Optimal. Leaf size=56 \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x+c x^2\right )}{2 c} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1368, 648, 632,
212, 642} \begin {gather*} \frac {b \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x+c x^2\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 1368
Rubi steps
\begin {align*} \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right ) x} \, dx &=\int \frac {x}{a+b x+c x^2} \, dx\\ &=\frac {\int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c}-\frac {b \int \frac {1}{a+b x+c x^2} \, dx}{2 c}\\ &=\frac {\log \left (a+b x+c x^2\right )}{2 c}+\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c}\\ &=\frac {b \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x+c x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 57, normalized size = 1.02 \begin {gather*} \frac {-\frac {2 b \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\log (a+x (b+c x))}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 56, normalized size = 1.00
method | result | size |
default | \(\frac {\ln \left (c \,x^{2}+b x +a \right )}{2 c}-\frac {b \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{c \sqrt {4 a c -b^{2}}}\) | \(56\) |
risch | \(\frac {2 \ln \left (-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, c x -4 a b c +b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) a}{4 a c -b^{2}}-\frac {\ln \left (-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, c x -4 a b c +b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) b^{2}}{2 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, c x -4 a b c +b^{3}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{2 c \left (4 a c -b^{2}\right )}+\frac {2 \ln \left (2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, c x -4 a b c +b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) a}{4 a c -b^{2}}-\frac {\ln \left (2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, c x -4 a b c +b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) b^{2}}{2 c \left (4 a c -b^{2}\right )}-\frac {\ln \left (2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, c x -4 a b c +b^{3}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{2 c \left (4 a c -b^{2}\right )}\) | \(443\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 185, normalized size = 3.30 \begin {gather*} \left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}\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. 216 vs.
\(2 (49) = 98\).
time = 0.16, size = 216, normalized size = 3.86 \begin {gather*} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac {1}{2 c}\right ) \log {\left (x + \frac {- 4 a c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac {1}{2 c}\right ) + 2 a + b^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac {1}{2 c}\right )}{b} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac {1}{2 c}\right ) \log {\left (x + \frac {- 4 a c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac {1}{2 c}\right ) + 2 a + b^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 c \left (4 a c - b^{2}\right )} + \frac {1}{2 c}\right )}{b} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.83, size = 55, normalized size = 0.98 \begin {gather*} -\frac {b \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c} + \frac {\log \left (c x^{2} + b x + a\right )}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 112, normalized size = 2.00 \begin {gather*} \frac {2\,a\,c\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^2-b^2\,c}-\frac {b\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c\,\sqrt {4\,a\,c-b^2}}-\frac {b^2\,\ln \left (c\,x^2+b\,x+a\right )}{2\,\left (4\,a\,c^2-b^2\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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