Optimal. Leaf size=54 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{4 a^3 b}+\frac {1}{4 a^2 b (a-b x)}+\frac {1}{4 a b (a-b x)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 54, 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} \[ \frac {1}{4 a^2 b (a-b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{4 a^3 b}+\frac {1}{4 a b (a-b x)^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 208
Rule 627
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{\left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a-b x)^3 (a+b x)} \, 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 = 62, normalized size = 1.15 \[ \frac {2 a (2 a-b x)+(a-b x)^2 (-\log (a-b x))+(a-b x)^2 \log (a+b x)}{8 a^3 b (a-b x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 89, normalized size = 1.65 \[ -\frac {2 \, a b x - 4 \, a^{2} - {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) + {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{8 \, {\left (a^{3} b^{3} x^{2} - 2 \, a^{4} b^{2} x + a^{5} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 60, normalized size = 1.11 \[ \frac {\log \left ({\left | b x + a \right |}\right )}{8 \, a^{3} b} - \frac {\log \left ({\left | b x - a \right |}\right )}{8 \, a^{3} b} - \frac {a b x - 2 \, a^{2}}{4 \, {\left (b x - a\right )}^{2} a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 66, normalized size = 1.22 \[ \frac {1}{4 \left (b x -a \right )^{2} a b}-\frac {1}{4 \left (b x -a \right ) a^{2} b}-\frac {\ln \left (b x -a \right )}{8 a^{3} b}+\frac {\ln \left (b x +a \right )}{8 a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 67, normalized size = 1.24 \[ -\frac {b x - 2 \, a}{4 \, {\left (a^{2} b^{3} x^{2} - 2 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {\log \left (b x + a\right )}{8 \, a^{3} b} - \frac {\log \left (b x - a\right )}{8 \, a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 51, normalized size = 0.94 \[ \frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{4\,a^3\,b}-\frac {\frac {x}{4\,a^2}-\frac {1}{2\,a\,b}}{a^2-2\,a\,b\,x+b^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 58, normalized size = 1.07 \[ - \frac {- 2 a + b x}{4 a^{4} b - 8 a^{3} b^{2} x + 4 a^{2} b^{3} x^{2}} - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{8} - \frac {\log {\left (\frac {a}{b} + x \right )}}{8}}{a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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