Optimal. Leaf size=211 \[ -\frac {g^2 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 A+3 B n\right )}{2 d^3 i}-\frac {g^2 (a+b x) (b c-a d) \left (2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 A+B n\right )}{2 d^2 i}+\frac {g^2 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d i}-\frac {B g^2 n (b c-a d)^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i} \]
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Rubi [A] time = 0.49, antiderivative size = 343, normalized size of antiderivative = 1.63, number of steps used = 18, 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} \[ -\frac {B g^2 n (b c-a d)^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^3 i}+\frac {g^2 (b c-a d)^2 \log (c i+d i x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^3 i}+\frac {g^2 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d i}-\frac {A b g^2 x (b c-a d)}{d^2 i}-\frac {B g^2 (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^2 i}-\frac {b B g^2 n x (b c-a d)}{2 d^2 i}+\frac {B g^2 n (b c-a d)^2 \log ^2(i (c+d x))}{2 d^3 i}+\frac {3 B g^2 n (b c-a d)^2 \log (c+d x)}{2 d^3 i}-\frac {B g^2 n (b c-a d)^2 \log (c i+d i x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^3 i} \]
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)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{136 c+136 d x} \, dx &=\int \left (-\frac {b (b c-a d) g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{136 d^2}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2 (136 c+136 d x)}+\frac {b g (a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{136 d}\right ) \, dx\\ &=\frac {(b g) \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}{136 d}-\frac {\left (b (b c-a d) g^2\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{136 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{136 c+136 d x} \, dx}{d^2}\\ &=-\frac {A b (b c-a d) g^2 x}{136 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}-\frac {\left (b B (b c-a d) g^2\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{136 d^2}-\frac {(B n) \int \frac {(b c-a d) g^2 (a+b x)}{c+d x} \, dx}{272 d}-\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (136 c+136 d x)}{a+b x} \, dx}{136 d^3}\\ &=-\frac {A b (b c-a d) g^2 x}{136 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{136 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}-\frac {\left (B (b c-a d) g^2 n\right ) \int \frac {a+b x}{c+d x} \, dx}{272 d}-\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \left (\frac {b \log (136 c+136 d x)}{a+b x}-\frac {d \log (136 c+136 d x)}{c+d x}\right ) \, dx}{136 d^3}+\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {1}{c+d x} \, dx}{136 d^2}\\ &=-\frac {A b (b c-a d) g^2 x}{136 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{136 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac {B (b c-a d)^2 g^2 n \log (c+d x)}{136 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}-\frac {\left (B (b c-a d) g^2 n\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{272 d}-\frac {\left (b B (b c-a d)^2 g^2 n\right ) \int \frac {\log (136 c+136 d x)}{a+b x} \, dx}{136 d^3}+\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {\log (136 c+136 d x)}{c+d x} \, dx}{136 d^2}\\ &=-\frac {A b (b c-a d) g^2 x}{136 d^2}-\frac {b B (b c-a d) g^2 n x}{272 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{136 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac {3 B (b c-a d)^2 g^2 n \log (c+d x)}{272 d^3}-\frac {B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (136 c+136 d x)}{136 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}+\frac {\left (B (b c-a d)^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {136 \log (x)}{x} \, dx,x,136 c+136 d x\right )}{18496 d^3}+\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {\log \left (\frac {136 d (a+b x)}{-136 b c+136 a d}\right )}{136 c+136 d x} \, dx}{d^2}\\ &=-\frac {A b (b c-a d) g^2 x}{136 d^2}-\frac {b B (b c-a d) g^2 n x}{272 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{136 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac {3 B (b c-a d)^2 g^2 n \log (c+d x)}{272 d^3}-\frac {B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (136 c+136 d x)}{136 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}+\frac {\left (B (b c-a d)^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,136 c+136 d x\right )}{136 d^3}+\frac {\left (B (b c-a d)^2 g^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-136 b c+136 a d}\right )}{x} \, dx,x,136 c+136 d x\right )}{136 d^3}\\ &=-\frac {A b (b c-a d) g^2 x}{136 d^2}-\frac {b B (b c-a d) g^2 n x}{272 d^2}-\frac {B (b c-a d) g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{136 d^2}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{272 d}+\frac {3 B (b c-a d)^2 g^2 n \log (c+d x)}{272 d^3}+\frac {B (b c-a d)^2 g^2 n \log ^2(136 (c+d x))}{272 d^3}-\frac {B (b c-a d)^2 g^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (136 c+136 d x)}{136 d^3}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (136 c+136 d x)}{136 d^3}-\frac {B (b c-a d)^2 g^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{136 d^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 266, normalized size = 1.26 \[ \frac {g^2 \left (d^2 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 (b c-a d)^2 \log (i (c+d x)) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-2 A b d x (b c-a d)+2 B d (a+b x) (a d-b c) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n (b c-a d)^2 \left (\log (i (c+d x)) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (i (c+d x))\right )+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 B n (b c-a d)^2 \log (c+d x)-B n (b c-a d) ((a d-b c) \log (c+d x)+b d x)\right )}{2 d^3 i} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A b^{2} g^{2} x^{2} + 2 \, A a b g^{2} x + A a^{2} g^{2} + {\left (B b^{2} g^{2} x^{2} + 2 \, B a b g^{2} x + B a^{2} g^{2}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{d i x + c i}, x\right ) \]
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 = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{d i x +c i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 4.69, size = 627, normalized size = 2.97 \[ 2 \, A a b g^{2} {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {1}{2} \, A b^{2} g^{2} {\left (\frac {2 \, c^{2} \log \left (d x + c\right )}{d^{3} i} + \frac {d x^{2} - 2 \, c x}{d^{2} i}\right )} + \frac {A a^{2} g^{2} \log \left (d i x + c i\right )}{d i} + \frac {{\left (b^{2} c^{2} g^{2} n - 2 \, a b c d g^{2} n + a^{2} d^{2} g^{2} n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{3} i} + \frac {{\left (2 \, a^{2} d^{2} g^{2} \log \relax (e) + {\left (3 \, g^{2} n + 2 \, g^{2} \log \relax (e)\right )} b^{2} c^{2} - 4 \, {\left (g^{2} n + g^{2} \log \relax (e)\right )} a b c d\right )} B \log \left (d x + c\right )}{2 \, d^{3} i} + \frac {B b^{2} d^{2} g^{2} x^{2} \log \relax (e) - 2 \, {\left (b^{2} c^{2} g^{2} n - 2 \, a b c d g^{2} n + a^{2} d^{2} g^{2} n\right )} B \log \left (b x + a\right ) \log \left (d x + c\right ) + {\left (b^{2} c^{2} g^{2} n - 2 \, a b c d g^{2} n + a^{2} d^{2} g^{2} n\right )} B \log \left (d x + c\right )^{2} - {\left ({\left (g^{2} n + 2 \, g^{2} \log \relax (e)\right )} b^{2} c d - {\left (g^{2} n + 4 \, g^{2} \log \relax (e)\right )} a b d^{2}\right )} B x - {\left (2 \, a b c d g^{2} n - 3 \, a^{2} d^{2} g^{2} n\right )} B \log \left (b x + a\right ) + {\left (B b^{2} d^{2} g^{2} x^{2} - 2 \, {\left (b^{2} c d g^{2} - 2 \, a b d^{2} g^{2}\right )} B x + 2 \, {\left (b^{2} c^{2} g^{2} - 2 \, a b c d g^{2} + a^{2} d^{2} g^{2}\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (B b^{2} d^{2} g^{2} x^{2} - 2 \, {\left (b^{2} c d g^{2} - 2 \, a b d^{2} g^{2}\right )} B x + 2 \, {\left (b^{2} c^{2} g^{2} - 2 \, a b c d g^{2} + a^{2} d^{2} g^{2}\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, d^{3} i} \]
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 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^2\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{c\,i+d\,i\,x} \,d x \]
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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