Optimal. Leaf size=69 \[ \frac{e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac{b e^2 (c+d x)^2}{6 d}+\frac{b e^2 \log \left (1-(c+d x)^2\right )}{6 d} \]
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Rubi [A] time = 0.0651115, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {6107, 12, 5916, 266, 43} \[ \frac{e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac{b e^2 (c+d x)^2}{6 d}+\frac{b e^2 \log \left (1-(c+d x)^2\right )}{6 d} \]
Antiderivative was successfully verified.
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Rule 6107
Rule 12
Rule 5916
Rule 266
Rule 43
Rubi steps
\begin{align*} \int (c e+d e x)^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int e^2 x^2 \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int x^2 \left (a+b \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{1-x^2} \, dx,x,c+d x\right )}{3 d}\\ &=\frac{e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \frac{x}{1-x} \, dx,x,(c+d x)^2\right )}{6 d}\\ &=\frac{e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}-\frac{\left (b e^2\right ) \operatorname{Subst}\left (\int \left (-1+\frac{1}{1-x}\right ) \, dx,x,(c+d x)^2\right )}{6 d}\\ &=\frac{b e^2 (c+d x)^2}{6 d}+\frac{e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac{b e^2 \log \left (1-(c+d x)^2\right )}{6 d}\\ \end{align*}
Mathematica [A] time = 0.0728182, size = 59, normalized size = 0.86 \[ \frac{e^2 \left ((c+d x)^2 (2 a (c+d x)+b)+b \log \left (1-(c+d x)^2\right )+2 b (c+d x)^3 \tanh ^{-1}(c+d x)\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 174, normalized size = 2.5 \begin{align*}{\frac{{d}^{2}{x}^{3}a{e}^{2}}{3}}+d{x}^{2}ac{e}^{2}+xa{c}^{2}{e}^{2}+{\frac{a{c}^{3}{e}^{2}}{3\,d}}+{\frac{{d}^{2}{\it Artanh} \left ( dx+c \right ){x}^{3}b{e}^{2}}{3}}+d{\it Artanh} \left ( dx+c \right ){x}^{2}bc{e}^{2}+{\it Artanh} \left ( dx+c \right ) xb{c}^{2}{e}^{2}+{\frac{{\it Artanh} \left ( dx+c \right ) b{c}^{3}{e}^{2}}{3\,d}}+{\frac{d{x}^{2}b{e}^{2}}{6}}+{\frac{xbc{e}^{2}}{3}}+{\frac{b{c}^{2}{e}^{2}}{6\,d}}+{\frac{{e}^{2}b\ln \left ( dx+c-1 \right ) }{6\,d}}+{\frac{{e}^{2}b\ln \left ( dx+c+1 \right ) }{6\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.962126, size = 304, normalized size = 4.41 \begin{align*} \frac{1}{3} \, a d^{2} e^{2} x^{3} + a c d e^{2} x^{2} + \frac{1}{2} \,{\left (2 \, x^{2} \operatorname{artanh}\left (d x + c\right ) + d{\left (\frac{2 \, x}{d^{2}} - \frac{{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac{{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b c d e^{2} + \frac{1}{6} \,{\left (2 \, x^{3} \operatorname{artanh}\left (d x + c\right ) + d{\left (\frac{d x^{2} - 4 \, c x}{d^{3}} + \frac{{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac{{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} b d^{2} e^{2} + a c^{2} e^{2} x + \frac{{\left (2 \,{\left (d x + c\right )} \operatorname{artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b c^{2} e^{2}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.43145, size = 323, normalized size = 4.68 \begin{align*} \frac{2 \, a d^{3} e^{2} x^{3} +{\left (6 \, a c + b\right )} d^{2} e^{2} x^{2} + 2 \,{\left (3 \, a c^{2} + b c\right )} d e^{2} x +{\left (b c^{3} + b\right )} e^{2} \log \left (d x + c + 1\right ) -{\left (b c^{3} - b\right )} e^{2} \log \left (d x + c - 1\right ) +{\left (b d^{3} e^{2} x^{3} + 3 \, b c d^{2} e^{2} x^{2} + 3 \, b c^{2} d e^{2} x\right )} \log \left (-\frac{d x + c + 1}{d x + c - 1}\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.20643, size = 180, normalized size = 2.61 \begin{align*} \begin{cases} a c^{2} e^{2} x + a c d e^{2} x^{2} + \frac{a d^{2} e^{2} x^{3}}{3} + \frac{b c^{3} e^{2} \operatorname{atanh}{\left (c + d x \right )}}{3 d} + b c^{2} e^{2} x \operatorname{atanh}{\left (c + d x \right )} + b c d e^{2} x^{2} \operatorname{atanh}{\left (c + d x \right )} + \frac{b c e^{2} x}{3} + \frac{b d^{2} e^{2} x^{3} \operatorname{atanh}{\left (c + d x \right )}}{3} + \frac{b d e^{2} x^{2}}{6} + \frac{b e^{2} \log{\left (\frac{c}{d} + x + \frac{1}{d} \right )}}{3 d} - \frac{b e^{2} \operatorname{atanh}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname{atanh}{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27452, size = 258, normalized size = 3.74 \begin{align*} \frac{b d^{3} x^{3} e^{2} \log \left (-\frac{d x + c + 1}{d x + c - 1}\right ) + 2 \, a d^{3} x^{3} e^{2} + 3 \, b c d^{2} x^{2} e^{2} \log \left (-\frac{d x + c + 1}{d x + c - 1}\right ) + 6 \, a c d^{2} x^{2} e^{2} + 3 \, b c^{2} d x e^{2} \log \left (-\frac{d x + c + 1}{d x + c - 1}\right ) + 6 \, a c^{2} d x e^{2} + b d^{2} x^{2} e^{2} + b c^{3} e^{2} \log \left (d x + c + 1\right ) - b c^{3} e^{2} \log \left (d x + c - 1\right ) + 2 \, b c d x e^{2} + b e^{2} \log \left (d x + c + 1\right ) + b e^{2} \log \left (d x + c - 1\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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