Optimal. Leaf size=269 \[ \frac {g^3 (b c-a d)^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+11 B n\right )}{6 d^4 i}+\frac {g^3 (a+b x) (b c-a d)^2 \left (6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+6 A+5 B n\right )}{6 d^3 i}-\frac {g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A+B n\right )}{6 d^2 i}+\frac {g^3 (a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d i}+\frac {B g^3 n (b c-a d)^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i} \]
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Rubi [A] time = 0.65, antiderivative size = 426, normalized size of antiderivative = 1.58, number of steps used = 22, number of rules used = 13, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.302, Rules used = {2528, 2486, 31, 2525, 12, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {B g^3 n (b c-a d)^3 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^4 i}-\frac {g^3 (a+b x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^2 i}-\frac {g^3 (b c-a d)^3 \log (c i+d i x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^4 i}+\frac {g^3 (a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d i}+\frac {A b g^3 x (b c-a d)^2}{d^3 i}+\frac {B g^3 (a+b x) (b c-a d)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^3 i}-\frac {B g^3 n (a+b x)^2 (b c-a d)}{6 d^2 i}+\frac {5 b B g^3 n x (b c-a d)^2}{6 d^3 i}-\frac {B g^3 n (b c-a d)^3 \log ^2(i (c+d x))}{2 d^4 i}-\frac {11 B g^3 n (b c-a d)^3 \log (c+d x)}{6 d^4 i}+\frac {B g^3 n (b c-a d)^3 \log (c i+d i x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^4 i} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 43
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 {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{135 c+135 d x} \, dx &=\int \left (\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{135 d^3}+\frac {(-b c+a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3 (135 c+135 d x)}-\frac {b (b c-a d) g^2 (a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{135 d^2}+\frac {b g (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{135 d}\right ) \, dx\\ &=\frac {(b g) \int (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{135 d}-\frac {\left (b (b c-a d) g^2\right ) \int (a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{135 d^2}+\frac {\left (b (b c-a d)^2 g^3\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{135 d^3}-\frac {\left ((b c-a d)^3 g^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{135 c+135 d x} \, dx}{d^3}\\ &=\frac {A b (b c-a d)^2 g^3 x}{135 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (135 c+135 d x)}{135 d^4}+\frac {\left (b B (b c-a d)^2 g^3\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{135 d^3}-\frac {(B n) \int \frac {(b c-a d) g^3 (a+b x)^2}{c+d x} \, dx}{405 d}+\frac {(B (b c-a d) g n) \int \frac {(b c-a d) g^2 (a+b x)}{c+d x} \, dx}{270 d^2}+\frac {\left (B (b c-a d)^3 g^3 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (135 c+135 d x)}{a+b x} \, dx}{135 d^4}\\ &=\frac {A b (b c-a d)^2 g^3 x}{135 d^3}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{135 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (135 c+135 d x)}{135 d^4}-\frac {\left (B (b c-a d) g^3 n\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{405 d}+\frac {\left (B (b c-a d)^2 g^3 n\right ) \int \frac {a+b x}{c+d x} \, dx}{270 d^2}+\frac {\left (B (b c-a d)^3 g^3 n\right ) \int \left (\frac {b \log (135 c+135 d x)}{a+b x}-\frac {d \log (135 c+135 d x)}{c+d x}\right ) \, dx}{135 d^4}-\frac {\left (B (b c-a d)^3 g^3 n\right ) \int \frac {1}{c+d x} \, dx}{135 d^3}\\ &=\frac {A b (b c-a d)^2 g^3 x}{135 d^3}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{135 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\frac {B (b c-a d)^3 g^3 n \log (c+d x)}{135 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (135 c+135 d x)}{135 d^4}-\frac {\left (B (b c-a d) g^3 n\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{405 d}+\frac {\left (B (b c-a d)^2 g^3 n\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{270 d^2}+\frac {\left (b B (b c-a d)^3 g^3 n\right ) \int \frac {\log (135 c+135 d x)}{a+b x} \, dx}{135 d^4}-\frac {\left (B (b c-a d)^3 g^3 n\right ) \int \frac {\log (135 c+135 d x)}{c+d x} \, dx}{135 d^3}\\ &=\frac {A b (b c-a d)^2 g^3 x}{135 d^3}+\frac {b B (b c-a d)^2 g^3 n x}{162 d^3}-\frac {B (b c-a d) g^3 n (a+b x)^2}{810 d^2}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{135 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\frac {11 B (b c-a d)^3 g^3 n \log (c+d x)}{810 d^4}+\frac {B (b c-a d)^3 g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (135 c+135 d x)}{135 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (135 c+135 d x)}{135 d^4}-\frac {\left (B (b c-a d)^3 g^3 n\right ) \operatorname {Subst}\left (\int \frac {135 \log (x)}{x} \, dx,x,135 c+135 d x\right )}{18225 d^4}-\frac {\left (B (b c-a d)^3 g^3 n\right ) \int \frac {\log \left (\frac {135 d (a+b x)}{-135 b c+135 a d}\right )}{135 c+135 d x} \, dx}{d^3}\\ &=\frac {A b (b c-a d)^2 g^3 x}{135 d^3}+\frac {b B (b c-a d)^2 g^3 n x}{162 d^3}-\frac {B (b c-a d) g^3 n (a+b x)^2}{810 d^2}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{135 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\frac {11 B (b c-a d)^3 g^3 n \log (c+d x)}{810 d^4}+\frac {B (b c-a d)^3 g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (135 c+135 d x)}{135 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (135 c+135 d x)}{135 d^4}-\frac {\left (B (b c-a d)^3 g^3 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,135 c+135 d x\right )}{135 d^4}-\frac {\left (B (b c-a d)^3 g^3 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-135 b c+135 a d}\right )}{x} \, dx,x,135 c+135 d x\right )}{135 d^4}\\ &=\frac {A b (b c-a d)^2 g^3 x}{135 d^3}+\frac {b B (b c-a d)^2 g^3 n x}{162 d^3}-\frac {B (b c-a d) g^3 n (a+b x)^2}{810 d^2}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{135 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{270 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{405 d}-\frac {11 B (b c-a d)^3 g^3 n \log (c+d x)}{810 d^4}-\frac {B (b c-a d)^3 g^3 n \log ^2(135 (c+d x))}{270 d^4}+\frac {B (b c-a d)^3 g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (135 c+135 d x)}{135 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (135 c+135 d x)}{135 d^4}+\frac {B (b c-a d)^3 g^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{135 d^4}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 370, normalized size = 1.38 \[ \frac {g^3 \left (2 d^3 (a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+3 d^2 (a+b x)^2 (a d-b c) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-6 (b c-a d)^3 \log (i (c+d x)) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+6 A b d x (b c-a d)^2+B n (b c-a d) \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 B n (b c-a d)^3 \left (\log (i (c+d x)) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (i (c+d x))\right )+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )-6 B n (b c-a d)^3 \log (c+d x)+3 B n (b c-a d)^2 ((a d-b c) \log (c+d x)+b d x)\right )}{6 d^4 i} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A b^{3} g^{3} x^{3} + 3 \, A a b^{2} g^{3} x^{2} + 3 \, A a^{2} b g^{3} x + A a^{3} g^{3} + {\left (B b^{3} g^{3} x^{3} + 3 \, B a b^{2} g^{3} x^{2} + 3 \, B a^{2} b g^{3} x + B a^{3} g^{3}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{d i x + c i}, x\right ) \]
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 [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{d i x +c i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 4.29, size = 1003, normalized size = 3.73 \[ 3 \, A a^{2} b g^{3} {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} - \frac {1}{6} \, A b^{3} g^{3} {\left (\frac {6 \, c^{3} \log \left (d x + c\right )}{d^{4} i} - \frac {2 \, d^{2} x^{3} - 3 \, c d x^{2} + 6 \, c^{2} x}{d^{3} i}\right )} + \frac {3}{2} \, A a b^{2} g^{3} {\left (\frac {2 \, c^{2} \log \left (d x + c\right )}{d^{3} i} + \frac {d x^{2} - 2 \, c x}{d^{2} i}\right )} + \frac {A a^{3} g^{3} \log \left (d i x + c i\right )}{d i} - \frac {{\left (b^{3} c^{3} g^{3} n - 3 \, a b^{2} c^{2} d g^{3} n + 3 \, a^{2} b c d^{2} g^{3} n - a^{3} d^{3} g^{3} n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{4} i} + \frac {{\left (6 \, a^{3} d^{3} g^{3} \log \relax (e) - {\left (11 \, g^{3} n + 6 \, g^{3} \log \relax (e)\right )} b^{3} c^{3} + 9 \, {\left (3 \, g^{3} n + 2 \, g^{3} \log \relax (e)\right )} a b^{2} c^{2} d - 18 \, {\left (g^{3} n + g^{3} \log \relax (e)\right )} a^{2} b c d^{2}\right )} B \log \left (d x + c\right )}{6 \, d^{4} i} + \frac {2 \, B b^{3} d^{3} g^{3} x^{3} \log \relax (e) - {\left ({\left (g^{3} n + 3 \, g^{3} \log \relax (e)\right )} b^{3} c d^{2} - {\left (g^{3} n + 9 \, g^{3} \log \relax (e)\right )} a b^{2} d^{3}\right )} B x^{2} + 6 \, {\left (b^{3} c^{3} g^{3} n - 3 \, a b^{2} c^{2} d g^{3} n + 3 \, a^{2} b c d^{2} g^{3} n - a^{3} d^{3} g^{3} n\right )} B \log \left (b x + a\right ) \log \left (d x + c\right ) - 3 \, {\left (b^{3} c^{3} g^{3} n - 3 \, a b^{2} c^{2} d g^{3} n + 3 \, a^{2} b c d^{2} g^{3} n - a^{3} d^{3} g^{3} n\right )} B \log \left (d x + c\right )^{2} + {\left ({\left (5 \, g^{3} n + 6 \, g^{3} \log \relax (e)\right )} b^{3} c^{2} d - 6 \, {\left (2 \, g^{3} n + 3 \, g^{3} \log \relax (e)\right )} a b^{2} c d^{2} + {\left (7 \, g^{3} n + 18 \, g^{3} \log \relax (e)\right )} a^{2} b d^{3}\right )} B x + {\left (6 \, a b^{2} c^{2} d g^{3} n - 15 \, a^{2} b c d^{2} g^{3} n + 11 \, a^{3} d^{3} g^{3} n\right )} B \log \left (b x + a\right ) + {\left (2 \, B b^{3} d^{3} g^{3} x^{3} - 3 \, {\left (b^{3} c d^{2} g^{3} - 3 \, a b^{2} d^{3} g^{3}\right )} B x^{2} + 6 \, {\left (b^{3} c^{2} d g^{3} - 3 \, a b^{2} c d^{2} g^{3} + 3 \, a^{2} b d^{3} g^{3}\right )} B x - 6 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (2 \, B b^{3} d^{3} g^{3} x^{3} - 3 \, {\left (b^{3} c d^{2} g^{3} - 3 \, a b^{2} d^{3} g^{3}\right )} B x^{2} + 6 \, {\left (b^{3} c^{2} d g^{3} - 3 \, a b^{2} c d^{2} g^{3} + 3 \, a^{2} b d^{3} g^{3}\right )} B x - 6 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{6 \, d^{4} i} \]
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 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^3\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{c\,i+d\,i\,x} \,d x \]
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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