Optimal. Leaf size=84 \[ \frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac{b (c d-e)^2 \log (c x+1)}{4 c^2 e}+\frac{b (c d+e)^2 \log (1-c x)}{4 c^2 e}+\frac{b e x}{2 c} \]
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Rubi [A] time = 0.0762702, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5926, 702, 633, 31} \[ \frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac{b (c d-e)^2 \log (c x+1)}{4 c^2 e}+\frac{b (c d+e)^2 \log (1-c x)}{4 c^2 e}+\frac{b e x}{2 c} \]
Antiderivative was successfully verified.
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Rule 5926
Rule 702
Rule 633
Rule 31
Rubi steps
\begin{align*} \int (d+e x) \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac{(b c) \int \frac{(d+e x)^2}{1-c^2 x^2} \, dx}{2 e}\\ &=\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac{(b c) \int \left (-\frac{e^2}{c^2}+\frac{c^2 d^2+e^2+2 c^2 d e x}{c^2 \left (1-c^2 x^2\right )}\right ) \, dx}{2 e}\\ &=\frac{b e x}{2 c}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}-\frac{b \int \frac{c^2 d^2+e^2+2 c^2 d e x}{1-c^2 x^2} \, dx}{2 c e}\\ &=\frac{b e x}{2 c}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}+\frac{\left (b (c d-e)^2\right ) \int \frac{1}{-c-c^2 x} \, dx}{4 e}-\frac{\left (b (c d+e)^2\right ) \int \frac{1}{c-c^2 x} \, dx}{4 e}\\ &=\frac{b e x}{2 c}+\frac{(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 e}+\frac{b (c d+e)^2 \log (1-c x)}{4 c^2 e}-\frac{b (c d-e)^2 \log (1+c x)}{4 c^2 e}\\ \end{align*}
Mathematica [A] time = 0.0096586, size = 96, normalized size = 1.14 \[ a d x+\frac{1}{2} a e x^2+\frac{b d \log \left (1-c^2 x^2\right )}{2 c}+\frac{b e \log (1-c x)}{4 c^2}-\frac{b e \log (c x+1)}{4 c^2}+b d x \tanh ^{-1}(c x)+\frac{1}{2} b e x^2 \tanh ^{-1}(c x)+\frac{b e x}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 92, normalized size = 1.1 \begin{align*}{\frac{a{x}^{2}e}{2}}+adx+{\frac{b{\it Artanh} \left ( cx \right ){x}^{2}e}{2}}+b{\it Artanh} \left ( cx \right ) xd+{\frac{bex}{2\,c}}+{\frac{b\ln \left ( cx-1 \right ) d}{2\,c}}+{\frac{b\ln \left ( cx-1 \right ) e}{4\,{c}^{2}}}+{\frac{b\ln \left ( cx+1 \right ) d}{2\,c}}-{\frac{b\ln \left ( cx+1 \right ) e}{4\,{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972936, size = 112, normalized size = 1.33 \begin{align*} \frac{1}{2} \, a e x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \, x}{c^{2}} - \frac{\log \left (c x + 1\right )}{c^{3}} + \frac{\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b e + a d x + \frac{{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64868, size = 228, normalized size = 2.71 \begin{align*} \frac{2 \, a c^{2} e x^{2} + 2 \,{\left (2 \, a c^{2} d + b c e\right )} x +{\left (2 \, b c d - b e\right )} \log \left (c x + 1\right ) +{\left (2 \, b c d + b e\right )} \log \left (c x - 1\right ) +{\left (b c^{2} e x^{2} + 2 \, b c^{2} d x\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.18385, size = 92, normalized size = 1.1 \begin{align*} \begin{cases} a d x + \frac{a e x^{2}}{2} + b d x \operatorname{atanh}{\left (c x \right )} + \frac{b e x^{2} \operatorname{atanh}{\left (c x \right )}}{2} + \frac{b d \log{\left (x - \frac{1}{c} \right )}}{c} + \frac{b d \operatorname{atanh}{\left (c x \right )}}{c} + \frac{b e x}{2 c} - \frac{b e \operatorname{atanh}{\left (c x \right )}}{2 c^{2}} & \text{for}\: c \neq 0 \\a \left (d x + \frac{e x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16253, size = 157, normalized size = 1.87 \begin{align*} \frac{b c^{2} x^{2} e \log \left (-\frac{c x + 1}{c x - 1}\right ) + 2 \, a c^{2} x^{2} e + 2 \, b c^{2} d x \log \left (-\frac{c x + 1}{c x - 1}\right ) + 4 \, a c^{2} d x + 2 \, b c x e + 2 \, b c d \log \left (c^{2} x^{2} - 1\right ) - b e \log \left (c x + 1\right ) + b e \log \left (c x - 1\right )}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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