Optimal. Leaf size=36 \[ -\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Rubi [A] time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1585, 1107, 618, 206} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1107
Rule 1585
Rubi steps
\begin {align*} \int \frac {x^2}{a x+b x^3+c x^5} \, dx &=\int \frac {x}{a+b x^2+c x^4} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 1.08 \begin {gather*} \frac {\tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}} \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^2}{a x+b x^3+c x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.23, size = 129, normalized size = 3.58 \begin {gather*} \left [\frac {\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 )}{2 \, \sqrt {b^{2} - 4 \, a c}}, -\frac {\sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{b^{2} - 4 \, a c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 35, normalized size = 0.97 \begin {gather*} \frac {\arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 1.00 \begin {gather*} \frac {\arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{c x^{5} + b x^{3} + a x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 41, normalized size = 1.14 \begin {gather*} \frac {\mathrm {atan}\left (\frac {2\,a\,c\,x^2+a\,b}{a\,\sqrt {4\,a\,c-b^2}}\right )}{\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 = 0.50, size = 131, normalized size = 3.64 \begin {gather*} - \frac {\sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x^{2} + \frac {- 4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} + b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 c} \right )}}{2} + \frac {\sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x^{2} + \frac {4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} - b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 c} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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