Optimal. Leaf size=63 \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c} \]
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Rubi [A] time = 0.05, antiderivative size = 63, 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 = {1114, 634, 618, 206, 628} \begin {gather*} \frac {b \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^3}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=\frac {\log \left (a+b x^2+c x^4\right )}{4 c}+\frac {b \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c}\\ &=\frac {b \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c}}+\frac {\log \left (a+b x^2+c x^4\right )}{4 c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.98 \begin {gather*} \frac {\log \left (a+b x^2+c x^4\right )-\frac {2 b \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}}{4 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+b x^2+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.98, size = 197, normalized size = 3.13 \begin {gather*} \left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\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 = 0.57, size = 59, normalized size = 0.94 \begin {gather*} -\frac {b \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c} + \frac {\log \left (c x^{4} + b x^{2} + a\right )}{4 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 60, normalized size = 0.95 \begin {gather*} -\frac {b \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c}+\frac {\ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 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 = 4.26, size = 118, normalized size = 1.87 \begin {gather*} \frac {4\,a\,c\,\ln \left (c\,x^4+b\,x^2+a\right )}{16\,a\,c^2-4\,b^2\,c}-\frac {b^2\,\ln \left (c\,x^4+b\,x^2+a\right )}{16\,a\,c^2-4\,b^2\,c}-\frac {b\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x^2}{\sqrt {4\,a\,c-b^2}}\right )}{2\,c\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.03, size = 223, normalized size = 3.54 \begin {gather*} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac {1}{4 c}\right ) \log {\left (x^{2} + \frac {- 8 a c \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac {1}{4 c}\right ) + 2 a + 2 b^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac {1}{4 c}\right )}{b} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac {1}{4 c}\right ) \log {\left (x^{2} + \frac {- 8 a c \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac {1}{4 c}\right ) + 2 a + 2 b^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c \left (4 a c - b^{2}\right )} + \frac {1}{4 c}\right )}{b} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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