Optimal. Leaf size=103 \[ \frac {\sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2} \sqrt {a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.08, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1917, 1914, 621, 206} \begin {gather*} \frac {\sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {b 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 )}{2 c^{3/2} \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
Rule 1917
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a x^2+b x^3+c x^4}} \, dx &=\frac {\sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {b \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{2 c}\\ &=\frac {\sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {\left (b x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {\sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {\left (b 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 )}{c \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {\sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {b 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 )}{2 c^{3/2} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 89, normalized size = 0.86 \begin {gather*} \frac {x \left (2 \sqrt {c} (a+x (b+c x))-b \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{2 c^{3/2} \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 91, normalized size = 0.88 \begin {gather*} \frac {b \log \left (-2 c^{3/2} \sqrt {a x^2+b x^3+c x^4}+b c x+2 c^2 x^2\right )}{2 c^{3/2}}+\frac {\sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {b \log (x)}{2 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 188, normalized size = 1.83 \begin {gather*} \left [\frac {b \sqrt {c} x \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 ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} c}{4 \, c^{2} x}, \frac {b \sqrt {-c} x \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 ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} c}{2 \, c^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 108, normalized size = 1.05 \begin {gather*} \frac {b \arctan \left (\frac {\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}} - \frac {\sqrt {a}}{x}}{\sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {b {\left (\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}} - \frac {\sqrt {a}}{x}\right )} - 2 \, \sqrt {a} c}{{\left ({\left (\sqrt {c + \frac {b}{x} + \frac {a}{x^{2}}} - \frac {\sqrt {a}}{x}\right )}^{2} - c\right )} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 88, normalized size = 0.85 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, \left (-b c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+2 \sqrt {c \,x^{2}+b x +a}\, c^{\frac {3}{2}}\right ) x}{2 \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, c^{\frac {5}{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}}{\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^2}{\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^{2}}{\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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