Optimal. Leaf size=220 \[ \frac {B i n (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d}-\frac {B i n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2}+\frac {i (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d}-\frac {B^2 i n^2 (b c-a d)^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 d}+\frac {B^2 i n^2 (b c-a d)^2 \log (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.48, antiderivative size = 307, normalized size of antiderivative = 1.40, 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, 2390, 2301, 2394, 2393, 2391} \[ -\frac {B^2 i n^2 (b c-a d)^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 d}-\frac {B i n (b c-a d)^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d}+\frac {i (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d}-\frac {A B i n x (b c-a d)}{b}-\frac {B^2 i n (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2}+\frac {B^2 i n^2 (b c-a d)^2 \log ^2(a+b x)}{2 b^2 d}+\frac {B^2 i n^2 (b c-a d)^2 \log (c+d x)}{b^2 d}-\frac {B^2 i n^2 (b c-a d)^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 d} \]
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 (162 c+162 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\frac {81 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac {(B n) \int \frac {26244 (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{162 d}\\ &=\frac {81 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac {(162 B (b c-a d) n) \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{d}\\ &=\frac {81 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac {(162 B (b c-a d) n) \int \left (\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (a+b x)}\right ) \, dx}{d}\\ &=\frac {81 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac {(162 B (b c-a d) n) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b}-\frac {\left (162 B (b c-a d)^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b d}\\ &=-\frac {162 A B (b c-a d) n x}{b}-\frac {162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac {81 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac {\left (162 B^2 (b c-a d) n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b}+\frac {\left (162 B^2 (b c-a d)^2 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 (a+b x)}{a+b x} \, dx}{b^2 d}\\ &=-\frac {162 A B (b c-a d) n x}{b}-\frac {162 B^2 (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2}-\frac {162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac {81 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac {\left (162 B^2 (b c-a d)^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^2}+\frac {\left (162 B^2 (b c-a d)^2 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^2 d}\\ &=-\frac {162 A B (b c-a d) n x}{b}-\frac {162 B^2 (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2}-\frac {162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac {81 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac {162 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^2 d}-\frac {\left (162 B^2 (b c-a d)^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2}+\frac {\left (162 B^2 (b c-a d)^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b d}\\ &=-\frac {162 A B (b c-a d) n x}{b}-\frac {162 B^2 (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2}-\frac {162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac {81 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac {162 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^2 d}-\frac {162 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 d}+\frac {\left (162 B^2 (b c-a d)^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 d}+\frac {\left (162 B^2 (b c-a d)^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b d}\\ &=-\frac {162 A B (b c-a d) n x}{b}+\frac {81 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^2 d}-\frac {162 B^2 (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2}-\frac {162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac {81 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac {162 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^2 d}-\frac {162 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 d}+\frac {\left (162 B^2 (b c-a d)^2 n^2\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^2 d}\\ &=-\frac {162 A B (b c-a d) n x}{b}+\frac {81 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^2 d}-\frac {162 B^2 (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^2}-\frac {162 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}+\frac {81 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac {162 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^2 d}-\frac {162 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 d}-\frac {162 B^2 (b c-a d)^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 216, normalized size = 0.98 \[ \frac {i \left (\frac {B n (b c-a d) \left (-2 (b c-a d) \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n \log \left (\frac {b (c+d x)}{b c-a d}\right )+A\right )-2 \left (B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n (a d-b c) \log (c+d x)+A b d x\right )+2 B n (a d-b c) \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )+B n (b c-a d) \log ^2(a+b x)\right )}{b^2}+(c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} d i x + A^{2} c i + {\left (B^{2} d i x + B^{2} c i\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B d i x + A B c i\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ), x\right ) \]
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 = 0.11, size = 0, normalized size = 0.00 \[ \int \left (d i x +c i \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 6.65, size = 825, normalized size = 3.75 \[ A B d i 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} d i x^{2} - A B d i 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 c i n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + 2 \, A B c i x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A^{2} c i x - \frac {{\left (a c d i n^{2} - {\left (i n^{2} - i n \log \relax (e)\right )} b c^{2}\right )} B^{2} \log \left (d x + c\right )}{b d} - \frac {{\left (b^{2} c^{2} i n^{2} - 2 \, a b c d i n^{2} + a^{2} d^{2} i 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^{2} d} + \frac {2 \, B^{2} b^{2} c^{2} i n^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) - B^{2} b^{2} c^{2} i n^{2} \log \left (d x + c\right )^{2} + B^{2} b^{2} d^{2} i x^{2} \log \relax (e)^{2} - {\left (2 \, a b c d i n^{2} - a^{2} d^{2} i n^{2}\right )} B^{2} \log \left (b x + a\right )^{2} + 2 \, {\left (a b d^{2} i n \log \relax (e) - {\left (i n \log \relax (e) - i \log \relax (e)^{2}\right )} b^{2} c d\right )} B^{2} x - 2 \, {\left ({\left (i n^{2} - 2 \, i n \log \relax (e)\right )} a b c d - {\left (i n^{2} - i 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} i x^{2} + 2 \, B^{2} b^{2} c d i x\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + 2 \, {\left (B^{2} b^{2} d^{2} i x^{2} \log \relax (e) - B^{2} b^{2} c^{2} i n \log \left (d x + c\right ) + {\left (a b d^{2} i n - {\left (i n - 2 \, i \log \relax (e)\right )} b^{2} c d\right )} B^{2} x + {\left (2 \, a b c d i n - a^{2} d^{2} i n\right )} B^{2} \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B^{2} b^{2} d^{2} i x^{2} \log \relax (e) - B^{2} b^{2} c^{2} i n \log \left (d x + c\right ) + {\left (a b d^{2} i n - {\left (i n - 2 \, i \log \relax (e)\right )} b^{2} c d\right )} B^{2} x + {\left (2 \, a b c d i n - a^{2} d^{2} i n\right )} B^{2} \log \left (b x + a\right ) + {\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, b^{2} d} \]
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 \[ \int \left (c\,i+d\,i\,x\right )\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]
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 \[ i \left (\int A^{2} c\, dx + \int A^{2} d x\, dx + \int B^{2} c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 A B c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx + \int B^{2} d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}\, dx + \int 2 A B d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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