Optimal. Leaf size=105 \[ \frac {5 \tanh ^{-1}\left (\frac {b x}{a}\right )}{16 a^6 b}+\frac {1}{8 a^5 b (a-b x)}-\frac {3}{16 a^5 b (a+b x)}+\frac {1}{32 a^4 b (a-b x)^2}-\frac {3}{32 a^4 b (a+b x)^2}-\frac {1}{24 a^3 b (a+b x)^3} \]
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Rubi [A] time = 0.07, antiderivative size = 105, 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}{8 a^5 b (a-b x)}-\frac {3}{16 a^5 b (a+b x)}+\frac {1}{32 a^4 b (a-b x)^2}-\frac {3}{32 a^4 b (a+b x)^2}-\frac {1}{24 a^3 b (a+b x)^3}+\frac {5 \tanh ^{-1}\left (\frac {b x}{a}\right )}{16 a^6 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) \left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a-b x)^3 (a+b x)^4} \, dx\\ &=\int \left (\frac {1}{16 a^4 (a-b x)^3}+\frac {1}{8 a^5 (a-b x)^2}+\frac {1}{8 a^3 (a+b x)^4}+\frac {3}{16 a^4 (a+b x)^3}+\frac {3}{16 a^5 (a+b x)^2}+\frac {5}{16 a^5 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac {1}{32 a^4 b (a-b x)^2}+\frac {1}{8 a^5 b (a-b x)}-\frac {1}{24 a^3 b (a+b x)^3}-\frac {3}{32 a^4 b (a+b x)^2}-\frac {3}{16 a^5 b (a+b x)}+\frac {5 \int \frac {1}{a^2-b^2 x^2} \, dx}{16 a^5}\\ &=\frac {1}{32 a^4 b (a-b x)^2}+\frac {1}{8 a^5 b (a-b x)}-\frac {1}{24 a^3 b (a+b x)^3}-\frac {3}{32 a^4 b (a+b x)^2}-\frac {3}{16 a^5 b (a+b x)}+\frac {5 \tanh ^{-1}\left (\frac {b x}{a}\right )}{16 a^6 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 87, normalized size = 0.83 \begin {gather*} \frac {-\frac {2 a \left (8 a^4-25 a^3 b x-25 a^2 b^2 x^2+15 a b^3 x^3+15 b^4 x^4\right )}{(a-b x)^2 (a+b x)^3}-15 \log (a-b x)+15 \log (a+b x)}{96 a^6 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) \left (a^2-b^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 216, normalized size = 2.06 \begin {gather*} -\frac {30 \, a b^{4} x^{4} + 30 \, a^{2} b^{3} x^{3} - 50 \, a^{3} b^{2} x^{2} - 50 \, a^{4} b x + 16 \, a^{5} - 15 \, {\left (b^{5} x^{5} + a b^{4} x^{4} - 2 \, a^{2} b^{3} x^{3} - 2 \, a^{3} b^{2} x^{2} + a^{4} b x + a^{5}\right )} \log \left (b x + a\right ) + 15 \, {\left (b^{5} x^{5} + a b^{4} x^{4} - 2 \, a^{2} b^{3} x^{3} - 2 \, a^{3} b^{2} x^{2} + a^{4} b x + a^{5}\right )} \log \left (b x - a\right )}{96 \, {\left (a^{6} b^{6} x^{5} + a^{7} b^{5} x^{4} - 2 \, a^{8} b^{4} x^{3} - 2 \, a^{9} b^{3} x^{2} + a^{10} b^{2} x + a^{11} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 101, normalized size = 0.96 \begin {gather*} \frac {5 \, \log \left ({\left | b x + a \right |}\right )}{32 \, a^{6} b} - \frac {5 \, \log \left ({\left | b x - a \right |}\right )}{32 \, a^{6} b} - \frac {15 \, a b^{4} x^{4} + 15 \, a^{2} b^{3} x^{3} - 25 \, a^{3} b^{2} x^{2} - 25 \, a^{4} b x + 8 \, a^{5}}{48 \, {\left (b x + a\right )}^{3} {\left (b x - a\right )}^{2} a^{6} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 111, normalized size = 1.06 \begin {gather*} -\frac {1}{24 \left (b x +a \right )^{3} a^{3} b}+\frac {1}{32 \left (b x -a \right )^{2} a^{4} b}-\frac {3}{32 \left (b x +a \right )^{2} a^{4} b}-\frac {1}{8 \left (b x -a \right ) a^{5} b}-\frac {3}{16 \left (b x +a \right ) a^{5} b}-\frac {5 \ln \left (b x -a \right )}{32 a^{6} b}+\frac {5 \ln \left (b x +a \right )}{32 a^{6} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 132, normalized size = 1.26 \begin {gather*} -\frac {15 \, b^{4} x^{4} + 15 \, a b^{3} x^{3} - 25 \, a^{2} b^{2} x^{2} - 25 \, a^{3} b x + 8 \, a^{4}}{48 \, {\left (a^{5} b^{6} x^{5} + a^{6} b^{5} x^{4} - 2 \, a^{7} b^{4} x^{3} - 2 \, a^{8} b^{3} x^{2} + a^{9} b^{2} x + a^{10} b\right )}} + \frac {5 \, \log \left (b x + a\right )}{32 \, a^{6} b} - \frac {5 \, \log \left (b x - a\right )}{32 \, a^{6} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 113, normalized size = 1.08 \begin {gather*} \frac {5\,\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{16\,a^6\,b}-\frac {\frac {1}{6\,a\,b}-\frac {25\,x}{48\,a^2}-\frac {25\,b\,x^2}{48\,a^3}+\frac {5\,b^2\,x^3}{16\,a^4}+\frac {5\,b^3\,x^4}{16\,a^5}}{a^5+a^4\,b\,x-2\,a^3\,b^2\,x^2-2\,a^2\,b^3\,x^3+a\,b^4\,x^4+b^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 134, normalized size = 1.28 \begin {gather*} - \frac {8 a^{4} - 25 a^{3} b x - 25 a^{2} b^{2} x^{2} + 15 a b^{3} x^{3} + 15 b^{4} x^{4}}{48 a^{10} b + 48 a^{9} b^{2} x - 96 a^{8} b^{3} x^{2} - 96 a^{7} b^{4} x^{3} + 48 a^{6} b^{5} x^{4} + 48 a^{5} b^{6} x^{5}} - \frac {\frac {5 \log {\left (- \frac {a}{b} + x \right )}}{32} - \frac {5 \log {\left (\frac {a}{b} + x \right )}}{32}}{a^{6} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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