Optimal. Leaf size=48 \[ -\frac {2 \log (b+2 c x)}{\left (b^2-4 a c\right ) d}+\frac {\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right ) d} \]
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Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {695, 31, 642}
\begin {gather*} \frac {\log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )}-\frac {2 \log (b+2 c x)}{d \left (b^2-4 a c\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 642
Rule 695
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx &=\frac {\int \frac {b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) d^2}-\frac {(4 c) \int \frac {1}{b+2 c x} \, dx}{\left (b^2-4 a c\right ) d}\\ &=-\frac {2 \log (b+2 c x)}{\left (b^2-4 a c\right ) d}+\frac {\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right ) d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 0.71 \begin {gather*} \frac {-2 \log (b+2 c x)+\log (a+x (b+c x))}{\left (b^2-4 a c\right ) d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 52, normalized size = 1.08
method | result | size |
default | \(\frac {-\frac {\ln \left (c \,x^{2}+b x +a \right )}{4 a c -b^{2}}+\frac {2 \ln \left (2 c x +b \right )}{4 a c -b^{2}}}{d}\) | \(52\) |
norman | \(\frac {2 \ln \left (2 c x +b \right )}{d \left (4 a c -b^{2}\right )}-\frac {\ln \left (c \,x^{2}+b x +a \right )}{d \left (4 a c -b^{2}\right )}\) | \(54\) |
risch | \(\frac {2 \ln \left (2 c x +b \right )}{d \left (4 a c -b^{2}\right )}-\frac {\ln \left (c \,x^{2}+b x +a \right )}{d \left (4 a c -b^{2}\right )}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 48, normalized size = 1.00 \begin {gather*} \frac {\log \left (c x^{2} + b x + a\right )}{{\left (b^{2} - 4 \, a c\right )} d} - \frac {2 \, \log \left (2 \, c x + b\right )}{{\left (b^{2} - 4 \, a c\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.40, size = 35, normalized size = 0.73 \begin {gather*} \frac {\log \left (c x^{2} + b x + a\right ) - 2 \, \log \left (2 \, c x + b\right )}{{\left (b^{2} - 4 \, a c\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.40, size = 42, normalized size = 0.88 \begin {gather*} \frac {2 \log {\left (\frac {b}{2 c} + x \right )}}{d \left (4 a c - b^{2}\right )} - \frac {\log {\left (\frac {a}{c} + \frac {b x}{c} + x^{2} \right )}}{d \left (4 a c - b^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.77, size = 57, normalized size = 1.19 \begin {gather*} -\frac {2 \, c^{2} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{2} c^{2} d - 4 \, a c^{3} d} + \frac {\log \left (c x^{2} + b x + a\right )}{b^{2} d - 4 \, a c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 47, normalized size = 0.98 \begin {gather*} \frac {2\,c\,d\,\ln \left (\frac {{\left (b+2\,c\,x\right )}^2}{c\,x^2+b\,x+a}\right )}{8\,a\,c^2\,d^2-2\,b^2\,c\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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