Optimal. Leaf size=34 \[ -\frac {1}{a x}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1598, 331, 211}
\begin {gather*} -\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 331
Rule 1598
Rubi steps
\begin {align*} \int \frac {1}{x \left (a x+b x^3\right )} \, dx &=\int \frac {1}{x^2 \left (a+b x^2\right )} \, dx\\ &=-\frac {1}{a x}-\frac {b \int \frac {1}{a+b x^2} \, dx}{a}\\ &=-\frac {1}{a x}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} -\frac {1}{a x}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 30, normalized size = 0.88
method | result | size |
default | \(-\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \sqrt {a b}}-\frac {1}{a x}\) | \(30\) |
risch | \(-\frac {1}{a x}+\frac {\sqrt {-a b}\, \ln \left (-b x +\sqrt {-a b}\right )}{2 a^{2}}-\frac {\sqrt {-a b}\, \ln \left (-b x -\sqrt {-a b}\right )}{2 a^{2}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 29, normalized size = 0.85 \begin {gather*} -\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {1}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.34, size = 82, normalized size = 2.41 \begin {gather*} \left [\frac {x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2}{2 \, a x}, -\frac {x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 1}{a x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (29) = 58\).
time = 0.07, size = 65, normalized size = 1.91 \begin {gather*} \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (- \frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (\frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {1}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.57, size = 29, normalized size = 0.85 \begin {gather*} -\frac {b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {1}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.96, size = 26, normalized size = 0.76 \begin {gather*} -\frac {1}{a\,x}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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