Optimal. Leaf size=42 \[ -\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 46}
\begin {gather*} \frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 1598
Rubi steps
\begin {align*} \int \frac {1}{x \left (a x^2+b x^3\right )} \, dx &=\int \frac {1}{x^3 (a+b x)} \, dx\\ &=\int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 42, normalized size = 1.00 \begin {gather*} -\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 41, normalized size = 0.98
method | result | size |
default | \(-\frac {1}{2 a \,x^{2}}+\frac {b}{a^{2} x}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}\) | \(41\) |
norman | \(\frac {\frac {b x}{a^{2}}-\frac {1}{2 a}}{x^{2}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}\) | \(41\) |
risch | \(\frac {\frac {b x}{a^{2}}-\frac {1}{2 a}}{x^{2}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}+\frac {b^{2} \ln \left (-x \right )}{a^{3}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 40, normalized size = 0.95 \begin {gather*} -\frac {b^{2} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} \log \left (x\right )}{a^{3}} + \frac {2 \, b x - a}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.90, size = 41, normalized size = 0.98 \begin {gather*} -\frac {2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 31, normalized size = 0.74 \begin {gather*} \frac {- a + 2 b x}{2 a^{2} x^{2}} + \frac {b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.62, size = 45, normalized size = 1.07 \begin {gather*} -\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 38, normalized size = 0.90 \begin {gather*} -\frac {\frac {a^2}{2}-a\,b\,x}{a^3\,x^2}-\frac {2\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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