Optimal. Leaf size=108 \[ \frac{1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b d x^2}{10 c^2}+\frac{b d x}{4 c^3}+\frac{9 b d \log (1-c x)}{40 c^4}-\frac{b d \log (c x+1)}{40 c^4}+\frac{b d x^3}{12 c}+\frac{1}{20} b d x^4 \]
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Rubi [A] time = 0.096842, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {43, 5936, 12, 801, 633, 31} \[ \frac{1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{b d x^2}{10 c^2}+\frac{b d x}{4 c^3}+\frac{9 b d \log (1-c x)}{40 c^4}-\frac{b d \log (c x+1)}{40 c^4}+\frac{b d x^3}{12 c}+\frac{1}{20} b d x^4 \]
Antiderivative was successfully verified.
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Rule 43
Rule 5936
Rule 12
Rule 801
Rule 633
Rule 31
Rubi steps
\begin{align*} \int x^3 (d+c d x) \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac{d x^4 (5+4 c x)}{20 \left (1-c^2 x^2\right )} \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{20} (b c d) \int \frac{x^4 (5+4 c x)}{1-c^2 x^2} \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{20} (b c d) \int \left (-\frac{5}{c^4}-\frac{4 x}{c^3}-\frac{5 x^2}{c^2}-\frac{4 x^3}{c}+\frac{5+4 c x}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac{b d x}{4 c^3}+\frac{b d x^2}{10 c^2}+\frac{b d x^3}{12 c}+\frac{1}{20} b d x^4+\frac{1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{(b d) \int \frac{5+4 c x}{1-c^2 x^2} \, dx}{20 c^3}\\ &=\frac{b d x}{4 c^3}+\frac{b d x^2}{10 c^2}+\frac{b d x^3}{12 c}+\frac{1}{20} b d x^4+\frac{1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{(b d) \int \frac{1}{-c-c^2 x} \, dx}{40 c^2}-\frac{(9 b d) \int \frac{1}{c-c^2 x} \, dx}{40 c^2}\\ &=\frac{b d x}{4 c^3}+\frac{b d x^2}{10 c^2}+\frac{b d x^3}{12 c}+\frac{1}{20} b d x^4+\frac{1}{4} d x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{5} c d x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{9 b d \log (1-c x)}{40 c^4}-\frac{b d \log (1+c x)}{40 c^4}\\ \end{align*}
Mathematica [A] time = 0.0737976, size = 97, normalized size = 0.9 \[ \frac{d \left (24 a c^5 x^5+30 a c^4 x^4+6 b c^4 x^4+10 b c^3 x^3+12 b c^2 x^2+6 b c^4 x^4 (4 c x+5) \tanh ^{-1}(c x)+30 b c x+27 b \log (1-c x)-3 b \log (c x+1)\right )}{120 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 101, normalized size = 0.9 \begin{align*}{\frac{cda{x}^{5}}{5}}+{\frac{da{x}^{4}}{4}}+{\frac{cdb{\it Artanh} \left ( cx \right ){x}^{5}}{5}}+{\frac{db{\it Artanh} \left ( cx \right ){x}^{4}}{4}}+{\frac{bd{x}^{4}}{20}}+{\frac{bd{x}^{3}}{12\,c}}+{\frac{bd{x}^{2}}{10\,{c}^{2}}}+{\frac{bdx}{4\,{c}^{3}}}+{\frac{9\,db\ln \left ( cx-1 \right ) }{40\,{c}^{4}}}-{\frac{db\ln \left ( cx+1 \right ) }{40\,{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957459, size = 163, normalized size = 1.51 \begin{align*} \frac{1}{5} \, a c d x^{5} + \frac{1}{4} \, a d 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 d + \frac{1}{24} \,{\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 d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04982, size = 275, normalized size = 2.55 \begin{align*} \frac{24 \, a c^{5} d x^{5} + 6 \,{\left (5 \, a + b\right )} c^{4} d x^{4} + 10 \, b c^{3} d x^{3} + 12 \, b c^{2} d x^{2} + 30 \, b c d x - 3 \, b d \log \left (c x + 1\right ) + 27 \, b d \log \left (c x - 1\right ) + 3 \,{\left (4 \, b c^{5} d x^{5} + 5 \, b c^{4} d x^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{120 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.05266, size = 124, normalized size = 1.15 \begin{align*} \begin{cases} \frac{a c d x^{5}}{5} + \frac{a d x^{4}}{4} + \frac{b c d x^{5} \operatorname{atanh}{\left (c x \right )}}{5} + \frac{b d x^{4} \operatorname{atanh}{\left (c x \right )}}{4} + \frac{b d x^{4}}{20} + \frac{b d x^{3}}{12 c} + \frac{b d x^{2}}{10 c^{2}} + \frac{b d x}{4 c^{3}} + \frac{b d \log{\left (x - \frac{1}{c} \right )}}{5 c^{4}} - \frac{b d \operatorname{atanh}{\left (c x \right )}}{20 c^{4}} & \text{for}\: c \neq 0 \\\frac{a d x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24833, size = 147, normalized size = 1.36 \begin{align*} \frac{1}{5} \, a c d x^{5} + \frac{1}{20} \,{\left (5 \, a d + b d\right )} x^{4} + \frac{b d x^{3}}{12 \, c} + \frac{b d x^{2}}{10 \, c^{2}} + \frac{1}{40} \,{\left (4 \, b c d x^{5} + 5 \, b d x^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) + \frac{b d x}{4 \, c^{3}} - \frac{b d \log \left (c x + 1\right )}{40 \, c^{4}} + \frac{9 \, b d \log \left (c x - 1\right )}{40 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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