Optimal. Leaf size=402 \[ -\frac {b e n}{2 (e h-d i) (g h-f i) (h+i x)}-\frac {b e g n \log (d+e x)}{(e h-d i) (g h-f i)^2}-\frac {b e^2 n \log (d+e x)}{2 (e h-d i)^2 (g h-f i)}+\frac {a+b \log \left (c (d+e x)^n\right )}{2 (g h-f i) (h+i x)^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^2 (h+i x)}+\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{(g h-f i)^3}+\frac {b e g n \log (h+i x)}{(e h-d i) (g h-f i)^2}+\frac {b e^2 n \log (h+i x)}{2 (e h-d i)^2 (g h-f i)}-\frac {g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (h+i x)}{e h-d i}\right )}{(g h-f i)^3}+\frac {b g^2 n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{(g h-f i)^3}-\frac {b g^2 n \text {Li}_2\left (-\frac {i (d+e x)}{e h-d i}\right )}{(g h-f i)^3} \]
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Rubi [A]
time = 0.27, 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 = {2465, 2441,
2440, 2438, 2442, 46, 36, 31} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 46
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2465
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 ) \text {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 ) \text {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 = 0.24, size = 311, normalized size = 0.77 \begin {gather*} \frac {\frac {(g h-f i)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x)^2}+\frac {2 g (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )}{h+i x}+2 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 )-\frac {2 b e g (g h-f i) n (\log (d+e x)-\log (h+i x))}{e h-d i}-\frac {b e (g h-f i)^2 n (e h-d i+e (h+i x) \log (d+e x)-e (h+i x) \log (h+i x))}{(e h-d i)^2 (h+i x)}-2 g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (h+i x)}{e h-d i}\right )+2 b g^2 n \text {Li}_2\left (\frac {g (d+e x)}{-e f+d g}\right )-2 b g^2 n \text {Li}_2\left (\frac {i (d+e x)}{-e h+d i}\right )}{2 (g h-f i)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.50, size = 1468, normalized size = 3.65
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1468\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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