Optimal. Leaf size=54 \[ \frac {(a+b x)^2}{6 b}+\frac {\log \left (1-(a+b x)^2\right )}{6 b}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b} \]
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Rubi [A] time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6108, 5917, 266, 43} \[ \frac {(a+b x)^2}{6 b}+\frac {\log \left (1-(a+b x)^2\right )}{6 b}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 5917
Rule 6108
Rubi steps
\begin {align*} \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^2 \coth ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac {\operatorname {Subst}\left (\int \frac {x^3}{1-x^2} \, dx,x,a+b x\right )}{3 b}\\ &=\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac {\operatorname {Subst}\left (\int \frac {x}{1-x} \, dx,x,(a+b x)^2\right )}{6 b}\\ &=\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac {\operatorname {Subst}\left (\int \left (-1+\frac {1}{1-x}\right ) \, dx,x,(a+b x)^2\right )}{6 b}\\ &=\frac {(a+b x)^2}{6 b}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}+\frac {\log \left (1-(a+b x)^2\right )}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 0.78 \[ \frac {(a+b x)^2+\log \left (1-(a+b x)^2\right )+2 (a+b x)^3 \coth ^{-1}(a+b x)}{6 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 86, normalized size = 1.59 \[ \frac {b^{2} x^{2} + 2 \, a b x + {\left (a^{3} + 1\right )} \log \left (b x + a + 1\right ) - {\left (a^{3} - 1\right )} \log \left (b x + a - 1\right ) + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x\right )} \log \left (\frac {b x + a + 1}{b x + a - 1}\right )}{6 \, b} \]
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 \[ \int {\left (b x + a\right )}^{2} \operatorname {arcoth}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 95, normalized size = 1.76 \[ \frac {b^{2} \mathrm {arccoth}\left (b x +a \right ) x^{3}}{3}+b \,\mathrm {arccoth}\left (b x +a \right ) x^{2} a +\mathrm {arccoth}\left (b x +a \right ) x \,a^{2}+\frac {\mathrm {arccoth}\left (b x +a \right ) a^{3}}{3 b}+\frac {b \,x^{2}}{6}+\frac {a x}{3}+\frac {a^{2}}{6 b}+\frac {\ln \left (b x +a -1\right )}{6 b}+\frac {\ln \left (b x +a +1\right )}{6 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 81, normalized size = 1.50 \[ \frac {1}{6} \, b {\left (\frac {b x^{2} + 2 \, a x}{b} + \frac {{\left (a^{3} + 1\right )} \log \left (b x + a + 1\right )}{b^{2}} - \frac {{\left (a^{3} - 1\right )} \log \left (b x + a - 1\right )}{b^{2}}\right )} + \frac {1}{3} \, {\left (b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x\right )} \operatorname {arcoth}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 114, normalized size = 2.11 \[ \frac {a\,x}{3}+\ln \left (\frac {1}{a+b\,x}+1\right )\,\left (\frac {a^2\,x}{2}+\frac {a\,b\,x^2}{2}+\frac {b^2\,x^3}{6}\right )+\frac {b\,x^2}{6}-\ln \left (1-\frac {1}{a+b\,x}\right )\,\left (\frac {a^2\,x}{2}+\frac {a\,b\,x^2}{2}+\frac {b^2\,x^3}{6}\right )-\frac {\ln \left (a+b\,x-1\right )\,\left (a^3-1\right )}{6\,b}+\frac {\ln \left (a+b\,x+1\right )\,\left (a^3+1\right )}{6\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 97, normalized size = 1.80 \[ \begin {cases} \frac {a^{3} \operatorname {acoth}{\left (a + b x \right )}}{3 b} + a^{2} x \operatorname {acoth}{\left (a + b x \right )} + a b x^{2} \operatorname {acoth}{\left (a + b x \right )} + \frac {a x}{3} + \frac {b^{2} x^{3} \operatorname {acoth}{\left (a + b x \right )}}{3} + \frac {b x^{2}}{6} + \frac {\log {\left (\frac {a}{b} + x + \frac {1}{b} \right )}}{3 b} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\a^{2} x \operatorname {acoth}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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