Optimal. Leaf size=402 \[ \frac {g^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^3}-\frac {g^2 \log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x) (g h-f i)^2}+\frac {a+b \log \left (c (d+e x)^n\right )}{2 (h+i x)^2 (g h-f i)}-\frac {b e^2 n \log (d+e x)}{2 (e h-d i)^2 (g h-f i)}+\frac {b e^2 n \log (h+i x)}{2 (e h-d i)^2 (g h-f i)}+\frac {b g^2 n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^3}-\frac {b g^2 n \text {Li}_2\left (-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^3}-\frac {b e n}{2 (h+i x) (e h-d i) (g h-f i)}-\frac {b e g n \log (d+e x)}{(e h-d i) (g h-f i)^2}+\frac {b e g n \log (h+i x)}{(e h-d i) (g h-f i)^2} \]
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Rubi [A] time = 0.37, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2418, 2394, 2393, 2391, 2395, 44, 36, 31} \[ \frac {b g^2 n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^3}-\frac {b g^2 n \text {PolyLog}\left (2,-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^3}+\frac {g^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^3}-\frac {g^2 \log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x) (g h-f i)^2}+\frac {a+b \log \left (c (d+e x)^n\right )}{2 (h+i x)^2 (g h-f i)}-\frac {b e^2 n \log (d+e x)}{2 (e h-d i)^2 (g h-f i)}+\frac {b e^2 n \log (h+i x)}{2 (e h-d i)^2 (g h-f i)}-\frac {b e n}{2 (h+i x) (e h-d i) (g h-f i)}-\frac {b e g n \log (d+e x)}{(e h-d i) (g h-f i)^2}+\frac {b e g n \log (h+i x)}{(e h-d i) (g h-f i)^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 44
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2418
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{(h+223 x)^3 (f+g x)} \, dx &=\int \left (\frac {223 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h) (h+223 x)^3}-\frac {223 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^2 (h+223 x)^2}+\frac {223 g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^3 (h+223 x)}-\frac {g^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^3 (f+g x)}\right ) \, dx\\ &=\frac {\left (223 g^2\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{h+223 x} \, dx}{(223 f-g h)^3}-\frac {g^3 \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{(223 f-g h)^3}-\frac {(223 g) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(h+223 x)^2} \, dx}{(223 f-g h)^2}+\frac {223 \int \frac {a+b \log \left (c (d+e x)^n\right )}{(h+223 x)^3} \, dx}{223 f-g h}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 (223 f-g h) (h+223 x)^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^2 (h+223 x)}+\frac {g^2 \log \left (-\frac {e (h+223 x)}{223 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^3}-\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{(223 f-g h)^3}-\frac {\left (b e g^2 n\right ) \int \frac {\log \left (\frac {e (h+223 x)}{-223 d+e h}\right )}{d+e x} \, dx}{(223 f-g h)^3}+\frac {\left (b e g^2 n\right ) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{(223 f-g h)^3}-\frac {(b e g n) \int \frac {1}{(h+223 x) (d+e x)} \, dx}{(223 f-g h)^2}+\frac {(b e n) \int \frac {1}{(h+223 x)^2 (d+e x)} \, dx}{2 (223 f-g h)}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 (223 f-g h) (h+223 x)^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^2 (h+223 x)}+\frac {g^2 \log \left (-\frac {e (h+223 x)}{223 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^3}-\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{(223 f-g h)^3}+\frac {\left (b g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{(223 f-g h)^3}-\frac {\left (b g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {223 x}{-223 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(223 f-g h)^3}-\frac {(223 b e g n) \int \frac {1}{h+223 x} \, dx}{(223 d-e h) (223 f-g h)^2}+\frac {\left (b e^2 g n\right ) \int \frac {1}{d+e x} \, dx}{(223 d-e h) (223 f-g h)^2}+\frac {(b e n) \int \left (\frac {223}{(223 d-e h) (h+223 x)^2}-\frac {223 e}{(223 d-e h)^2 (h+223 x)}+\frac {e^2}{(223 d-e h)^2 (d+e x)}\right ) \, dx}{2 (223 f-g h)}\\ &=-\frac {b e n}{2 (223 d-e h) (223 f-g h) (h+223 x)}-\frac {b e g n \log (h+223 x)}{(223 d-e h) (223 f-g h)^2}-\frac {b e^2 n \log (h+223 x)}{2 (223 d-e h)^2 (223 f-g h)}+\frac {b e g n \log (d+e x)}{(223 d-e h) (223 f-g h)^2}+\frac {b e^2 n \log (d+e x)}{2 (223 d-e h)^2 (223 f-g h)}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 (223 f-g h) (h+223 x)^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^2 (h+223 x)}+\frac {g^2 \log \left (-\frac {e (h+223 x)}{223 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^3}-\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{(223 f-g h)^3}-\frac {b g^2 n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{(223 f-g h)^3}+\frac {b g^2 n \text {Li}_2\left (\frac {223 (d+e x)}{223 d-e h}\right )}{(223 f-g h)^3}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 311, normalized size = 0.77 \[ \frac {2 g^2 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {2 g (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )}{h+i x}+\frac {(g h-f i)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x)^2}-2 g^2 \log \left (\frac {e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+2 b g^2 n \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )-\frac {2 b e g n (g h-f i) (\log (d+e x)-\log (h+i x))}{e h-d i}-\frac {b e n (g h-f i)^2 (e (h+i x) \log (d+e x)-d i-e (h+i x) \log (h+i x)+e h)}{(h+i x) (e h-d i)^2}-2 b g^2 n \text {Li}_2\left (\frac {i (d+e x)}{d i-e h}\right )}{2 (g h-f i)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{g i^{3} x^{4} + f h^{3} + {\left (3 \, g h i^{2} + f i^{3}\right )} x^{3} + 3 \, {\left (g h^{2} i + f h i^{2}\right )} x^{2} + {\left (g h^{3} + 3 \, f h^{2} i\right )} x}, x\right ) \]
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 \[ \int \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} {\left (i x + h\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 1468, normalized size = 3.65 \[ \text {result too large to display} \]
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 \[ \frac {1}{2} \, {\left (\frac {2 \, g^{2} \log \left (g x + f\right )}{g^{3} h^{3} - 3 \, f g^{2} h^{2} i + 3 \, f^{2} g h i^{2} - f^{3} i^{3}} - \frac {2 \, g^{2} \log \left (i x + h\right )}{g^{3} h^{3} - 3 \, f g^{2} h^{2} i + 3 \, f^{2} g h i^{2} - f^{3} i^{3}} + \frac {2 \, g i x + 3 \, g h - f i}{g^{2} h^{4} - 2 \, f g h^{3} i + f^{2} h^{2} i^{2} + {\left (g^{2} h^{2} i^{2} - 2 \, f g h i^{3} + f^{2} i^{4}\right )} x^{2} + 2 \, {\left (g^{2} h^{3} i - 2 \, f g h^{2} i^{2} + f^{2} h i^{3}\right )} x}\right )} a + b \int \frac {\log \left ({\left (e x + d\right )}^{n}\right ) + \log \relax (c)}{g i^{3} x^{4} + f h^{3} + {\left (3 \, g h i^{2} + f i^{3}\right )} x^{3} + 3 \, {\left (g h^{2} i + f h i^{2}\right )} x^{2} + {\left (g h^{3} + 3 \, f h^{2} i\right )} x}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\left (f+g\,x\right )\,{\left (h+i\,x\right )}^3} \,d x \]
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 \[ \int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{\left (f + g x\right ) \left (h + i x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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