Optimal. Leaf size=104 \[ -\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\log (x) \left (b^2-a c\right )}{a^3}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c}}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \]
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Rubi [A] time = 0.15, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1585, 709, 800, 634, 618, 206, 628} \begin {gather*} -\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\log (x) \left (b^2-a c\right )}{a^3}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c}}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 709
Rule 800
Rule 1585
Rubi steps
\begin {align*} \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )} \, dx &=\int \frac {1}{x^3 \left (a+b x+c x^2\right )} \, dx\\ &=-\frac {1}{2 a x^2}+\frac {\int \frac {-b-c x}{x^2 \left (a+b x+c x^2\right )} \, dx}{a}\\ &=-\frac {1}{2 a x^2}+\frac {\int \left (-\frac {b}{a x^2}+\frac {b^2-a c}{a^2 x}+\frac {-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx}{a}\\ &=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {\left (b^2-a c\right ) \log (x)}{a^3}+\frac {\int \frac {-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x}{a+b x+c x^2} \, dx}{a^3}\\ &=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {\left (b^2-a c\right ) \log (x)}{a^3}-\frac {\left (b \left (b^2-3 a c\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 a^3}-\frac {\left (b^2-a c\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^3}\\ &=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {\left (b^2-a c\right ) \log (x)}{a^3}-\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}+\frac {\left (b \left (b^2-3 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^3}\\ &=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log (x)}{a^3}-\frac {\left (b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 102, normalized size = 0.98 \begin {gather*} \frac {-\frac {a^2}{x^2}+2 \log (x) \left (b^2-a c\right )+\left (a c-b^2\right ) \log (a+x (b+c x))-\frac {2 b \left (b^2-3 a c\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+\frac {2 a b}{x}}{2 a^3} \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 {1}{x \left (a x^2+b x^3+c x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.91, size = 358, normalized size = 3.44 \begin {gather*} \left [-\frac {{\left (b^{3} - 3 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} x^{2} \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 ) + a^{2} b^{2} - 4 \, a^{3} c + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \relax (x) - 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}}, \frac {2 \, {\left (b^{3} - 3 \, a b c\right )} \sqrt {-b^{2} + 4 \, a c} x^{2} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - a^{2} b^{2} + 4 \, a^{3} c - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \left (c x^{2} + b x + a\right ) + 2 \, {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} \log \relax (x) + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 105, normalized size = 1.01 \begin {gather*} -\frac {{\left (b^{2} - a c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} + \frac {{\left (b^{2} - a c\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} a^{3}} + \frac {2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 150, normalized size = 1.44 \begin {gather*} \frac {3 b c \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{2}}-\frac {b^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{3}}-\frac {c \ln \relax (x )}{a^{2}}+\frac {c \ln \left (c \,x^{2}+b x +a \right )}{2 a^{2}}+\frac {b^{2} \ln \relax (x )}{a^{3}}-\frac {b^{2} \ln \left (c \,x^{2}+b x +a \right )}{2 a^{3}}+\frac {b}{a^{2} x}-\frac {1}{2 a \,x^{2}} \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 = 0.59, size = 447, normalized size = 4.30 \begin {gather*} \frac {\ln \left (2\,a\,b^4+2\,b^5\,x+6\,a^3\,c^2+2\,a\,b^3\,\sqrt {b^2-4\,a\,c}+2\,b^4\,x\,\sqrt {b^2-4\,a\,c}-9\,a^2\,b^2\,c-10\,a\,b^3\,c\,x-3\,a^2\,b\,c\,\sqrt {b^2-4\,a\,c}+9\,a^2\,b\,c^2\,x+3\,a^2\,c^2\,x\,\sqrt {b^2-4\,a\,c}-6\,a\,b^2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {b^4}{2}-a\,\left (\frac {5\,b^2\,c}{2}+\frac {3\,b\,c\,\sqrt {b^2-4\,a\,c}}{2}\right )+\frac {b^3\,\sqrt {b^2-4\,a\,c}}{2}+2\,a^2\,c^2\right )}{4\,a^4\,c-a^3\,b^2}-\frac {\ln \left (2\,a\,b^4+2\,b^5\,x+6\,a^3\,c^2-2\,a\,b^3\,\sqrt {b^2-4\,a\,c}-2\,b^4\,x\,\sqrt {b^2-4\,a\,c}-9\,a^2\,b^2\,c-10\,a\,b^3\,c\,x+3\,a^2\,b\,c\,\sqrt {b^2-4\,a\,c}+9\,a^2\,b\,c^2\,x-3\,a^2\,c^2\,x\,\sqrt {b^2-4\,a\,c}+6\,a\,b^2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (a\,\left (\frac {5\,b^2\,c}{2}-\frac {3\,b\,c\,\sqrt {b^2-4\,a\,c}}{2}\right )-\frac {b^4}{2}+\frac {b^3\,\sqrt {b^2-4\,a\,c}}{2}-2\,a^2\,c^2\right )}{4\,a^4\,c-a^3\,b^2}-\frac {\frac {1}{2\,a}-\frac {b\,x}{a^2}}{x^2}-\frac {\ln \relax (x)\,\left (a\,c-b^2\right )}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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