Optimal. Leaf size=87 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{4 a^5 b}+\frac {1}{16 a^4 b (a-b x)}-\frac {3}{16 a^4 b (a+b x)}-\frac {1}{8 a^3 b (a+b x)^2}-\frac {1}{12 a^2 b (a+b x)^3} \]
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Rubi [A] time = 0.05, antiderivative size = 87, 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}{16 a^4 b (a-b x)}-\frac {3}{16 a^4 b (a+b x)}-\frac {1}{8 a^3 b (a+b x)^2}-\frac {1}{12 a^2 b (a+b x)^3}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{4 a^5 b} \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 )^2} \, dx &=\int \frac {1}{(a-b x)^2 (a+b x)^4} \, dx\\ &=\int \left (\frac {1}{16 a^4 (a-b x)^2}+\frac {1}{4 a^2 (a+b x)^4}+\frac {1}{4 a^3 (a+b x)^3}+\frac {3}{16 a^4 (a+b x)^2}+\frac {1}{4 a^4 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac {1}{16 a^4 b (a-b x)}-\frac {1}{12 a^2 b (a+b x)^3}-\frac {1}{8 a^3 b (a+b x)^2}-\frac {3}{16 a^4 b (a+b x)}+\frac {\int \frac {1}{a^2-b^2 x^2} \, dx}{4 a^4}\\ &=\frac {1}{16 a^4 b (a-b x)}-\frac {1}{12 a^2 b (a+b x)^3}-\frac {1}{8 a^3 b (a+b x)^2}-\frac {3}{16 a^4 b (a+b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{4 a^5 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 75, normalized size = 0.86 \begin {gather*} \frac {\frac {2 a \left (-4 a^3+a^2 b x+6 a b^2 x^2+3 b^3 x^3\right )}{(a-b x) (a+b x)^3}-3 \log (a-b x)+3 \log (a+b x)}{24 a^5 b} \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 )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 150, normalized size = 1.72 \begin {gather*} -\frac {6 \, a b^{3} x^{3} + 12 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b x - 8 \, a^{4} - 3 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} - 2 \, a^{3} b x - a^{4}\right )} \log \left (b x + a\right ) + 3 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} - 2 \, a^{3} b x - a^{4}\right )} \log \left (b x - a\right )}{24 \, {\left (a^{5} b^{5} x^{4} + 2 \, a^{6} b^{4} x^{3} - 2 \, a^{8} b^{2} x - a^{9} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 99, normalized size = 1.14 \begin {gather*} -\frac {\log \left ({\left | -\frac {2 \, a}{b x + a} + 1 \right |}\right )}{8 \, a^{5} b} + \frac {1}{32 \, a^{5} b {\left (\frac {2 \, a}{b x + a} - 1\right )}} - \frac {\frac {9 \, a^{2} b^{5}}{b x + a} + \frac {6 \, a^{3} b^{5}}{{\left (b x + a\right )}^{2}} + \frac {4 \, a^{4} b^{5}}{{\left (b x + a\right )}^{3}}}{48 \, a^{6} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 94, normalized size = 1.08 \begin {gather*} -\frac {1}{12 \left (b x +a \right )^{3} a^{2} b}-\frac {1}{8 \left (b x +a \right )^{2} a^{3} b}-\frac {1}{16 \left (b x -a \right ) a^{4} b}-\frac {3}{16 \left (b x +a \right ) a^{4} b}-\frac {\ln \left (b x -a \right )}{8 a^{5} b}+\frac {\ln \left (b x +a \right )}{8 a^{5} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 101, normalized size = 1.16 \begin {gather*} -\frac {3 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + a^{2} b x - 4 \, a^{3}}{12 \, {\left (a^{4} b^{5} x^{4} + 2 \, a^{5} b^{4} x^{3} - 2 \, a^{7} b^{2} x - a^{8} b\right )}} + \frac {\log \left (b x + a\right )}{8 \, a^{5} b} - \frac {\log \left (b x - a\right )}{8 \, a^{5} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 82, normalized size = 0.94 \begin {gather*} \frac {\frac {x}{12\,a^2}-\frac {1}{3\,a\,b}+\frac {b\,x^2}{2\,a^3}+\frac {b^2\,x^3}{4\,a^4}}{a^4+2\,a^3\,b\,x-2\,a\,b^3\,x^3-b^4\,x^4}+\frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{4\,a^5\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 92, normalized size = 1.06 \begin {gather*} \frac {4 a^{3} - a^{2} b x - 6 a b^{2} x^{2} - 3 b^{3} x^{3}}{- 12 a^{8} b - 24 a^{7} b^{2} x + 24 a^{5} b^{4} x^{3} + 12 a^{4} b^{5} x^{4}} + \frac {- \frac {\log {\left (- \frac {a}{b} + x \right )}}{8} + \frac {\log {\left (\frac {a}{b} + x \right )}}{8}}{a^{5} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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