Optimal. Leaf size=62 \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}-\frac {\log \left (a+b x+c x^2\right )}{2 a}+\frac {\log (x)}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1585, 705, 29, 634, 618, 206, 628} \begin {gather*} \frac {b \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}-\frac {\log \left (a+b x+c x^2\right )}{2 a}+\frac {\log (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 206
Rule 618
Rule 628
Rule 634
Rule 705
Rule 1585
Rubi steps
\begin {align*} \int \frac {x}{a x^2+b x^3+c x^4} \, dx &=\int \frac {1}{x \left (a+b x+c x^2\right )} \, dx\\ &=\frac {\int \frac {1}{x} \, dx}{a}+\frac {\int \frac {-b-c x}{a+b x+c x^2} \, dx}{a}\\ &=\frac {\log (x)}{a}-\frac {\int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a}-\frac {b \int \frac {1}{a+b x+c x^2} \, dx}{2 a}\\ &=\frac {\log (x)}{a}-\frac {\log \left (a+b x+c x^2\right )}{2 a}+\frac {b \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a}\\ &=\frac {b \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c}}+\frac {\log (x)}{a}-\frac {\log \left (a+b x+c x^2\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 61, normalized size = 0.98 \begin {gather*} -\frac {\frac {2 b \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+\log (a+x (b+c x))-2 \log (x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{a x^2+b x^3+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.92, size = 211, normalized size = 3.40 \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} - 4 \, a c\right )} \log \relax (x)}{2 \, {\left (a b^{2} - 4 \, a^{2} c\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} - 4 \, a c\right )} \log \relax (x)}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 62, normalized size = 1.00 \begin {gather*} -\frac {b \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} a} - \frac {\log \left (c x^{2} + b x + a\right )}{2 \, a} + \frac {\log \left ({\left | x \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 62, normalized size = 1.00 \begin {gather*} -\frac {b \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a}+\frac {\ln \relax (x )}{a}-\frac {\ln \left (c \,x^{2}+b x +a \right )}{2 a} \end {gather*}
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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mupad [B] time = 2.30, size = 213, normalized size = 3.44 \begin {gather*} \frac {\ln \relax (x)}{a}-\ln \left (b\,c-\left (x\,\left (6\,a\,c^2-2\,b^2\,c\right )-a\,b\,c\right )\,\left (\frac {1}{2\,a}-\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right )+3\,c^2\,x\right )\,\left (\frac {1}{2\,a}-\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right )-\ln \left (\left (x\,\left (6\,a\,c^2-2\,b^2\,c\right )-a\,b\,c\right )\,\left (\frac {1}{2\,a}+\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right )-b\,c-3\,c^2\,x\right )\,\left (\frac {1}{2\,a}+\frac {b\,\sqrt {b^2-4\,a\,c}}{2\,\left (a\,b^2-4\,a^2\,c\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.36, size = 564, normalized size = 9.10 \begin {gather*} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) \log {\left (x + \frac {24 a^{4} c^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) + 2 a^{2} b^{4} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) \log {\left (x + \frac {24 a^{4} c^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 14 a^{3} b^{2} c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} - 12 a^{3} c^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) + 2 a^{2} b^{4} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right )^{2} + 3 a^{2} b^{2} c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{2 a \left (4 a c - b^{2}\right )} - \frac {1}{2 a}\right ) - 12 a^{2} c^{2} + 11 a b^{2} c - 2 b^{4}}{9 a b c^{2} - 2 b^{3} c} \right )} + \frac {\log {\relax (x )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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