Optimal. Leaf size=178 \[ \frac{1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{30} b c^2 d^3 x^5+\frac{11 b d^3 x}{12 c^2}+\frac{37 b d^3 \log (1-c x)}{40 c^3}+\frac{b d^3 \log (c x+1)}{120 c^3}+\frac{3}{20} b c d^3 x^4+\frac{7 b d^3 x^2}{15 c}+\frac{11}{36} b d^3 x^3 \]
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Rubi [A] time = 0.175664, antiderivative size = 178, 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} \[ \frac{1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{30} b c^2 d^3 x^5+\frac{11 b d^3 x}{12 c^2}+\frac{37 b d^3 \log (1-c x)}{40 c^3}+\frac{b d^3 \log (c x+1)}{120 c^3}+\frac{3}{20} b c d^3 x^4+\frac{7 b d^3 x^2}{15 c}+\frac{11}{36} b d^3 x^3 \]
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)^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac{d^3 x^3 \left (20+45 c x+36 c^2 x^2+10 c^3 x^3\right )}{60 \left (1-c^2 x^2\right )} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{60} \left (b c d^3\right ) \int \frac{x^3 \left (20+45 c x+36 c^2 x^2+10 c^3 x^3\right )}{1-c^2 x^2} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{60} \left (b c d^3\right ) \int \left (-\frac{55}{c^3}-\frac{56 x}{c^2}-\frac{55 x^2}{c}-36 x^3-10 c x^4+\frac{55+56 c x}{c^3 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac{11 b d^3 x}{12 c^2}+\frac{7 b d^3 x^2}{15 c}+\frac{11}{36} b d^3 x^3+\frac{3}{20} b c d^3 x^4+\frac{1}{30} b c^2 d^3 x^5+\frac{1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{\left (b d^3\right ) \int \frac{55+56 c x}{1-c^2 x^2} \, dx}{60 c^2}\\ &=\frac{11 b d^3 x}{12 c^2}+\frac{7 b d^3 x^2}{15 c}+\frac{11}{36} b d^3 x^3+\frac{3}{20} b c d^3 x^4+\frac{1}{30} b c^2 d^3 x^5+\frac{1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac{\left (b d^3\right ) \int \frac{1}{-c-c^2 x} \, dx}{120 c}-\frac{\left (37 b d^3\right ) \int \frac{1}{c-c^2 x} \, dx}{40 c}\\ &=\frac{11 b d^3 x}{12 c^2}+\frac{7 b d^3 x^2}{15 c}+\frac{11}{36} b d^3 x^3+\frac{3}{20} b c d^3 x^4+\frac{1}{30} b c^2 d^3 x^5+\frac{1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{37 b d^3 \log (1-c x)}{40 c^3}+\frac{b d^3 \log (1+c x)}{120 c^3}\\ \end{align*}
Mathematica [A] time = 0.125615, size = 142, normalized size = 0.8 \[ \frac{d^3 \left (60 a c^6 x^6+216 a c^5 x^5+270 a c^4 x^4+120 a c^3 x^3+12 b c^5 x^5+54 b c^4 x^4+110 b c^3 x^3+168 b c^2 x^2+6 b c^3 x^3 \left (10 c^3 x^3+36 c^2 x^2+45 c x+20\right ) \tanh ^{-1}(c x)+330 b c x+333 b \log (1-c x)+3 b \log (c x+1)\right )}{360 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 187, normalized size = 1.1 \begin{align*}{\frac{{c}^{3}{d}^{3}a{x}^{6}}{6}}+{\frac{3\,{c}^{2}{d}^{3}a{x}^{5}}{5}}+{\frac{3\,c{d}^{3}a{x}^{4}}{4}}+{\frac{{d}^{3}a{x}^{3}}{3}}+{\frac{{c}^{3}{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{6}}{6}}+{\frac{3\,{c}^{2}{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{5}}{5}}+{\frac{3\,c{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{4}}{4}}+{\frac{{d}^{3}b{\it Artanh} \left ( cx \right ){x}^{3}}{3}}+{\frac{b{c}^{2}{d}^{3}{x}^{5}}{30}}+{\frac{3\,bc{d}^{3}{x}^{4}}{20}}+{\frac{11\,b{d}^{3}{x}^{3}}{36}}+{\frac{7\,{d}^{3}b{x}^{2}}{15\,c}}+{\frac{11\,b{d}^{3}x}{12\,{c}^{2}}}+{\frac{37\,{d}^{3}b\ln \left ( cx-1 \right ) }{40\,{c}^{3}}}+{\frac{{d}^{3}b\ln \left ( cx+1 \right ) }{120\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965809, size = 358, normalized size = 2.01 \begin{align*} \frac{1}{6} \, a c^{3} d^{3} x^{6} + \frac{3}{5} \, a c^{2} d^{3} x^{5} + \frac{3}{4} \, a c d^{3} x^{4} + \frac{1}{180} \,{\left (30 \, x^{6} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \,{\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac{15 \, \log \left (c x + 1\right )}{c^{7}} + \frac{15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{3} d^{3} + \frac{3}{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^{3} + \frac{1}{3} \, a d^{3} x^{3} + \frac{1}{8} \,{\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^{3} + \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^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22022, size = 413, normalized size = 2.32 \begin{align*} \frac{60 \, a c^{6} d^{3} x^{6} + 12 \,{\left (18 \, a + b\right )} c^{5} d^{3} x^{5} + 54 \,{\left (5 \, a + b\right )} c^{4} d^{3} x^{4} + 10 \,{\left (12 \, a + 11 \, b\right )} c^{3} d^{3} x^{3} + 168 \, b c^{2} d^{3} x^{2} + 330 \, b c d^{3} x + 3 \, b d^{3} \log \left (c x + 1\right ) + 333 \, b d^{3} \log \left (c x - 1\right ) + 3 \,{\left (10 \, b c^{6} d^{3} x^{6} + 36 \, b c^{5} d^{3} x^{5} + 45 \, b c^{4} d^{3} x^{4} + 20 \, b c^{3} d^{3} x^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{360 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.32664, size = 235, normalized size = 1.32 \begin{align*} \begin{cases} \frac{a c^{3} d^{3} x^{6}}{6} + \frac{3 a c^{2} d^{3} x^{5}}{5} + \frac{3 a c d^{3} x^{4}}{4} + \frac{a d^{3} x^{3}}{3} + \frac{b c^{3} d^{3} x^{6} \operatorname{atanh}{\left (c x \right )}}{6} + \frac{3 b c^{2} d^{3} x^{5} \operatorname{atanh}{\left (c x \right )}}{5} + \frac{b c^{2} d^{3} x^{5}}{30} + \frac{3 b c d^{3} x^{4} \operatorname{atanh}{\left (c x \right )}}{4} + \frac{3 b c d^{3} x^{4}}{20} + \frac{b d^{3} x^{3} \operatorname{atanh}{\left (c x \right )}}{3} + \frac{11 b d^{3} x^{3}}{36} + \frac{7 b d^{3} x^{2}}{15 c} + \frac{11 b d^{3} x}{12 c^{2}} + \frac{14 b d^{3} \log{\left (x - \frac{1}{c} \right )}}{15 c^{3}} + \frac{b d^{3} \operatorname{atanh}{\left (c x \right )}}{60 c^{3}} & \text{for}\: c \neq 0 \\\frac{a d^{3} 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 = 1.25597, size = 251, normalized size = 1.41 \begin{align*} \frac{1}{6} \, a c^{3} d^{3} x^{6} + \frac{1}{30} \,{\left (18 \, a c^{2} d^{3} + b c^{2} d^{3}\right )} x^{5} + \frac{7 \, b d^{3} x^{2}}{15 \, c} + \frac{3}{20} \,{\left (5 \, a c d^{3} + b c d^{3}\right )} x^{4} + \frac{1}{36} \,{\left (12 \, a d^{3} + 11 \, b d^{3}\right )} x^{3} + \frac{11 \, b d^{3} x}{12 \, c^{2}} + \frac{b d^{3} \log \left (c x + 1\right )}{120 \, c^{3}} + \frac{37 \, b d^{3} \log \left (c x - 1\right )}{40 \, c^{3}} + \frac{1}{120} \,{\left (10 \, b c^{3} d^{3} x^{6} + 36 \, b c^{2} d^{3} x^{5} + 45 \, b c d^{3} x^{4} + 20 \, b d^{3} x^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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