Optimal. Leaf size=144 \[ -\frac{2 b n^2 \text{PolyLog}\left (2,\frac{c x}{b}+1\right )}{c}+x \log ^2\left (d \left (b x+c x^2\right )^n\right )-4 n x \log \left (d \left (b x+c x^2\right )^n\right )+\frac{2 b n \log (b+c x) \log \left (d \left (b x+c x^2\right )^n\right )}{c}-\frac{b n^2 \log ^2(b+c x)}{c}-\frac{2 b n^2 \log \left (-\frac{c x}{b}\right ) \log (b+c x)}{c}-\frac{4 b n^2 \log (b+c x)}{c}+8 n^2 x \]
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Rubi [A] time = 0.284003, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {2523, 2528, 43, 2524, 1593, 2418, 2394, 2315, 2390, 2301} \[ -\frac{2 b n^2 \text{PolyLog}\left (2,\frac{c x}{b}+1\right )}{c}+x \log ^2\left (d \left (b x+c x^2\right )^n\right )-4 n x \log \left (d \left (b x+c x^2\right )^n\right )+\frac{2 b n \log (b+c x) \log \left (d \left (b x+c x^2\right )^n\right )}{c}-\frac{b n^2 \log ^2(b+c x)}{c}-\frac{2 b n^2 \log \left (-\frac{c x}{b}\right ) \log (b+c x)}{c}-\frac{4 b n^2 \log (b+c x)}{c}+8 n^2 x \]
Antiderivative was successfully verified.
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Rule 2523
Rule 2528
Rule 43
Rule 2524
Rule 1593
Rule 2418
Rule 2394
Rule 2315
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int \log ^2\left (d \left (b x+c x^2\right )^n\right ) \, dx &=x \log ^2\left (d \left (b x+c x^2\right )^n\right )-(2 n) \int \frac{(b+2 c x) \log \left (d \left (b x+c x^2\right )^n\right )}{b+c x} \, dx\\ &=x \log ^2\left (d \left (b x+c x^2\right )^n\right )-(2 n) \int \left (2 \log \left (d \left (b x+c x^2\right )^n\right )-\frac{b \log \left (d \left (b x+c x^2\right )^n\right )}{b+c x}\right ) \, dx\\ &=x \log ^2\left (d \left (b x+c x^2\right )^n\right )-(4 n) \int \log \left (d \left (b x+c x^2\right )^n\right ) \, dx+(2 b n) \int \frac{\log \left (d \left (b x+c x^2\right )^n\right )}{b+c x} \, dx\\ &=-4 n x \log \left (d \left (b x+c x^2\right )^n\right )+\frac{2 b n \log (b+c x) \log \left (d \left (b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (b x+c x^2\right )^n\right )+\left (4 n^2\right ) \int \frac{b+2 c x}{b+c x} \, dx-\frac{\left (2 b n^2\right ) \int \frac{(b+2 c x) \log (b+c x)}{b x+c x^2} \, dx}{c}\\ &=-4 n x \log \left (d \left (b x+c x^2\right )^n\right )+\frac{2 b n \log (b+c x) \log \left (d \left (b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (b x+c x^2\right )^n\right )+\left (4 n^2\right ) \int \left (2-\frac{b}{b+c x}\right ) \, dx-\frac{\left (2 b n^2\right ) \int \frac{(b+2 c x) \log (b+c x)}{x (b+c x)} \, dx}{c}\\ &=8 n^2 x-\frac{4 b n^2 \log (b+c x)}{c}-4 n x \log \left (d \left (b x+c x^2\right )^n\right )+\frac{2 b n \log (b+c x) \log \left (d \left (b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (b x+c x^2\right )^n\right )-\frac{\left (2 b n^2\right ) \int \left (\frac{\log (b+c x)}{x}+\frac{c \log (b+c x)}{b+c x}\right ) \, dx}{c}\\ &=8 n^2 x-\frac{4 b n^2 \log (b+c x)}{c}-4 n x \log \left (d \left (b x+c x^2\right )^n\right )+\frac{2 b n \log (b+c x) \log \left (d \left (b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (b x+c x^2\right )^n\right )-\left (2 b n^2\right ) \int \frac{\log (b+c x)}{b+c x} \, dx-\frac{\left (2 b n^2\right ) \int \frac{\log (b+c x)}{x} \, dx}{c}\\ &=8 n^2 x-\frac{4 b n^2 \log (b+c x)}{c}-\frac{2 b n^2 \log \left (-\frac{c x}{b}\right ) \log (b+c x)}{c}-4 n x \log \left (d \left (b x+c x^2\right )^n\right )+\frac{2 b n \log (b+c x) \log \left (d \left (b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (b x+c x^2\right )^n\right )+\left (2 b n^2\right ) \int \frac{\log \left (-\frac{c x}{b}\right )}{b+c x} \, dx-\frac{\left (2 b n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,b+c x\right )}{c}\\ &=8 n^2 x-\frac{4 b n^2 \log (b+c x)}{c}-\frac{2 b n^2 \log \left (-\frac{c x}{b}\right ) \log (b+c x)}{c}-\frac{b n^2 \log ^2(b+c x)}{c}-4 n x \log \left (d \left (b x+c x^2\right )^n\right )+\frac{2 b n \log (b+c x) \log \left (d \left (b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (b x+c x^2\right )^n\right )-\frac{2 b n^2 \text{Li}_2\left (1+\frac{c x}{b}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0637361, size = 111, normalized size = 0.77 \[ \frac{-2 b n^2 \text{PolyLog}\left (2,\frac{c x}{b}+1\right )+c x \left (\log ^2\left (d (x (b+c x))^n\right )-4 n \log \left (d (x (b+c x))^n\right )+8 n^2\right )-2 b n \log (b+c x) \left (-\log \left (d (x (b+c x))^n\right )+n \log \left (-\frac{c x}{b}\right )+2 n\right )-b n^2 \log ^2(b+c x)}{c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23172, size = 166, normalized size = 1.15 \begin{align*} -{\left (\frac{2 \,{\left (\log \left (c x + b\right ) \log \left (-\frac{c x + b}{b} + 1\right ) +{\rm Li}_2\left (\frac{c x + b}{b}\right )\right )} b}{c} + \frac{b \log \left (c x + b\right )^{2} - 8 \, c x + 4 \, b \log \left (c x + b\right )}{c}\right )} n^{2} - 2 \, n{\left (2 \, x - \frac{b \log \left (c x + b\right )}{c}\right )} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) + x \log \left ({\left (c x^{2} + b x\right )}^{n} d\right )^{2} \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 (\log \left ({\left (c x^{2} + b x\right )}^{n} d\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (d \left (b x + c x^{2}\right )^{n} \right )}^{2}\, dx \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 \log \left ({\left (c x^{2} + b x\right )}^{n} d\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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