[ -\frac {a+b \log \left (c (d+e x)^n\right )-\frac {b e n (f+g x) (e (f+g x) \log (d+e x)-d g-e (f+g x) \log (f+g x)+e f)}{(e f-d g)^2}}{2 g (f+g x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 274, normalized size = 2.45 \[ -\frac {a e^{2} f^{2} - 2 \, a d e f g + a d^{2} g^{2} - {\left (b e^{2} f g - b d e g^{2}\right )} n x - {\left (b e^{2} f^{2} - b d e f g\right )} n - {\left (b e^{2} g^{2} n x^{2} + 2 \, b e^{2} f g n x + {\left (2 \, b d e f g - b d^{2} g^{2}\right )} n\right )} \log \left (e x + d\right ) + {\left (b e^{2} g^{2} n x^{2} + 2 \, b e^{2} f g n x + b e^{2} f^{2} n\right )} \log \left (g x + f\right ) + {\left (b e^{2} f^{2} - 2 \, b d e f g + b d^{2} g^{2}\right )} \log \relax (c)}{2 \, {\left (e^{2} f^{4} g - 2 \, d e f^{3} g^{2} + d^{2} f^{2} g^{3} + {\left (e^{2} f^{2} g^{3} - 2 \, d e f g^{4} + d^{2} g^{5}\right )} x^{2} + 2 \, {\left (e^{2} f^{3} g^{2} - 2 \, d e f^{2} g^{3} + d^{2} f g^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 302, normalized size = 2.70 \[ -\frac {b g^{2} n x^{2} e^{2} \log \left (g x + f\right ) - b g^{2} n x^{2} e^{2} \log \left (x e + d\right ) + b d g^{2} n x e + 2 \, b f g n x e^{2} \log \left (g x + f\right ) + b d^{2} g^{2} n \log \left (x e + d\right ) - 2 \, b f g n x e^{2} \log \left (x e + d\right ) - 2 \, b d f g n e \log \left (x e + d\right ) - b f g n x e^{2} + b d f g n e + b f^{2} n e^{2} \log \left (g x + f\right ) + b d^{2} g^{2} \log \relax (c) - 2 \, b d f g e \log \relax (c) + a d^{2} g^{2} - b f^{2} n e^{2} - 2 \, a d f g e + b f^{2} e^{2} \log \relax (c) + a f^{2} e^{2}}{2 \, {\left (d^{2} g^{5} x^{2} - 2 \, d f g^{4} x^{2} e + 2 \, d^{2} f g^{4} x + f^{2} g^{3} x^{2} e^{2} - 4 \, d f^{2} g^{3} x e + d^{2} f^{2} g^{3} + 2 \, f^{3} g^{2} x e^{2} - 2 \, d f^{3} g^{2} e + f^{4} g e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 633, normalized size = 5.65 \[ -\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{2 \left (g x +f \right )^{2} g}-\frac {2 a \,d^{2} g^{2}+2 b \,d^{2} g^{2} \ln \relax (c )+2 b \,e^{2} f^{2} \ln \relax (c )+2 b d e f g n +2 b d e \,g^{2} n x -2 b \,e^{2} f g n x +2 a \,e^{2} f^{2}-2 b \,e^{2} f^{2} n -4 a d e f g -i \pi b \,d^{2} g^{2} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b \,e^{2} f^{2} \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 i \pi b d e f g \,\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 )+i \pi b \,d^{2} g^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,d^{2} g^{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}+i \pi b \,e^{2} f^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,e^{2} 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}+4 b \,e^{2} f g n x \ln \left (g x +f \right )-4 b \,e^{2} f g n x \ln \left (-e x -d \right )+2 b \,e^{2} f^{2} n \ln \left (g x +f \right )-2 b \,e^{2} f^{2} n \ln \left (-e x -d \right )-i \pi b \,e^{2} 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 i \pi b d e f g \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b \,d^{2} g^{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 i \pi b d e f g \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-2 i \pi b d e f g \,\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 b \,e^{2} g^{2} n \,x^{2} \ln \left (g x +f \right )-2 b \,e^{2} g^{2} n \,x^{2} \ln \left (-e x -d \right )-4 b d e f g \ln \relax (c )}{4 \left (g x +f \right )^{2} \left (d g -e f \right )^{2} g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 167, normalized size = 1.49 \[ \frac {1}{2} \, b e n {\left (\frac {e \log \left (e x + d\right )}{e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}} - \frac {e \log \left (g x + f\right )}{e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}} + \frac {1}{e f^{2} g - d f g^{2} + {\left (e f g^{2} - d g^{3}\right )} x}\right )} - \frac {b \log \left ({\left (e x + d\right )}^{n} c\right )}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} - \frac {a}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 173, normalized size = 1.54 \[ \frac {b\,e^2\,n\,\mathrm {atanh}\left (\frac {2\,d^2\,g^3-2\,e^2\,f^2\,g}{2\,g\,{\left (d\,g-e\,f\right )}^2}+\frac {2\,e\,g\,x}{d\,g-e\,f}\right )}{g\,{\left (d\,g-e\,f\right )}^2}-\frac {b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{2\,g\,\left (f^2+2\,f\,g\,x+g^2\,x^2\right )}-\frac {\frac {a\,d\,g-a\,e\,f+b\,e\,f\,n}{d\,g-e\,f}+\frac {b\,e\,g\,n\,x}{d\,g-e\,f}}{2\,f^2\,g+4\,f\,g^2\,x+2\,g^3\,x^2} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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