Optimal. Leaf size=345 \[ \frac {d^3 i^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^3}-\frac {3 d^2 i^3 (b c-a d) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^3}-\frac {2 d i^3 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3 (a+b x)}-\frac {i^3 (c+d x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^3 (a+b x)^2}+\frac {3 B d^2 i^3 (b c-a d) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^3}-\frac {B d^2 i^3 (b c-a d) \log (c+d x)}{b^4 g^3}-\frac {2 B d i^3 (c+d x) (b c-a d)}{b^3 g^3 (a+b x)}-\frac {B i^3 (c+d x)^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2} \]
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Rubi [A] time = 0.72, antiderivative size = 442, normalized size of antiderivative = 1.28, number of steps used = 22, number of rules used = 13, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2528, 2486, 31, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {3 B d^2 i^3 (b c-a d) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^3}+\frac {3 d^2 i^3 (b c-a d) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^3}-\frac {3 d i^3 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^3 (a+b x)}-\frac {i^3 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^4 g^3 (a+b x)^2}+\frac {B d^3 i^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {3 B d^2 i^3 (b c-a d) \log ^2(a+b x)}{2 b^4 g^3}-\frac {5 B d^2 i^3 (b c-a d) \log (a+b x)}{2 b^4 g^3}+\frac {3 B d^2 i^3 (b c-a d) \log (c+d x)}{2 b^4 g^3}+\frac {3 B d^2 i^3 (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^3}-\frac {5 B d i^3 (b c-a d)^2}{2 b^4 g^3 (a+b x)}-\frac {B i^3 (b c-a d)^3}{4 b^4 g^3 (a+b x)^2}+\frac {A d^3 i^3 x}{b^3 g^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 44
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2486
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {(26 c+26 d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac {17576 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {17576 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^3}+\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}+\frac {52728 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}\right ) \, dx\\ &=\frac {\left (17576 d^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^3 g^3}+\frac {\left (52728 d^2 (b c-a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 g^3}+\frac {\left (52728 d (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^3 g^3}+\frac {\left (17576 (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^3 g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac {\left (17576 B d^3\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^3 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^4 g^3}+\frac {\left (52728 B d (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^3}+\frac {\left (8788 B (b c-a d)^3\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}-\frac {\left (17576 B d^3 (b c-a d)\right ) \int \frac {1}{c+d x} \, dx}{b^4 g^3}+\frac {\left (52728 B d (b c-a d)^3\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^3}+\frac {\left (8788 B (b c-a d)^4\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 e g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}-\frac {17576 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac {\left (52728 B d (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^4 g^3}+\frac {\left (8788 B (b c-a d)^4\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^4 e g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}-\frac {4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac {43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac {43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac {26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^3 g^3}+\frac {\left (52728 B d^3 (b c-a d)\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^4 g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}-\frac {4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac {43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac {43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac {26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac {52728 B d^2 (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}-\frac {4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac {43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac {43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}-\frac {26364 B d^2 (b c-a d) \log ^2(a+b x)}{b^4 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac {26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac {52728 B d^2 (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^3}-\frac {\left (52728 B d^2 (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^3}\\ &=\frac {17576 A d^3 x}{b^3 g^3}-\frac {4394 B (b c-a d)^3}{b^4 g^3 (a+b x)^2}-\frac {43940 B d (b c-a d)^2}{b^4 g^3 (a+b x)}-\frac {43940 B d^2 (b c-a d) \log (a+b x)}{b^4 g^3}-\frac {26364 B d^2 (b c-a d) \log ^2(a+b x)}{b^4 g^3}+\frac {17576 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^4 g^3}-\frac {8788 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)^2}-\frac {52728 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3 (a+b x)}+\frac {52728 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}+\frac {26364 B d^2 (b c-a d) \log (c+d x)}{b^4 g^3}+\frac {52728 B d^2 (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^4 g^3}+\frac {52728 B d^2 (b c-a d) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^4 g^3}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 314, normalized size = 0.91 \[ \frac {i^3 \left (12 d^2 (b c-a d) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {12 d (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {2 (b c-a d)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}+4 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+6 B d^2 (a d-b c) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+10 B d^2 (a d-b c) \log (a+b x)+6 B d^2 (b c-a d) \log (c+d x)-\frac {10 B d (b c-a d)^2}{a+b x}-\frac {B (b c-a d)^3}{(a+b x)^2}+4 A b d^3 x\right )}{4 b^4 g^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A d^{3} i^{3} x^{3} + 3 \, A c d^{2} i^{3} x^{2} + 3 \, A c^{2} d i^{3} x + A c^{3} i^{3} + {\left (B d^{3} i^{3} x^{3} + 3 \, B c d^{2} i^{3} x^{2} + 3 \, B c^{2} d i^{3} x + B c^{3} i^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{3} g^{3} x^{3} + 3 \, a b^{2} g^{3} x^{2} + 3 \, a^{2} b g^{3} x + a^{3} g^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 1855, normalized size = 5.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.51, size = 2302, normalized size = 6.67 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (c\,i+d\,i\,x\right )}^3\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (a\,g+b\,g\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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