Optimal. Leaf size=171 \[ \frac{d^4 (c x+1)^7 \left (a+b \tanh ^{-1}(c x)\right )}{7 c^3}-\frac{d^4 (c x+1)^6 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac{d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}+\frac{1}{42} b c^3 d^4 x^6+\frac{2}{15} b c^2 d^4 x^5+\frac{5 b d^4 x}{3 c^2}+\frac{176 b d^4 \log (1-c x)}{105 c^3}+\frac{47}{140} b c d^4 x^4+\frac{88 b d^4 x^2}{105 c}+\frac{5}{9} b d^4 x^3 \]
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Rubi [A] time = 0.180548, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {43, 5936, 12, 893} \[ \frac{d^4 (c x+1)^7 \left (a+b \tanh ^{-1}(c x)\right )}{7 c^3}-\frac{d^4 (c x+1)^6 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac{d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}+\frac{1}{42} b c^3 d^4 x^6+\frac{2}{15} b c^2 d^4 x^5+\frac{5 b d^4 x}{3 c^2}+\frac{176 b d^4 \log (1-c x)}{105 c^3}+\frac{47}{140} b c d^4 x^4+\frac{88 b d^4 x^2}{105 c}+\frac{5}{9} b d^4 x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 5936
Rule 12
Rule 893
Rubi steps
\begin{align*} \int x^2 (d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}-\frac{d^4 (1+c x)^6 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac{d^4 (1+c x)^7 \left (a+b \tanh ^{-1}(c x)\right )}{7 c^3}-(b c) \int \frac{(d+c d x)^4 \left (1-5 c x+15 c^2 x^2\right )}{105 c^3 (1-c x)} \, dx\\ &=\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}-\frac{d^4 (1+c x)^6 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac{d^4 (1+c x)^7 \left (a+b \tanh ^{-1}(c x)\right )}{7 c^3}-\frac{b \int \frac{(d+c d x)^4 \left (1-5 c x+15 c^2 x^2\right )}{1-c x} \, dx}{105 c^2}\\ &=\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}-\frac{d^4 (1+c x)^6 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac{d^4 (1+c x)^7 \left (a+b \tanh ^{-1}(c x)\right )}{7 c^3}-\frac{b \int \left (-175 d^4-176 c d^4 x-175 c^2 d^4 x^2-141 c^3 d^4 x^3-70 c^4 d^4 x^4-15 c^5 d^4 x^5-\frac{176 d^4}{-1+c x}\right ) \, dx}{105 c^2}\\ &=\frac{5 b d^4 x}{3 c^2}+\frac{88 b d^4 x^2}{105 c}+\frac{5}{9} b d^4 x^3+\frac{47}{140} b c d^4 x^4+\frac{2}{15} b c^2 d^4 x^5+\frac{1}{42} b c^3 d^4 x^6+\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}-\frac{d^4 (1+c x)^6 \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac{d^4 (1+c x)^7 \left (a+b \tanh ^{-1}(c x)\right )}{7 c^3}+\frac{176 b d^4 \log (1-c x)}{105 c^3}\\ \end{align*}
Mathematica [A] time = 0.146447, size = 168, normalized size = 0.98 \[ \frac{d^4 \left (180 a c^7 x^7+840 a c^6 x^6+1512 a c^5 x^5+1260 a c^4 x^4+420 a c^3 x^3+30 b c^6 x^6+168 b c^5 x^5+423 b c^4 x^4+700 b c^3 x^3+1056 b c^2 x^2+12 b c^3 x^3 \left (15 c^4 x^4+70 c^3 x^3+126 c^2 x^2+105 c x+35\right ) \tanh ^{-1}(c x)+2100 b c x+2106 b \log (1-c x)+6 b \log (c x+1)\right )}{1260 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 225, normalized size = 1.3 \begin{align*}{\frac{{c}^{4}{d}^{4}a{x}^{7}}{7}}+{\frac{2\,{c}^{3}{d}^{4}a{x}^{6}}{3}}+{\frac{6\,{c}^{2}{d}^{4}a{x}^{5}}{5}}+c{d}^{4}a{x}^{4}+{\frac{{d}^{4}a{x}^{3}}{3}}+{\frac{{c}^{4}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{7}}{7}}+{\frac{2\,{c}^{3}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{6}}{3}}+{\frac{6\,{c}^{2}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{5}}{5}}+c{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{4}+{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{3}}{3}}+{\frac{b{c}^{3}{d}^{4}{x}^{6}}{42}}+{\frac{2\,b{c}^{2}{d}^{4}{x}^{5}}{15}}+{\frac{47\,bc{d}^{4}{x}^{4}}{140}}+{\frac{5\,b{d}^{4}{x}^{3}}{9}}+{\frac{88\,{d}^{4}b{x}^{2}}{105\,c}}+{\frac{5\,b{d}^{4}x}{3\,{c}^{2}}}+{\frac{117\,{d}^{4}b\ln \left ( cx-1 \right ) }{70\,{c}^{3}}}+{\frac{{d}^{4}b\ln \left ( cx+1 \right ) }{210\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.97381, size = 458, normalized size = 2.68 \begin{align*} \frac{1}{7} \, a c^{4} d^{4} x^{7} + \frac{2}{3} \, a c^{3} d^{4} x^{6} + \frac{6}{5} \, a c^{2} d^{4} x^{5} + \frac{1}{84} \,{\left (12 \, x^{7} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac{6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b c^{4} d^{4} + a c d^{4} x^{4} + \frac{1}{45} \,{\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^{4} + \frac{3}{10} \,{\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^{4} + \frac{1}{3} \, a d^{4} x^{3} + \frac{1}{6} \,{\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^{4} + \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^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20207, size = 490, normalized size = 2.87 \begin{align*} \frac{180 \, a c^{7} d^{4} x^{7} + 30 \,{\left (28 \, a + b\right )} c^{6} d^{4} x^{6} + 168 \,{\left (9 \, a + b\right )} c^{5} d^{4} x^{5} + 9 \,{\left (140 \, a + 47 \, b\right )} c^{4} d^{4} x^{4} + 140 \,{\left (3 \, a + 5 \, b\right )} c^{3} d^{4} x^{3} + 1056 \, b c^{2} d^{4} x^{2} + 2100 \, b c d^{4} x + 6 \, b d^{4} \log \left (c x + 1\right ) + 2106 \, b d^{4} \log \left (c x - 1\right ) + 6 \,{\left (15 \, b c^{7} d^{4} x^{7} + 70 \, b c^{6} d^{4} x^{6} + 126 \, b c^{5} d^{4} x^{5} + 105 \, b c^{4} d^{4} x^{4} + 35 \, b c^{3} d^{4} x^{3}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{1260 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.84074, size = 279, normalized size = 1.63 \begin{align*} \begin{cases} \frac{a c^{4} d^{4} x^{7}}{7} + \frac{2 a c^{3} d^{4} x^{6}}{3} + \frac{6 a c^{2} d^{4} x^{5}}{5} + a c d^{4} x^{4} + \frac{a d^{4} x^{3}}{3} + \frac{b c^{4} d^{4} x^{7} \operatorname{atanh}{\left (c x \right )}}{7} + \frac{2 b c^{3} d^{4} x^{6} \operatorname{atanh}{\left (c x \right )}}{3} + \frac{b c^{3} d^{4} x^{6}}{42} + \frac{6 b c^{2} d^{4} x^{5} \operatorname{atanh}{\left (c x \right )}}{5} + \frac{2 b c^{2} d^{4} x^{5}}{15} + b c d^{4} x^{4} \operatorname{atanh}{\left (c x \right )} + \frac{47 b c d^{4} x^{4}}{140} + \frac{b d^{4} x^{3} \operatorname{atanh}{\left (c x \right )}}{3} + \frac{5 b d^{4} x^{3}}{9} + \frac{88 b d^{4} x^{2}}{105 c} + \frac{5 b d^{4} x}{3 c^{2}} + \frac{176 b d^{4} \log{\left (x - \frac{1}{c} \right )}}{105 c^{3}} + \frac{b d^{4} \operatorname{atanh}{\left (c x \right )}}{105 c^{3}} & \text{for}\: c \neq 0 \\\frac{a d^{4} 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 = 1.26854, size = 300, normalized size = 1.75 \begin{align*} \frac{1}{7} \, a c^{4} d^{4} x^{7} + \frac{1}{42} \,{\left (28 \, a c^{3} d^{4} + b c^{3} d^{4}\right )} x^{6} + \frac{88 \, b d^{4} x^{2}}{105 \, c} + \frac{2}{15} \,{\left (9 \, a c^{2} d^{4} + b c^{2} d^{4}\right )} x^{5} + \frac{1}{140} \,{\left (140 \, a c d^{4} + 47 \, b c d^{4}\right )} x^{4} + \frac{5 \, b d^{4} x}{3 \, c^{2}} + \frac{1}{9} \,{\left (3 \, a d^{4} + 5 \, b d^{4}\right )} x^{3} + \frac{b d^{4} \log \left (c x + 1\right )}{210 \, c^{3}} + \frac{117 \, b d^{4} \log \left (c x - 1\right )}{70 \, c^{3}} + \frac{1}{210} \,{\left (15 \, b c^{4} d^{4} x^{7} + 70 \, b c^{3} d^{4} x^{6} + 126 \, b c^{2} d^{4} x^{5} + 105 \, b c d^{4} x^{4} + 35 \, b d^{4} 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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