Optimal. Leaf size=143 \[ \frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b d^2 x}{2 c^2}+\frac{31 b d^2 \log (1-c x)}{60 c^3}+\frac{b d^2 \log (c x+1)}{60 c^3}+\frac{1}{20} b c d^2 x^4+\frac{4 b d^2 x^2}{15 c}+\frac{1}{6} b d^2 x^3 \]
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Rubi [A] time = 0.154411, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {43, 5936, 12, 1802, 633, 31} \[ \frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b d^2 x}{2 c^2}+\frac{31 b d^2 \log (1-c x)}{60 c^3}+\frac{b d^2 \log (c x+1)}{60 c^3}+\frac{1}{20} b c d^2 x^4+\frac{4 b d^2 x^2}{15 c}+\frac{1}{6} b d^2 x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 5936
Rule 12
Rule 1802
Rule 633
Rule 31
Rubi steps
\begin{align*} \int x^2 (d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac{d^2 x^3 \left (10+15 c x+6 c^2 x^2\right )}{30 \left (1-c^2 x^2\right )} \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{30} \left (b c d^2\right ) \int \frac{x^3 \left (10+15 c x+6 c^2 x^2\right )}{1-c^2 x^2} \, dx\\ &=\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{30} \left (b c d^2\right ) \int \left (-\frac{15}{c^3}-\frac{16 x}{c^2}-\frac{15 x^2}{c}-6 x^3+\frac{15+16 c x}{c^3 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac{b d^2 x}{2 c^2}+\frac{4 b d^2 x^2}{15 c}+\frac{1}{6} b d^2 x^3+\frac{1}{20} b c d^2 x^4+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{\left (b d^2\right ) \int \frac{15+16 c x}{1-c^2 x^2} \, dx}{30 c^2}\\ &=\frac{b d^2 x}{2 c^2}+\frac{4 b d^2 x^2}{15 c}+\frac{1}{6} b d^2 x^3+\frac{1}{20} b c d^2 x^4+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{\left (b d^2\right ) \int \frac{1}{-c-c^2 x} \, dx}{60 c}-\frac{\left (31 b d^2\right ) \int \frac{1}{c-c^2 x} \, dx}{60 c}\\ &=\frac{b d^2 x}{2 c^2}+\frac{4 b d^2 x^2}{15 c}+\frac{1}{6} b d^2 x^3+\frac{1}{20} b c d^2 x^4+\frac{1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{31 b d^2 \log (1-c x)}{60 c^3}+\frac{b d^2 \log (1+c x)}{60 c^3}\\ \end{align*}
Mathematica [A] time = 0.0936998, size = 115, normalized size = 0.8 \[ \frac{d^2 \left (12 a c^5 x^5+30 a c^4 x^4+20 a c^3 x^3+3 b c^4 x^4+10 b c^3 x^3+16 b c^2 x^2+2 b c^3 x^3 \left (6 c^2 x^2+15 c x+10\right ) \tanh ^{-1}(c x)+30 b c x+31 b \log (1-c x)+b \log (c x+1)\right )}{60 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 147, normalized size = 1. \begin{align*}{\frac{{c}^{2}{d}^{2}a{x}^{5}}{5}}+{\frac{c{d}^{2}a{x}^{4}}{2}}+{\frac{{d}^{2}a{x}^{3}}{3}}+{\frac{{c}^{2}{d}^{2}b{\it Artanh} \left ( cx \right ){x}^{5}}{5}}+{\frac{c{d}^{2}b{\it Artanh} \left ( cx \right ){x}^{4}}{2}}+{\frac{{d}^{2}b{\it Artanh} \left ( cx \right ){x}^{3}}{3}}+{\frac{bc{d}^{2}{x}^{4}}{20}}+{\frac{b{d}^{2}{x}^{3}}{6}}+{\frac{4\,b{d}^{2}{x}^{2}}{15\,c}}+{\frac{b{d}^{2}x}{2\,{c}^{2}}}+{\frac{31\,{d}^{2}b\ln \left ( cx-1 \right ) }{60\,{c}^{3}}}+{\frac{{d}^{2}b\ln \left ( cx+1 \right ) }{60\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975962, size = 248, normalized size = 1.73 \begin{align*} \frac{1}{5} \, a c^{2} d^{2} x^{5} + \frac{1}{2} \, a c d^{2} x^{4} + \frac{1}{20} \,{\left (4 \, x^{5} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c^{2} d^{2} + \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{12} \,{\left (6 \, x^{4} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \,{\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac{3 \, \log \left (c x + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b c d^{2} + \frac{1}{6} \,{\left (2 \, x^{3} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{x^{2}}{c^{2}} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b d^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05371, size = 332, normalized size = 2.32 \begin{align*} \frac{12 \, a c^{5} d^{2} x^{5} + 3 \,{\left (10 \, a + b\right )} c^{4} d^{2} x^{4} + 10 \,{\left (2 \, a + b\right )} c^{3} d^{2} x^{3} + 16 \, b c^{2} d^{2} x^{2} + 30 \, b c d^{2} x + b d^{2} \log \left (c x + 1\right ) + 31 \, b d^{2} \log \left (c x - 1\right ) +{\left (6 \, b c^{5} d^{2} x^{5} + 15 \, b c^{4} d^{2} x^{4} + 10 \, b c^{3} d^{2} x^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{60 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.03088, size = 177, normalized size = 1.24 \begin{align*} \begin{cases} \frac{a c^{2} d^{2} x^{5}}{5} + \frac{a c d^{2} x^{4}}{2} + \frac{a d^{2} x^{3}}{3} + \frac{b c^{2} d^{2} x^{5} \operatorname{atanh}{\left (c x \right )}}{5} + \frac{b c d^{2} x^{4} \operatorname{atanh}{\left (c x \right )}}{2} + \frac{b c d^{2} x^{4}}{20} + \frac{b d^{2} x^{3} \operatorname{atanh}{\left (c x \right )}}{3} + \frac{b d^{2} x^{3}}{6} + \frac{4 b d^{2} x^{2}}{15 c} + \frac{b d^{2} x}{2 c^{2}} + \frac{8 b d^{2} \log{\left (x - \frac{1}{c} \right )}}{15 c^{3}} + \frac{b d^{2} \operatorname{atanh}{\left (c x \right )}}{30 c^{3}} & \text{for}\: c \neq 0 \\\frac{a d^{2} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32498, size = 203, normalized size = 1.42 \begin{align*} \frac{1}{5} \, a c^{2} d^{2} x^{5} + \frac{1}{20} \,{\left (10 \, a c d^{2} + b c d^{2}\right )} x^{4} + \frac{4 \, b d^{2} x^{2}}{15 \, c} + \frac{1}{6} \,{\left (2 \, a d^{2} + b d^{2}\right )} x^{3} + \frac{b d^{2} x}{2 \, c^{2}} + \frac{1}{60} \,{\left (6 \, b c^{2} d^{2} x^{5} + 15 \, b c d^{2} x^{4} + 10 \, b d^{2} x^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) + \frac{b d^{2} \log \left (c x + 1\right )}{60 \, c^{3}} + \frac{31 \, b d^{2} \log \left (c x - 1\right )}{60 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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