Optimal. Leaf size=196 \[ -\frac {B g n (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A+B n\right )}{b d^2}-\frac {B g n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b d}+\frac {g (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b}-\frac {B^2 g n^2 (b c-a d)^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b d^2} \]
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Rubi [A] time = 0.44, antiderivative size = 309, normalized size of antiderivative = 1.58, number of steps used = 15, number of rules used = 12, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ -\frac {B^2 g n^2 (b c-a d)^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b d^2}+\frac {B g n (b c-a d)^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b d^2}+\frac {g (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b}-\frac {A B g n x (b c-a d)}{d}+\frac {B^2 g n^2 (b c-a d)^2 \log ^2(c+d x)}{2 b d^2}+\frac {B^2 g n^2 (b c-a d)^2 \log (c+d x)}{b d^2}-\frac {B^2 g n^2 (b c-a d)^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b d^2}-\frac {B^2 g n (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
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 (a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}-\frac {(B n) \int \frac {(b c-a d) g^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b g}\\ &=\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}-\frac {(B (b c-a d) g n) \int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b}\\ &=\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}-\frac {(B (b c-a d) g n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac {(-b c+a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x)}\right ) \, dx}{b}\\ &=\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}-\frac {(B (b c-a d) g n) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d}+\frac {\left (B (b c-a d)^2 g n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b d}\\ &=-\frac {A B (b c-a d) g n x}{d}+\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac {B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}-\frac {\left (B^2 (b c-a d) g n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{d}-\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b d^2}\\ &=-\frac {A B (b c-a d) g n x}{d}-\frac {B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac {B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}-\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{b d^2}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac {1}{c+d x} \, dx}{b d}\\ &=-\frac {A B (b c-a d) g n x}{d}-\frac {B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac {B^2 (b c-a d)^2 g n^2 \log (c+d x)}{b d^2}+\frac {B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}-\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{d^2}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b d}\\ &=-\frac {A B (b c-a d) g n x}{d}-\frac {B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac {B^2 (b c-a d)^2 g n^2 \log (c+d x)}{b d^2}-\frac {B^2 (b c-a d)^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b d^2}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b d}\\ &=-\frac {A B (b c-a d) g n x}{d}-\frac {B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac {B^2 (b c-a d)^2 g n^2 \log (c+d x)}{b d^2}-\frac {B^2 (b c-a d)^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}+\frac {B^2 (b c-a d)^2 g n^2 \log ^2(c+d x)}{2 b d^2}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b d^2}\\ &=-\frac {A B (b c-a d) g n x}{d}-\frac {B^2 (b c-a d) g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b d}+\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b}+\frac {B^2 (b c-a d)^2 g n^2 \log (c+d x)}{b d^2}-\frac {B^2 (b c-a d)^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b d^2}+\frac {B (b c-a d)^2 g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b d^2}+\frac {B^2 (b c-a d)^2 g n^2 \log ^2(c+d x)}{2 b d^2}-\frac {B^2 (b c-a d)^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b d^2}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 215, normalized size = 1.10 \[ \frac {g \left ((a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2-\frac {B n (b c-a d) \left (-2 (b c-a d) \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n (b c-a d) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B n (b c-a d) \log (c+d x)+2 A b d x\right )}{d^2}\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b g x + A^{2} a g + {\left (B^{2} b g x + B^{2} a g\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b g x + A B a g\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ), x\right ) \]
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 = 0.12, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 7.48, size = 828, normalized size = 4.22 \[ A B b g x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{2} \, A^{2} b g x^{2} - A B b g n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + 2 \, A B a g n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + 2 \, A B a g x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A^{2} a g x + \frac {{\left ({\left (g n^{2} + g n \log \relax (e)\right )} b c^{2} - {\left (g n^{2} + 2 \, g n \log \relax (e)\right )} a c d\right )} B^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b^{2} c^{2} g n^{2} - 2 \, a b c d g n^{2} + a^{2} d^{2} g n^{2}\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^{2}}{b d^{2}} - \frac {B^{2} a^{2} d^{2} g n^{2} \log \left (b x + a\right )^{2} - B^{2} b^{2} d^{2} g x^{2} \log \relax (e)^{2} + 2 \, {\left (b^{2} c^{2} g n^{2} - 2 \, a b c d g n^{2}\right )} B^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b^{2} c^{2} g n^{2} - 2 \, a b c d g n^{2}\right )} B^{2} \log \left (d x + c\right )^{2} + 2 \, {\left (b^{2} c d g n \log \relax (e) - {\left (g n \log \relax (e) + g \log \relax (e)^{2}\right )} a b d^{2}\right )} B^{2} x + 2 \, {\left (a b c d g n^{2} - {\left (g n^{2} + g n \log \relax (e)\right )} a^{2} d^{2}\right )} B^{2} \log \left (b x + a\right ) - {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} a b d^{2} g x\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} - {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} a b d^{2} g x\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2} - 2 \, {\left (B^{2} b^{2} d^{2} g x^{2} \log \relax (e) + B^{2} a^{2} d^{2} g n \log \left (b x + a\right ) - {\left (b^{2} c d g n - {\left (g n + 2 \, g \log \relax (e)\right )} a b d^{2}\right )} B^{2} x + {\left (b^{2} c^{2} g n - 2 \, a b c d g n\right )} B^{2} \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) + 2 \, {\left (B^{2} b^{2} d^{2} g x^{2} \log \relax (e) + B^{2} a^{2} d^{2} g n \log \left (b x + a\right ) - {\left (b^{2} c d g n - {\left (g n + 2 \, g \log \relax (e)\right )} a b d^{2}\right )} B^{2} x + {\left (b^{2} c^{2} g n - 2 \, a b c d g n\right )} B^{2} \log \left (d x + c\right ) + {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} a b d^{2} g x\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a\,g+b\,g\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]
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 \[ g \left (\int A^{2} a\, dx + \int A^{2} b x\, dx + \int B^{2} a \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 A B a \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int B^{2} b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 A B b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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