Optimal. Leaf size=163 \[ \frac {b \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt {a+b x+c x^2}}-\frac {b (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{8 c^2 x}+\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x} \]
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Rubi [A] time = 0.06, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1903, 640, 612, 621, 206} \[ \frac {b \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt {a+b x+c x^2}}-\frac {b (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{8 c^2 x}+\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1903
Rubi steps
\begin {align*} \int \sqrt {a x^2+b x^3+c x^4} \, dx &=\frac {\sqrt {a x^2+b x^3+c x^4} \int x \sqrt {a+b x+c x^2} \, dx}{x \sqrt {a+b x+c x^2}}\\ &=\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x}-\frac {\left (b \sqrt {a x^2+b x^3+c x^4}\right ) \int \sqrt {a+b x+c x^2} \, dx}{2 c x \sqrt {a+b x+c x^2}}\\ &=-\frac {b (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{8 c^2 x}+\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x}+\frac {\left (b \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^2 x \sqrt {a+b x+c x^2}}\\ &=-\frac {b (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{8 c^2 x}+\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x}+\frac {\left (b \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}\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 )}{8 c^2 x \sqrt {a+b x+c x^2}}\\ &=-\frac {b (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{8 c^2 x}+\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x}+\frac {b \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 119, normalized size = 0.73 \[ \frac {2 \sqrt {c} x (a+x (b+c x)) \left (8 c \left (a+c x^2\right )-3 b^2+2 b c x\right )+3 b x \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{48 c^{5/2} \sqrt {x^2 (a+x (b+c x))}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 260, normalized size = 1.60 \[ \left [-\frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, {\left (8 \, c^{3} x^{2} + 2 \, b c^{2} x - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{96 \, c^{3} x}, -\frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, {\left (8 \, c^{3} x^{2} + 2 \, b c^{2} x - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{48 \, c^{3} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 166, normalized size = 1.02 \[ \frac {1}{24} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, x \mathrm {sgn}\relax (x) + \frac {b \mathrm {sgn}\relax (x)}{c}\right )} x - \frac {3 \, b^{2} \mathrm {sgn}\relax (x) - 8 \, a c \mathrm {sgn}\relax (x)}{c^{2}}\right )} - \frac {{\left (b^{3} \mathrm {sgn}\relax (x) - 4 \, a b c \mathrm {sgn}\relax (x)\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {5}{2}}} + \frac {{\left (3 \, b^{3} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 12 \, a b c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6 \, \sqrt {a} b^{2} \sqrt {c} - 16 \, a^{\frac {3}{2}} c^{\frac {3}{2}}\right )} \mathrm {sgn}\relax (x)}{48 \, c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 167, normalized size = 1.02 \[ \frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (-12 a b \,c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+3 b^{3} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-12 \sqrt {c \,x^{2}+b x +a}\, b \,c^{\frac {5}{2}} x -6 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{\frac {3}{2}}+16 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {5}{2}}\right )}{48 \sqrt {c \,x^{2}+b x +a}\, c^{\frac {7}{2}} x} \]
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 \[ \int \sqrt {c x^{4} + b x^{3} + a x^{2}}\,{d x} \]
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 \[ \int \sqrt {c\,x^4+b\,x^3+a\,x^2} \,d x \]
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 \[ \int \sqrt {a x^{2} + b x^{3} + c x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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