Optimal. Leaf size=361 \[ -\frac{2 B^2 i^2 n^2 (b c-a d)^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{3 b^3 d}+\frac{2 B i^2 n (b c-a d)^3 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 d}-\frac{2 B i^2 n (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3}-\frac{B i^2 n (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b d}+\frac{i^2 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d}+\frac{B^2 i^2 n^2 x (b c-a d)^2}{3 b^2}+\frac{B^2 i^2 n^2 (b c-a d)^3 \log \left (\frac{a+b x}{c+d x}\right )}{3 b^3 d}+\frac{B^2 i^2 n^2 (b c-a d)^3 \log (c+d x)}{b^3 d} \]
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Rubi [A] time = 0.543237, antiderivative size = 454, normalized size of antiderivative = 1.26, number of steps used = 19, number of rules used = 13, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.371, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 43} \[ -\frac{2 B^2 i^2 n^2 (b c-a d)^3 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{3 b^3 d}-\frac{2 B i^2 n (b c-a d)^3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^3 d}-\frac{2 A B i^2 n x (b c-a d)^2}{3 b^2}-\frac{B i^2 n (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b d}+\frac{i^2 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d}-\frac{2 B^2 i^2 n (a+b x) (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^3}+\frac{B^2 i^2 n^2 x (b c-a d)^2}{3 b^2}+\frac{B^2 i^2 n^2 (b c-a d)^3 \log ^2(a+b x)}{3 b^3 d}+\frac{B^2 i^2 n^2 (b c-a d)^3 \log (a+b x)}{3 b^3 d}+\frac{2 B^2 i^2 n^2 (b c-a d)^3 \log (c+d x)}{3 b^3 d}-\frac{2 B^2 i^2 n^2 (b c-a d)^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 d} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 2528
Rule 2486
Rule 31
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rule 43
Rubi steps
\begin{align*} \int (171 c+171 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\frac{9747 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac{(2 B n) \int \frac{5000211 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{513 d}\\ &=\frac{9747 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac{(19494 B (b c-a d) n) \int \frac{(c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{d}\\ &=\frac{9747 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac{(19494 B (b c-a d) n) \int \left (\frac{d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{(b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 (a+b x)}+\frac{d (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{d}\\ &=\frac{9747 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac{(19494 B (b c-a d) n) \int (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b}-\frac{\left (19494 B (b c-a d)^2 n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2}-\frac{\left (19494 B (b c-a d)^3 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^2 d}\\ &=-\frac{19494 A B (b c-a d)^2 n x}{b^2}-\frac{9747 B (b c-a d) n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d}-\frac{19494 B (b c-a d)^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 d}+\frac{9747 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}-\frac{\left (19494 B^2 (b c-a d)^2 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^2}+\frac{\left (9747 B^2 (b c-a d) n^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{b d}+\frac{\left (19494 B^2 (b c-a d)^3 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^3 d}\\ &=-\frac{19494 A B (b c-a d)^2 n x}{b^2}-\frac{19494 B^2 (b c-a d)^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3}-\frac{9747 B (b c-a d) n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d}-\frac{19494 B (b c-a d)^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 d}+\frac{9747 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{\left (9747 B^2 (b c-a d)^2 n^2\right ) \int \frac{c+d x}{a+b x} \, dx}{b d}+\frac{\left (19494 B^2 (b c-a d)^3 n^2\right ) \int \frac{1}{c+d x} \, dx}{b^3}+\frac{\left (19494 B^2 (b c-a d)^3 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^3 d}\\ &=-\frac{19494 A B (b c-a d)^2 n x}{b^2}-\frac{19494 B^2 (b c-a d)^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3}-\frac{9747 B (b c-a d) n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d}-\frac{19494 B (b c-a d)^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 d}+\frac{9747 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{19494 B^2 (b c-a d)^3 n^2 \log (c+d x)}{b^3 d}+\frac{\left (9747 B^2 (b c-a d)^2 n^2\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{b d}-\frac{\left (19494 B^2 (b c-a d)^3 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3}+\frac{\left (19494 B^2 (b c-a d)^3 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 d}\\ &=-\frac{19494 A B (b c-a d)^2 n x}{b^2}+\frac{9747 B^2 (b c-a d)^2 n^2 x}{b^2}+\frac{9747 B^2 (b c-a d)^3 n^2 \log (a+b x)}{b^3 d}-\frac{19494 B^2 (b c-a d)^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3}-\frac{9747 B (b c-a d) n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d}-\frac{19494 B (b c-a d)^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 d}+\frac{9747 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{19494 B^2 (b c-a d)^3 n^2 \log (c+d x)}{b^3 d}-\frac{19494 B^2 (b c-a d)^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 d}+\frac{\left (19494 B^2 (b c-a d)^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 d}+\frac{\left (19494 B^2 (b c-a d)^3 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 d}\\ &=-\frac{19494 A B (b c-a d)^2 n x}{b^2}+\frac{9747 B^2 (b c-a d)^2 n^2 x}{b^2}+\frac{9747 B^2 (b c-a d)^3 n^2 \log (a+b x)}{b^3 d}+\frac{9747 B^2 (b c-a d)^3 n^2 \log ^2(a+b x)}{b^3 d}-\frac{19494 B^2 (b c-a d)^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3}-\frac{9747 B (b c-a d) n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d}-\frac{19494 B (b c-a d)^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 d}+\frac{9747 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{19494 B^2 (b c-a d)^3 n^2 \log (c+d x)}{b^3 d}-\frac{19494 B^2 (b c-a d)^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 d}+\frac{\left (19494 B^2 (b c-a d)^3 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^3 d}\\ &=-\frac{19494 A B (b c-a d)^2 n x}{b^2}+\frac{9747 B^2 (b c-a d)^2 n^2 x}{b^2}+\frac{9747 B^2 (b c-a d)^3 n^2 \log (a+b x)}{b^3 d}+\frac{9747 B^2 (b c-a d)^3 n^2 \log ^2(a+b x)}{b^3 d}-\frac{19494 B^2 (b c-a d)^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3}-\frac{9747 B (b c-a d) n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d}-\frac{19494 B (b c-a d)^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 d}+\frac{9747 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{19494 B^2 (b c-a d)^3 n^2 \log (c+d x)}{b^3 d}-\frac{19494 B^2 (b c-a d)^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 d}-\frac{19494 B^2 (b c-a d)^3 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 d}\\ \end{align*}
Mathematica [A] time = 0.238118, size = 303, normalized size = 0.84 \[ \frac{i^2 \left ((c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2-\frac{B n (b c-a d) \left (-B n (b c-a d)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+b^2 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 (b c-a d)^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 A b d x (b c-a d)+2 B d (a+b x) (b c-a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 B n (b c-a d)^2 \log (c+d x)-B n (b c-a d) ((b c-a d) \log (a+b x)+b d x)\right )}{b^3}\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.527, size = 0, normalized size = 0. \begin{align*} \int \left ( dix+ci \right ) ^{2} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 3.69403, size = 1989, normalized size = 5.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} d^{2} i^{2} x^{2} + 2 \, A^{2} c d i^{2} x + A^{2} c^{2} i^{2} +{\left (B^{2} d^{2} i^{2} x^{2} + 2 \, B^{2} c d i^{2} x + B^{2} c^{2} i^{2}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B d^{2} i^{2} x^{2} + 2 \, A B c d i^{2} x + A B c^{2} i^{2}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d i x + c i\right )}^{2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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