Optimal. Leaf size=71 \[ \frac {x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} \sqrt {a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1914, 621, 206} \begin {gather*} \frac {x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} \sqrt {a x^2+b x^3+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 1914
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx &=\frac {\left (x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{\sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {\left (2 x \sqrt {a+b x+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 0.93 \begin {gather*} \frac {x \sqrt {a+b x+c x^2} \log \left (2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{\sqrt {c} \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 54, normalized size = 0.76 \begin {gather*} \frac {\log (x)}{\sqrt {c}}-\frac {\log \left (-2 \sqrt {c} \sqrt {a x^2+b x^3+c x^4}+b x+2 c x^2\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 129, normalized size = 1.82 \begin {gather*} \left [\frac {\log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right )}{2 \, \sqrt {c}}, -\frac {\sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right )}{c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 37, normalized size = 0.52 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}} - \frac {\sqrt {a}}{x}}{\sqrt {-c}}\right )}{\sqrt {-c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 0.92 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, x \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )}{\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \sqrt {c}} \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}{\sqrt {c x^{4} + b x^{3} + a x^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{\sqrt {c\,x^4+b\,x^3+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x^{2} \left (a + b x + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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