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 = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1585, 634, 618, 206, 628} \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 206
Rule 618
Rule 628
Rule 634
Rule 1585
Rubi steps
\begin {align*} \int \frac {x^3}{a x^2+b x^3+c x^4} \, 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 \operatorname {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 = 0.03, size = 57, normalized size = 1.02 \begin {gather*} \frac {\log (a+x (b+c x))-\frac {2 b \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}}{2 c} \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^3}{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 = 0.76, 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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giac [A] time = 0.42, 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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maple [A] time = 0.00, size = 56, 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}}\, c}+\frac {\ln \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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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 = 0.13, 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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sympy [B] time = 0.33, 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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