Optimal. Leaf size=22 \[ \frac {\log (x)}{b}-\frac {\log \left (b+d x^2\right )}{2 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {1607, 272, 36,
29, 31} \begin {gather*} \frac {\log (x)}{b}-\frac {\log \left (b+d x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 1607
Rubi steps
\begin {align*} \int \frac {1}{b x+d x^3} \, dx &=\int \frac {1}{x \left (b+d x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (b+d x)} \, dx,x,x^2\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 b}-\frac {d \text {Subst}\left (\int \frac {1}{b+d x} \, dx,x,x^2\right )}{2 b}\\ &=\frac {\log (x)}{b}-\frac {\log \left (b+d x^2\right )}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{b}-\frac {\log \left (b+d x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 21, normalized size = 0.95
method | result | size |
default | \(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (d \,x^{2}+b \right )}{2 b}\) | \(21\) |
norman | \(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (d \,x^{2}+b \right )}{2 b}\) | \(21\) |
risch | \(\frac {\ln \left (x \right )}{b}-\frac {\ln \left (d \,x^{2}+b \right )}{2 b}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 20, normalized size = 0.91 \begin {gather*} -\frac {\log \left (d x^{2} + b\right )}{2 \, b} + \frac {\log \left (x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 18, normalized size = 0.82 \begin {gather*} -\frac {\log \left (d x^{2} + b\right ) - 2 \, \log \left (x\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 15, normalized size = 0.68 \begin {gather*} \frac {\log {\left (x \right )}}{b} - \frac {\log {\left (\frac {b}{d} + x^{2} \right )}}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.48, size = 24, normalized size = 1.09 \begin {gather*} \frac {\log \left (x^{2}\right )}{2 \, b} - \frac {\log \left ({\left | d x^{2} + b \right |}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.13, size = 18, normalized size = 0.82 \begin {gather*} -\frac {\ln \left (d\,x^2+b\right )-2\,\ln \left (x\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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