Optimal. Leaf size=387 \[ -\frac {6 b^2 d e \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2}+\frac {a b^2 e^2 x}{c^2}-\frac {b^2 \left (3 c^2 d^2+e^2\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}-\frac {d \left (\frac {3 e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {b \left (3 c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {3 b^3 d e \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^2}+\frac {b^3 e^2 x \tanh ^{-1}(c x)}{c^2}+\frac {b^3 \left (3 c^2 d^2+e^2\right ) \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c^3}+\frac {b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3} \]
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Rubi [A] time = 0.80, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {5928, 5910, 5984, 5918, 2402, 2315, 5916, 5980, 260, 5948, 6048, 6058, 6610} \[ -\frac {b^2 \left (3 c^2 d^2+e^2\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}+\frac {b^3 \left (3 c^2 d^2+e^2\right ) \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^3}-\frac {3 b^3 d e \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^2}-\frac {6 b^2 d e \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2}+\frac {a b^2 e^2 x}{c^2}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {d \left (\frac {3 e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {b \left (3 c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3}+\frac {b^3 e^2 x \tanh ^{-1}(c x)}{c^2} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5928
Rule 5948
Rule 5980
Rule 5984
Rule 6048
Rule 6058
Rule 6610
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {(b c) \int \left (-\frac {3 d e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {e^3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}+\frac {\left (c^2 d^3+3 d e^2+e \left (3 c^2 d^2+e^2\right ) x\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 \left (1-c^2 x^2\right )}\right ) \, dx}{e}\\ &=\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {b \int \frac {\left (c^2 d^3+3 d e^2+e \left (3 c^2 d^2+e^2\right ) x\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{c e}+\frac {(3 b d e) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c}+\frac {\left (b e^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c}\\ &=\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {b \int \left (\frac {c^2 d^3 \left (1+\frac {3 e^2}{c^2 d^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}+\frac {e \left (3 c^2 d^2+e^2\right ) x \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}\right ) \, dx}{c e}-\left (6 b^2 d e\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (b^2 e^2\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {\left (6 b^2 d e\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c}+\frac {\left (b^2 e^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2}-\frac {\left (b^2 e^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^2}-\left (b d \left (\frac {c d^2}{e}+\frac {3 e}{c}\right )\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx-\frac {\left (b \left (3 c^2 d^2+e^2\right )\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{c}\\ &=\frac {a b^2 e^2 x}{c^2}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {6 b^2 d e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}+\frac {\left (6 b^3 d e\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c}+\frac {\left (b^3 e^2\right ) \int \tanh ^{-1}(c x) \, dx}{c^2}-\frac {\left (b \left (3 c^2 d^2+e^2\right )\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx}{c^2}\\ &=\frac {a b^2 e^2 x}{c^2}+\frac {b^3 e^2 x \tanh ^{-1}(c x)}{c^2}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {6 b^2 d e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {b \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}-\frac {\left (6 b^3 d e\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^2}-\frac {\left (b^3 e^2\right ) \int \frac {x}{1-c^2 x^2} \, dx}{c}+\frac {\left (2 b^2 \left (3 c^2 d^2+e^2\right )\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2}\\ &=\frac {a b^2 e^2 x}{c^2}+\frac {b^3 e^2 x \tanh ^{-1}(c x)}{c^2}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {6 b^2 d e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {b \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {3 b^3 d e \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^2}-\frac {b^2 \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3}+\frac {\left (b^3 \left (3 c^2 d^2+e^2\right )\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2}\\ &=\frac {a b^2 e^2 x}{c^2}+\frac {b^3 e^2 x \tanh ^{-1}(c x)}{c^2}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {6 b^2 d e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {b \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {3 b^3 d e \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^2}-\frac {b^2 \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \left (3 c^2 d^2+e^2\right ) \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 1.31, size = 591, normalized size = 1.53 \[ \frac {2 a^3 c^3 e^2 x^3+6 a^2 b c^3 x \tanh ^{-1}(c x) \left (3 d^2+3 d e x+e^2 x^2\right )+3 a^2 b \left (3 c^2 d^2+3 c d e+e^2\right ) \log (1-c x)+3 a^2 b \left (3 c^2 d^2-3 c d e+e^2\right ) \log (c x+1)+3 a^2 c^2 e x^2 (2 a c d+b e)+6 a^2 c^2 d x (a c d+3 b e)+18 a b^2 c^2 d^2 \left (\text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x) \left ((c x-1) \tanh ^{-1}(c x)-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )\right )+18 a b^2 c d e \left (\log \left (1-c^2 x^2\right )+\left (c^2 x^2-1\right ) \tanh ^{-1}(c x)^2+2 c x \tanh ^{-1}(c x)\right )+6 a b^2 e^2 \left (\left (c^3 x^3-1\right ) \tanh ^{-1}(c x)^2+\tanh ^{-1}(c x) \left (c^2 x^2-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-1\right )+\text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+c x\right )+6 b^3 c^2 d^2 \left (3 \tanh ^{-1}(c x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+\frac {3}{2} \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right )+\tanh ^{-1}(c x)^2 \left ((c x-1) \tanh ^{-1}(c x)-3 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )\right )-6 b^3 c d e \left (\tanh ^{-1}(c x) \left (\left (1-c^2 x^2\right ) \tanh ^{-1}(c x)^2+(3-3 c x) \tanh ^{-1}(c x)+6 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )-3 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )\right )+b^3 e^2 \left (2 c^3 x^3 \tanh ^{-1}(c x)^3+3 \log \left (1-c^2 x^2\right )+3 c^2 x^2 \tanh ^{-1}(c x)^2+6 \tanh ^{-1}(c x) \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+3 \text {Li}_3\left (-e^{-2 \tanh ^{-1}(c x)}\right )-2 \tanh ^{-1}(c x)^3-3 \tanh ^{-1}(c x)^2+6 c x \tanh ^{-1}(c x)-6 \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )}{6 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{3} e^{2} x^{2} + 2 \, a^{3} d e x + a^{3} d^{2} + {\left (b^{3} e^{2} x^{2} + 2 \, b^{3} d e x + b^{3} d^{2}\right )} \operatorname {artanh}\left (c x\right )^{3} + 3 \, {\left (a b^{2} e^{2} x^{2} + 2 \, a b^{2} d e x + a b^{2} d^{2}\right )} \operatorname {artanh}\left (c x\right )^{2} + 3 \, {\left (a^{2} b e^{2} x^{2} + 2 \, a^{2} b d e x + a^{2} b d^{2}\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.21, size = 4600, normalized size = 11.89 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a^{3} e^{2} x^{3} + a^{3} d e x^{2} + \frac {3}{2} \, {\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 )} a^{2} b d 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 )} a^{2} b e^{2} + a^{3} d^{2} x + \frac {3 \, {\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a^{2} b d^{2}}{2 \, c} - \frac {{\left (b^{3} c^{3} e^{2} x^{3} + 3 \, b^{3} c^{3} d e x^{2} + 3 \, b^{3} c^{3} d^{2} x - {\left (3 \, c^{2} d^{2} + 3 \, c d e + e^{2}\right )} b^{3}\right )} \log \left (-c x + 1\right )^{3} - 3 \, {\left (2 \, a b^{2} c^{3} e^{2} x^{3} + {\left (6 \, a b^{2} c^{3} d e + b^{3} c^{2} e^{2}\right )} x^{2} + 6 \, {\left (a b^{2} c^{3} d^{2} + b^{3} c^{2} d e\right )} x + {\left (b^{3} c^{3} e^{2} x^{3} + 3 \, b^{3} c^{3} d e x^{2} + 3 \, b^{3} c^{3} d^{2} x + {\left (3 \, c^{2} d^{2} - 3 \, c d e + e^{2}\right )} b^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )^{2}}{24 \, c^{3}} - \int -\frac {{\left (b^{3} c^{3} e^{2} x^{3} - b^{3} c^{2} d^{2} + {\left (2 \, c^{3} d e - c^{2} e^{2}\right )} b^{3} x^{2} + {\left (c^{3} d^{2} - 2 \, c^{2} d e\right )} b^{3} x\right )} \log \left (c x + 1\right )^{3} + 6 \, {\left (a b^{2} c^{3} e^{2} x^{3} - a b^{2} c^{2} d^{2} + {\left (2 \, c^{3} d e - c^{2} e^{2}\right )} a b^{2} x^{2} + {\left (c^{3} d^{2} - 2 \, c^{2} d e\right )} a b^{2} x\right )} \log \left (c x + 1\right )^{2} - {\left (4 \, a b^{2} c^{3} e^{2} x^{3} + 2 \, {\left (6 \, a b^{2} c^{3} d e + b^{3} c^{2} e^{2}\right )} x^{2} + 3 \, {\left (b^{3} c^{3} e^{2} x^{3} - b^{3} c^{2} d^{2} + {\left (2 \, c^{3} d e - c^{2} e^{2}\right )} b^{3} x^{2} + {\left (c^{3} d^{2} - 2 \, c^{2} d e\right )} b^{3} x\right )} \log \left (c x + 1\right )^{2} + 12 \, {\left (a b^{2} c^{3} d^{2} + b^{3} c^{2} d e\right )} x - 2 \, {\left (6 \, a b^{2} c^{2} d^{2} - {\left (3 \, c^{2} d^{2} - 3 \, c d e + e^{2}\right )} b^{3} - {\left (6 \, a b^{2} c^{3} e^{2} + b^{3} c^{3} e^{2}\right )} x^{3} - 3 \, {\left (b^{3} c^{3} d e + 2 \, {\left (2 \, c^{3} d e - c^{2} e^{2}\right )} a b^{2}\right )} x^{2} - 3 \, {\left (b^{3} c^{3} d^{2} + 2 \, {\left (c^{3} d^{2} - 2 \, c^{2} d e\right )} a b^{2}\right )} x\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \, {\left (c^{3} x - c^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,{\left (d+e\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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