Optimal. Leaf size=52 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{4 a^3 b}-\frac {1}{4 a^2 b (a+b x)}-\frac {1}{4 a b (a+b x)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {627, 44, 208} \begin {gather*} -\frac {1}{4 a^2 b (a+b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{4 a^3 b}-\frac {1}{4 a b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 208
Rule 627
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^2 \left (a^2-b^2 x^2\right )} \, dx &=\int \frac {1}{(a-b x) (a+b x)^3} \, dx\\ &=\int \left (\frac {1}{2 a (a+b x)^3}+\frac {1}{4 a^2 (a+b x)^2}+\frac {1}{4 a^2 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=-\frac {1}{4 a b (a+b x)^2}-\frac {1}{4 a^2 b (a+b x)}+\frac {\int \frac {1}{a^2-b^2 x^2} \, dx}{4 a^2}\\ &=-\frac {1}{4 a b (a+b x)^2}-\frac {1}{4 a^2 b (a+b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{4 a^3 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 1.12 \begin {gather*} \frac {-2 a (2 a+b x)+(a+b x)^2 (-\log (a-b x))+(a+b x)^2 \log (a+b x)}{8 a^3 b (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a+b x)^2 \left (a^2-b^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 89, normalized size = 1.71 \begin {gather*} -\frac {2 \, a b x + 4 \, a^{2} - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{8 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 51, normalized size = 0.98 \begin {gather*} -\frac {\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}}{4 \, a^{2} b^{2}} - \frac {\log \left ({\left | -\frac {2 \, a}{b x + a} + 1 \right |}\right )}{8 \, a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 62, normalized size = 1.19 \begin {gather*} -\frac {1}{4 \left (b x +a \right )^{2} a b}-\frac {1}{4 \left (b x +a \right ) a^{2} b}-\frac {\ln \left (b x -a \right )}{8 a^{3} b}+\frac {\ln \left (b x +a \right )}{8 a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 67, normalized size = 1.29 \begin {gather*} -\frac {b x + 2 \, a}{4 \, {\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {\log \left (b x + a\right )}{8 \, a^{3} b} - \frac {\log \left (b x - a\right )}{8 \, a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 51, normalized size = 0.98 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{4\,a^3\,b}-\frac {\frac {x}{4\,a^2}+\frac {1}{2\,a\,b}}{a^2+2\,a\,b\,x+b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 58, normalized size = 1.12 \begin {gather*} - \frac {2 a + b x}{4 a^{4} b + 8 a^{3} b^{2} x + 4 a^{2} b^{3} x^{2}} - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{8} - \frac {\log {\left (\frac {a}{b} + x \right )}}{8}}{a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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