Optimal. Leaf size=86 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{16 a^5 b}-\frac {1}{16 a^4 b (a+b x)}-\frac {1}{16 a^3 b (a+b x)^2}-\frac {1}{12 a^2 b (a+b x)^3}-\frac {1}{8 a b (a+b x)^4} \]
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Rubi [A] time = 0.05, antiderivative size = 86, 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} \[ -\frac {1}{16 a^4 b (a+b x)}-\frac {1}{16 a^3 b (a+b x)^2}-\frac {1}{12 a^2 b (a+b x)^3}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{16 a^5 b}-\frac {1}{8 a b (a+b x)^4} \]
Antiderivative was successfully verified.
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Rule 44
Rule 208
Rule 627
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^4 \left (a^2-b^2 x^2\right )} \, dx &=\int \frac {1}{(a-b x) (a+b x)^5} \, dx\\ &=\int \left (\frac {1}{2 a (a+b x)^5}+\frac {1}{4 a^2 (a+b x)^4}+\frac {1}{8 a^3 (a+b x)^3}+\frac {1}{16 a^4 (a+b x)^2}+\frac {1}{16 a^4 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=-\frac {1}{8 a b (a+b x)^4}-\frac {1}{12 a^2 b (a+b x)^3}-\frac {1}{16 a^3 b (a+b x)^2}-\frac {1}{16 a^4 b (a+b x)}+\frac {\int \frac {1}{a^2-b^2 x^2} \, dx}{16 a^4}\\ &=-\frac {1}{8 a b (a+b x)^4}-\frac {1}{12 a^2 b (a+b x)^3}-\frac {1}{16 a^3 b (a+b x)^2}-\frac {1}{16 a^4 b (a+b x)}+\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{16 a^5 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 82, normalized size = 0.95 \[ \frac {-2 a \left (16 a^3+19 a^2 b x+12 a b^2 x^2+3 b^3 x^3\right )-3 (a+b x)^4 \log (a-b x)+3 (a+b x)^4 \log (a+b x)}{96 a^5 b (a+b x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 178, normalized size = 2.07 \[ -\frac {6 \, a b^{3} x^{3} + 24 \, a^{2} b^{2} x^{2} + 38 \, a^{3} b x + 32 \, a^{4} - 3 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right ) + 3 \, {\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x - a\right )}{96 \, {\left (a^{5} b^{5} x^{4} + 4 \, a^{6} b^{4} x^{3} + 6 \, a^{7} b^{3} x^{2} + 4 \, a^{8} b^{2} x + a^{9} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 81, normalized size = 0.94 \[ \frac {\log \left ({\left | b x + a \right |}\right )}{32 \, a^{5} b} - \frac {\log \left ({\left | b x - a \right |}\right )}{32 \, a^{5} b} - \frac {3 \, a b^{3} x^{3} + 12 \, a^{2} b^{2} x^{2} + 19 \, a^{3} b x + 16 \, a^{4}}{48 \, {\left (b x + a\right )}^{4} a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 92, normalized size = 1.07 \[ -\frac {1}{8 \left (b x +a \right )^{4} a b}-\frac {1}{12 \left (b x +a \right )^{3} a^{2} b}-\frac {1}{16 \left (b x +a \right )^{2} a^{3} b}-\frac {1}{16 \left (b x +a \right ) a^{4} b}-\frac {\ln \left (b x -a \right )}{32 a^{5} b}+\frac {\ln \left (b x +a \right )}{32 a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 112, normalized size = 1.30 \[ -\frac {3 \, b^{3} x^{3} + 12 \, a b^{2} x^{2} + 19 \, a^{2} b x + 16 \, a^{3}}{48 \, {\left (a^{4} b^{5} x^{4} + 4 \, a^{5} b^{4} x^{3} + 6 \, a^{6} b^{3} x^{2} + 4 \, a^{7} b^{2} x + a^{8} b\right )}} + \frac {\log \left (b x + a\right )}{32 \, a^{5} b} - \frac {\log \left (b x - a\right )}{32 \, a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 93, normalized size = 1.08 \[ \frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{16\,a^5\,b}-\frac {\frac {19\,x}{48\,a^2}+\frac {1}{3\,a\,b}+\frac {b\,x^2}{4\,a^3}+\frac {b^2\,x^3}{16\,a^4}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 107, normalized size = 1.24 \[ - \frac {16 a^{3} + 19 a^{2} b x + 12 a b^{2} x^{2} + 3 b^{3} x^{3}}{48 a^{8} b + 192 a^{7} b^{2} x + 288 a^{6} b^{3} x^{2} + 192 a^{5} b^{4} x^{3} + 48 a^{4} b^{5} x^{4}} - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{32} - \frac {\log {\left (\frac {a}{b} + x \right )}}{32}}{a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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