Optimal. Leaf size=361 \[ -\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4 \log \left (1-\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {4 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 B^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {24 B^3 n^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \text {Li}_4\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {24 B^4 n^4 \text {Li}_5\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h} \]
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Rubi [A]
time = 0.44, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {2573, 2567, 12,
2379, 2421, 2430, 6724} \begin {gather*} \frac {24 B^3 n^3 \text {PolyLog}\left (4,\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h (b f-a g)}+\frac {12 B^2 n^2 \text {PolyLog}\left (3,\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{h (b f-a g)}+\frac {4 B n \text {PolyLog}\left (2,\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{h (b f-a g)}+\frac {24 B^4 n^4 \text {PolyLog}\left (5,\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right )}{h (b f-a g)}-\frac {\log \left (1-\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^4}{h (b f-a g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2379
Rule 2421
Rule 2430
Rule 2567
Rule 2573
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4}{(f+g x) (a h+b h x)} \, dx &=\int \left (\frac {A^4}{h (a+b x) (f+g x)}+\frac {4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}+\frac {6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}+\frac {4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}+\frac {B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}\right ) \, dx\\ &=\frac {A^4 \int \frac {1}{(a+b x) (f+g x)} \, dx}{h}+\frac {\left (4 A^3 B\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}+\frac {\left (6 A^2 B^2\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}+\frac {\left (4 A B^3\right ) \int \frac {\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}+\frac {B^4 \int \frac {\log ^4\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}\\ &=-\frac {4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {\left (A^4 b\right ) \int \frac {1}{a+b x} \, dx}{(b f-a g) h}-\frac {\left (A^4 g\right ) \int \frac {1}{f+g x} \, dx}{(b f-a g) h}+\frac {\left (4 A^3 B (b c-a d) n\right ) \int \frac {\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}+\frac {\left (12 A^2 B^2 (b c-a d) n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}+\frac {\left (12 A B^3 (b c-a d) n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}+\frac {\left (4 B^4 (b c-a d) n\right ) \int \frac {\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac {A^4 \log (a+b x)}{(b f-a g) h}-\frac {A^4 \log (f+g x)}{(b f-a g) h}-\frac {4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A^2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {4 B^4 n \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {\left (4 A^3 B (b c-a d) n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d f-c g}\right )}{1+\frac {(b c-a d) x}{d f-c g}} \, dx,x,\frac {f+g x}{a+b x}\right )}{(b f-a g) (d f-c g) h}-\frac {\left (12 A^2 B^2 (b c-a d) n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}-\frac {\left (24 A B^3 (b c-a d) n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}-\frac {\left (12 B^4 (b c-a d) n^2\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac {A^4 \log (a+b x)}{(b f-a g) h}-\frac {A^4 \log (f+g x)}{(b f-a g) h}-\frac {4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {4 A^3 B n \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A^2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {4 B^4 n \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A^2 B^2 n^2 \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {24 A B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 B^4 n^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {\left (24 A B^3 (b c-a d) n^3\right ) \int \frac {\text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}-\frac {\left (24 B^4 (b c-a d) n^3\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac {A^4 \log (a+b x)}{(b f-a g) h}-\frac {A^4 \log (f+g x)}{(b f-a g) h}-\frac {4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {4 A^3 B n \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A^2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {4 B^4 n \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A^2 B^2 n^2 \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {24 A B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 B^4 n^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {24 A B^3 n^3 \text {Li}_4\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {24 B^4 n^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_4\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {\left (24 B^4 (b c-a d) n^4\right ) \int \frac {\text {Li}_4\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac {A^4 \log (a+b x)}{(b f-a g) h}-\frac {A^4 \log (f+g x)}{(b f-a g) h}-\frac {4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {4 A^3 B n \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A^2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {4 B^4 n \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 A^2 B^2 n^2 \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {24 A B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {12 B^4 n^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {24 A B^3 n^3 \text {Li}_4\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {24 B^4 n^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_4\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {24 B^4 n^4 \text {Li}_5\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}\\ \end {align*}
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Mathematica [F]
time = 2.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4}{(f+g x) (a h+b h x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )^{4}}{\left (g x +f \right ) \left (b h x +a h \right )}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^4}{\left (f+g\,x\right )\,\left (a\,h+b\,h\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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