Optimal. Leaf size=125 \[ \frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 e}-\frac {b (c d-e)^4 \log (c x+1)}{8 c^4 e}+\frac {b (c d+e)^4 \log (1-c x)}{8 c^4 e}+\frac {b e x \left (6 c^2 d^2+e^2\right )}{4 c^3}+\frac {b d e^2 x^2}{2 c}+\frac {b e^3 x^3}{12 c} \]
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Rubi [A] time = 0.14, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5926, 702, 633, 31} \[ \frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 e}+\frac {b e x \left (6 c^2 d^2+e^2\right )}{4 c^3}-\frac {b (c d-e)^4 \log (c x+1)}{8 c^4 e}+\frac {b (c d+e)^4 \log (1-c x)}{8 c^4 e}+\frac {b d e^2 x^2}{2 c}+\frac {b e^3 x^3}{12 c} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 702
Rule 5926
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 e}-\frac {(b c) \int \frac {(d+e x)^4}{1-c^2 x^2} \, dx}{4 e}\\ &=\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 e}-\frac {(b c) \int \left (-\frac {e^2 \left (6 c^2 d^2+e^2\right )}{c^4}-\frac {4 d e^3 x}{c^2}-\frac {e^4 x^2}{c^2}+\frac {c^4 d^4+6 c^2 d^2 e^2+e^4+4 c^2 d e \left (c^2 d^2+e^2\right ) x}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx}{4 e}\\ &=\frac {b e \left (6 c^2 d^2+e^2\right ) x}{4 c^3}+\frac {b d e^2 x^2}{2 c}+\frac {b e^3 x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 e}-\frac {b \int \frac {c^4 d^4+6 c^2 d^2 e^2+e^4+4 c^2 d e \left (c^2 d^2+e^2\right ) x}{1-c^2 x^2} \, dx}{4 c^3 e}\\ &=\frac {b e \left (6 c^2 d^2+e^2\right ) x}{4 c^3}+\frac {b d e^2 x^2}{2 c}+\frac {b e^3 x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 e}+\frac {\left (b (c d-e)^4\right ) \int \frac {1}{-c-c^2 x} \, dx}{8 c^2 e}-\frac {\left (b (c d+e)^4\right ) \int \frac {1}{c-c^2 x} \, dx}{8 c^2 e}\\ &=\frac {b e \left (6 c^2 d^2+e^2\right ) x}{4 c^3}+\frac {b d e^2 x^2}{2 c}+\frac {b e^3 x^3}{12 c}+\frac {(d+e x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{4 e}+\frac {b (c d+e)^4 \log (1-c x)}{8 c^4 e}-\frac {b (c d-e)^4 \log (1+c x)}{8 c^4 e}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 205, normalized size = 1.64 \[ \frac {2 c^3 e^2 x^3 (12 a c d+b e)+12 c^3 d e x^2 (3 a c d+b e)+6 c x \left (4 a c^3 d^3+b e \left (6 c^2 d^2+e^2\right )\right )+6 a c^4 e^3 x^4+6 b c^4 x \tanh ^{-1}(c x) \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+3 b \left (4 c^3 d^3+6 c^2 d^2 e+4 c d e^2+e^3\right ) \log (1-c x)+3 b \left (4 c^3 d^3-6 c^2 d^2 e+4 c d e^2-e^3\right ) \log (c x+1)}{24 c^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 244, normalized size = 1.95 \[ \frac {6 \, a c^{4} e^{3} x^{4} + 2 \, {\left (12 \, a c^{4} d e^{2} + b c^{3} e^{3}\right )} x^{3} + 12 \, {\left (3 \, a c^{4} d^{2} e + b c^{3} d e^{2}\right )} x^{2} + 6 \, {\left (4 \, a c^{4} d^{3} + 6 \, b c^{3} d^{2} e + b c e^{3}\right )} x + 3 \, {\left (4 \, b c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 4 \, b c d e^{2} - b e^{3}\right )} \log \left (c x + 1\right ) + 3 \, {\left (4 \, b c^{3} d^{3} + 6 \, b c^{2} d^{2} e + 4 \, b c d e^{2} + b e^{3}\right )} \log \left (c x - 1\right ) + 3 \, {\left (b c^{4} e^{3} x^{4} + 4 \, b c^{4} d e^{2} x^{3} + 6 \, b c^{4} d^{2} e x^{2} + 4 \, b c^{4} d^{3} x\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{24 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 1375, normalized size = 11.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 308, normalized size = 2.46 \[ \frac {a \,x^{4} e^{3}}{4}+a d \,e^{2} x^{3}+\frac {3 a \,d^{2} e \,x^{2}}{2}+a x \,d^{3}+\frac {a \,d^{4}}{4 e}+\frac {b \,e^{3} \arctanh \left (c x \right ) x^{4}}{4}+b \,e^{2} \arctanh \left (c x \right ) x^{3} d +\frac {3 b e \arctanh \left (c x \right ) x^{2} d^{2}}{2}+b \arctanh \left (c x \right ) x \,d^{3}+\frac {b \arctanh \left (c x \right ) d^{4}}{4 e}+\frac {b \,e^{3} x^{3}}{12 c}+\frac {b d \,e^{2} x^{2}}{2 c}+\frac {3 b e x \,d^{2}}{2 c}+\frac {b x \,e^{3}}{4 c^{3}}+\frac {b \ln \left (c x -1\right ) d^{4}}{8 e}+\frac {b \ln \left (c x -1\right ) d^{3}}{2 c}+\frac {3 b e \ln \left (c x -1\right ) d^{2}}{4 c^{2}}+\frac {b \,e^{2} \ln \left (c x -1\right ) d}{2 c^{3}}+\frac {b \,e^{3} \ln \left (c x -1\right )}{8 c^{4}}-\frac {b \ln \left (c x +1\right ) d^{4}}{8 e}+\frac {b \ln \left (c x +1\right ) d^{3}}{2 c}-\frac {3 b e \ln \left (c x +1\right ) d^{2}}{4 c^{2}}+\frac {b \,e^{2} \ln \left (c x +1\right ) d}{2 c^{3}}-\frac {b \,e^{3} \ln \left (c x +1\right )}{8 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 209, normalized size = 1.67 \[ \frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b d^{2} e + \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b d e^{2} + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b e^{3} + a d^{3} x + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{3}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 197, normalized size = 1.58 \[ \frac {a\,e^3\,x^4}{4}+a\,d^3\,x+\frac {b\,d^3\,\ln \left (c^2\,x^2-1\right )}{2\,c}+\frac {b\,e^3\,x^3}{12\,c}+b\,d^3\,x\,\mathrm {atanh}\left (c\,x\right )+\frac {3\,a\,d^2\,e\,x^2}{2}+a\,d\,e^2\,x^3+\frac {b\,e^3\,x}{4\,c^3}-\frac {b\,e^3\,\mathrm {atanh}\left (c\,x\right )}{4\,c^4}+\frac {b\,e^3\,x^4\,\mathrm {atanh}\left (c\,x\right )}{4}+\frac {3\,b\,d^2\,e\,x}{2\,c}-\frac {3\,b\,d^2\,e\,\mathrm {atanh}\left (c\,x\right )}{2\,c^2}+\frac {3\,b\,d^2\,e\,x^2\,\mathrm {atanh}\left (c\,x\right )}{2}+b\,d\,e^2\,x^3\,\mathrm {atanh}\left (c\,x\right )+\frac {b\,d\,e^2\,\ln \left (c^2\,x^2-1\right )}{2\,c^3}+\frac {b\,d\,e^2\,x^2}{2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.98, size = 279, normalized size = 2.23 \[ \begin {cases} a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac {a e^{3} x^{4}}{4} + b d^{3} x \operatorname {atanh}{\left (c x \right )} + \frac {3 b d^{2} e x^{2} \operatorname {atanh}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname {atanh}{\left (c x \right )} + \frac {b e^{3} x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {b d^{3} \log {\left (x - \frac {1}{c} \right )}}{c} + \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{c} + \frac {3 b d^{2} e x}{2 c} + \frac {b d e^{2} x^{2}}{2 c} + \frac {b e^{3} x^{3}}{12 c} - \frac {3 b d^{2} e \operatorname {atanh}{\left (c x \right )}}{2 c^{2}} + \frac {b d e^{2} \log {\left (x - \frac {1}{c} \right )}}{c^{3}} + \frac {b d e^{2} \operatorname {atanh}{\left (c x \right )}}{c^{3}} + \frac {b e^{3} x}{4 c^{3}} - \frac {b e^{3} \operatorname {atanh}{\left (c x \right )}}{4 c^{4}} & \text {for}\: c \neq 0 \\a \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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