Optimal. Leaf size=70 \[ -\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {x}{c} \]
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Rubi [A] time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1585, 703, 634, 618, 206, 628} \begin {gather*} -\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 703
Rule 1585
Rubi steps
\begin {align*} \int \frac {x^4}{a x^2+b x^3+c x^4} \, dx &=\int \frac {x^2}{a+b x+c x^2} \, dx\\ &=\frac {x}{c}+\frac {\int \frac {-a-b x}{a+b x+c x^2} \, dx}{c}\\ &=\frac {x}{c}-\frac {b \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^2}+\frac {\left (b^2-2 a c\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^2}\\ &=\frac {x}{c}-\frac {b \log \left (a+b x+c x^2\right )}{2 c^2}-\frac {\left (b^2-2 a c\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^2}\\ &=\frac {x}{c}-\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 73, normalized size = 1.04 \begin {gather*} \frac {\left (b^2-2 a c\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{c^2 \sqrt {4 a c-b^2}}-\frac {b \log \left (a+b x+c x^2\right )}{2 c^2}+\frac {x}{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^4}{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 = 0.80, size = 235, normalized size = 3.36 \begin {gather*} \left [-\frac {{\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} \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 ) - 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x + {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, -\frac {2 \, {\left (b^{2} - 2 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x + {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 67, normalized size = 0.96 \begin {gather*} \frac {x}{c} - \frac {b \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 101, normalized size = 1.44 \begin {gather*} -\frac {2 a \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {b^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}-\frac {b \ln \left (c \,x^{2}+b x +a \right )}{2 c^{2}}+\frac {x}{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 = 2.03, size = 172, normalized size = 2.46 \begin {gather*} \frac {x}{c}+\frac {b^3\,\ln \left (c\,x^2+b\,x+a\right )}{2\,\left (4\,a\,c^3-b^2\,c^2\right )}-\frac {2\,a\,\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\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )}{c^2\,\sqrt {4\,a\,c-b^2}}-\frac {2\,a\,b\,c\,\ln \left (c\,x^2+b\,x+a\right )}{4\,a\,c^3-b^2\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.62, size = 306, normalized size = 4.37 \begin {gather*} \left (- \frac {b}{2 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- a b - 4 a c^{2} \left (- \frac {b}{2 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + b^{2} c \left (- \frac {b}{2 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (- \frac {b}{2 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x + \frac {- a b - 4 a c^{2} \left (- \frac {b}{2 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + b^{2} c \left (- \frac {b}{2 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac {x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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