Optimal. Leaf size=51 \[ \frac {-\frac {9 b}{2 a^2}-\frac {1}{a x}-\frac {3 b^2 x}{a^3}}{(a+b x)^2}+\frac {3 b \log \left (\frac {a+b x}{x}\right )}{a^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.12, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46}
\begin {gather*} -\frac {3 b \log (x)}{a^4}+\frac {3 b \log (a+b x)}{a^4}-\frac {2 b}{a^3 (a+b x)}-\frac {1}{a^3 x}-\frac {b}{2 a^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\int \left (\frac {1}{a^3 x^2}-\frac {3 b}{a^4 x}+\frac {b^2}{a^2 (a+b x)^3}+\frac {2 b^2}{a^3 (a+b x)^2}+\frac {3 b^2}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac {1}{a^3 x}-\frac {b}{2 a^2 (a+b x)^2}-\frac {2 b}{a^3 (a+b x)}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log (a+b x)}{a^4}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 53, normalized size = 1.04 \begin {gather*} -\frac {\frac {a \left (2 a^2+9 a b x+6 b^2 x^2\right )}{x (a+b x)^2}+6 b \log (x)-6 b \log (a+b x)}{2 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 56, normalized size = 1.10
method | result | size |
default | \(-\frac {b}{2 a^{2} \left (b x +a \right )^{2}}+\frac {3 b \ln \left (b x +a \right )}{a^{4}}-\frac {2 b}{a^{3} \left (b x +a \right )}-\frac {1}{a^{3} x}-\frac {3 b \ln \left (x \right )}{a^{4}}\) | \(56\) |
risch | \(\frac {-\frac {3 b^{2} x^{2}}{a^{3}}-\frac {9 b x}{2 a^{2}}-\frac {1}{a}}{x \left (b x +a \right )^{2}}+\frac {3 b \ln \left (-b x -a \right )}{a^{4}}-\frac {3 b \ln \left (x \right )}{a^{4}}\) | \(60\) |
norman | \(\frac {-\frac {1}{a}+\frac {6 b^{2} x^{2}}{a^{3}}+\frac {9 b^{3} x^{3}}{2 a^{4}}}{x \left (b x +a \right )^{2}}-\frac {3 b \ln \left (x \right )}{a^{4}}+\frac {3 b \ln \left (b x +a \right )}{a^{4}}\) | \(61\) |
parallelrisch | \(-\frac {6 \ln \left (x \right ) x^{3} b^{3}-6 \ln \left (b x +a \right ) x^{3} b^{3}+12 \ln \left (x \right ) x^{2} a \,b^{2}-12 \ln \left (b x +a \right ) x^{2} a \,b^{2}-9 b^{3} x^{3}+6 \ln \left (x \right ) x \,a^{2} b -6 \ln \left (b x +a \right ) x \,a^{2} b -12 a \,x^{2} b^{2}+2 a^{3}}{2 a^{4} x \left (b x +a \right )^{2}}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 69, normalized size = 1.35 \begin {gather*} -\frac {6 \, b^{2} x^{2} + 9 \, a b x + 2 \, a^{2}}{2 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} + \frac {3 \, b \log \left (b x + a\right )}{a^{4}} - \frac {3 \, b \log \left (x\right )}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs.
\(2 (50) = 100\).
time = 0.60, size = 109, normalized size = 2.14 \begin {gather*} -\frac {6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3} - 6 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (x\right )}{2 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 66, normalized size = 1.29 \begin {gather*} \frac {- 2 a^{2} - 9 a b x - 6 b^{2} x^{2}}{2 a^{5} x + 4 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac {3 b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 60, normalized size = 1.18 \begin {gather*} \frac {3 \, b \log \left ({\left | b x + a \right |}\right )}{a^{4}} - \frac {3 \, b \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3}}{2 \, {\left (b x + a\right )}^{2} a^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 63, normalized size = 1.24 \begin {gather*} \frac {6\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^4}-\frac {\frac {1}{a}+\frac {3\,b^2\,x^2}{a^3}+\frac {9\,b\,x}{2\,a^2}}{a^2\,x+2\,a\,b\,x^2+b^2\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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