Optimal. Leaf size=34 \[ \frac {c \log (x)}{a}-\frac {(b c-a d) \log \left (a+b x^2\right )}{2 a b} \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 78}
\begin {gather*} \frac {c \log (x)}{a}-\frac {(b c-a d) \log \left (a+b x^2\right )}{2 a b} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {c+d x^2}{x \left (a+b x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {c+d x}{x (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {c}{a x}+\frac {-b c+a d}{a (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {c \log (x)}{a}-\frac {(b c-a d) \log \left (a+b x^2\right )}{2 a b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} \frac {c \log (x)}{a}+\frac {(-b c+a d) \log \left (a+b x^2\right )}{2 a b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 33, normalized size = 0.97
method | result | size |
default | \(\frac {\left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a b}+\frac {c \ln \left (x \right )}{a}\) | \(33\) |
norman | \(\frac {\left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a b}+\frac {c \ln \left (x \right )}{a}\) | \(33\) |
risch | \(\frac {c \ln \left (x \right )}{a}+\frac {\ln \left (-b \,x^{2}-a \right ) d}{2 b}-\frac {\ln \left (-b \,x^{2}-a \right ) c}{2 a}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 1.03 \begin {gather*} \frac {c \log \left (x^{2}\right )}{2 \, a} - \frac {{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.89, size = 33, normalized size = 0.97 \begin {gather*} \frac {2 \, b c \log \left (x\right ) - {\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.40, size = 26, normalized size = 0.76 \begin {gather*} \frac {c \log {\left (x \right )}}{a} + \frac {\left (a d - b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 36, normalized size = 1.06 \begin {gather*} \frac {c \log \left (x^{2}\right )}{2 \, a} - \frac {{\left (b c - a d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 32, normalized size = 0.94 \begin {gather*} \frac {c\,\ln \left (x\right )}{a}+\frac {\ln \left (b\,x^2+a\right )\,\left (a\,d-b\,c\right )}{2\,a\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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