Optimal. Leaf size=119 \[ \frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 c x}-\frac {x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2} \sqrt {a x^2+b x^3+c x^4}} \]
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Rubi [A] time = 0.08, antiderivative size = 119, 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 = {1918, 1914, 621, 206} \begin {gather*} \frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 c x}-\frac {x \left (b^2-4 a c\right ) \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 )}{8 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 1918
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x} \, dx &=\frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 c x}-\frac {\left (b^2-4 a c\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 c}\\ &=\frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 c x}-\frac {\left (\left (b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 c x}-\frac {\left (\left (b^2-4 a c\right ) 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 )}{4 c \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 c x}-\frac {\left (b^2-4 a c\right ) 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 )}{8 c^{3/2} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 100, normalized size = 0.84 \begin {gather*} \frac {x \left (2 \sqrt {c} (b+2 c x) (a+x (b+c x))-\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \log \left (2 \sqrt {c} \sqrt {a+x (b+c x)}+b+2 c x\right )\right )}{8 c^{3/2} \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 116, normalized size = 0.97 \begin {gather*} \frac {\log (x) \left (4 a c-b^2\right )}{8 c^{3/2}}+\frac {\left (b^2-4 a c\right ) \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 )}{8 c^{3/2}}+\frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 c x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 220, normalized size = 1.85 \begin {gather*} \left [-\frac {{\left (b^{2} - 4 \, a c\right )} \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}} {\left (2 \, c^{2} x + b c\right )}}{16 \, c^{2} x}, \frac {{\left (b^{2} - 4 \, a c\right )} \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}} {\left (2 \, c^{2} x + b c\right )}}{8 \, c^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 125, normalized size = 1.05 \begin {gather*} \frac {1}{8} \, {\left (2 \, \sqrt {c x^{2} + b x + a} {\left (2 \, x + \frac {b}{c}\right )} + \frac {{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{c^{\frac {3}{2}}}\right )} \mathrm {sgn}\relax (x) - \frac {{\left (b^{2} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 4 \, a c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 2 \, \sqrt {a} b \sqrt {c}\right )} \mathrm {sgn}\relax (x)}{8 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 146, normalized size = 1.23 \begin {gather*} \frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (4 a \,c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-b^{2} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+4 \sqrt {c \,x^{2}+b x +a}\, c^{\frac {5}{2}} x +2 \sqrt {c \,x^{2}+b x +a}\, b \,c^{\frac {3}{2}}\right )}{8 \sqrt {c \,x^{2}+b x +a}\, c^{\frac {5}{2}} x} \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 {\sqrt {c x^{4} + b x^{3} + a x^{2}}}{x}\,{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 {\sqrt {c\,x^4+b\,x^3+a\,x^2}}{x} \,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 {\sqrt {x^{2} \left (a + b x + c x^{2}\right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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