Optimal. Leaf size=205 \[ -\frac {x \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 c^3 x}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 c^2}+\frac {x (b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{24 c} \]
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Rubi [A] time = 0.37, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1919, 1949, 12, 1914, 621, 206} \begin {gather*} \frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 c^3 x}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 c^2}-\frac {x \left (b^2-4 a c\right ) \left (5 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 )}{128 c^{7/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {x (b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{24 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 621
Rule 1914
Rule 1919
Rule 1949
Rubi steps
\begin {align*} \int x \sqrt {a x^2+b x^3+c x^4} \, dx &=\frac {x (b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{24 c}+\frac {\int \frac {x^2 \left (-2 a b-\frac {1}{2} \left (5 b^2-12 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{24 c}\\ &=-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 c^2}+\frac {x (b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\int \frac {x \left (-\frac {1}{2} a \left (5 b^2-12 a c\right )-\frac {1}{4} b \left (15 b^2-52 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{48 c^2}\\ &=-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 c^2}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 c^3 x}+\frac {x (b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{24 c}+\frac {\int -\frac {3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) x}{8 \sqrt {a x^2+b x^3+c x^4}} \, dx}{48 c^3}\\ &=-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 c^2}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 c^3 x}+\frac {x (b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{128 c^3}\\ &=-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 c^2}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 c^3 x}+\frac {x (b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\left (\left (b^2-4 a c\right ) \left (5 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}{128 c^3 \sqrt {a x^2+b x^3+c x^4}}\\ &=-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 c^2}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 c^3 x}+\frac {x (b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\left (\left (b^2-4 a c\right ) \left (5 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 )}{64 c^3 \sqrt {a x^2+b x^3+c x^4}}\\ &=-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{96 c^2}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{192 c^3 x}+\frac {x (b+6 c x) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\left (b^2-4 a c\right ) \left (5 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 )}{128 c^{7/2} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 150, normalized size = 0.73 \begin {gather*} \frac {2 \sqrt {c} x (a+x (b+c x)) \left (b \left (8 c^2 x^2-52 a c\right )+24 c^2 x \left (a+2 c x^2\right )+15 b^3-10 b^2 c x\right )-3 x \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \sqrt {a+x (b+c x)} \log \left (2 \sqrt {c} \sqrt {a+x (b+c x)}+b+2 c x\right )}{384 c^{7/2} \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.58, size = 173, normalized size = 0.84 \begin {gather*} \frac {\left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \log \left (-2 \sqrt {c} \sqrt {a x^2+b x^3+c x^4}+b x+2 c x^2\right )}{128 c^{7/2}}+\frac {\log (x) \left (-16 a^2 c^2+24 a b^2 c-5 b^4\right )}{128 c^{7/2}}+\frac {\sqrt {a x^2+b x^3+c x^4} \left (-52 a b c+24 a c^2 x+15 b^3-10 b^2 c x+8 b c^2 x^2+48 c^3 x^3\right )}{192 c^3 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 326, normalized size = 1.59 \begin {gather*} \left [\frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (48 \, c^{4} x^{3} + 8 \, b c^{3} x^{2} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \, {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{768 \, c^{4} x}, \frac {3 \, {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (48 \, c^{4} x^{3} + 8 \, b c^{3} x^{2} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \, {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{384 \, c^{4} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.75, size = 230, normalized size = 1.12 \begin {gather*} \frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, x \mathrm {sgn}\relax (x) + \frac {b \mathrm {sgn}\relax (x)}{c}\right )} x - \frac {5 \, b^{2} c \mathrm {sgn}\relax (x) - 12 \, a c^{2} \mathrm {sgn}\relax (x)}{c^{3}}\right )} x + \frac {15 \, b^{3} \mathrm {sgn}\relax (x) - 52 \, a b c \mathrm {sgn}\relax (x)}{c^{3}}\right )} + \frac {{\left (5 \, b^{4} \mathrm {sgn}\relax (x) - 24 \, a b^{2} c \mathrm {sgn}\relax (x) + 16 \, a^{2} c^{2} \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 )}{128 \, c^{\frac {7}{2}}} - \frac {{\left (15 \, b^{4} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 72 \, a b^{2} c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 48 \, a^{2} c^{2} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 30 \, \sqrt {a} b^{3} \sqrt {c} - 104 \, a^{\frac {3}{2}} b c^{\frac {3}{2}}\right )} \mathrm {sgn}\relax (x)}{384 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 265, normalized size = 1.29 \begin {gather*} \frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (-48 a^{2} c^{3} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+72 a \,b^{2} c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-15 b^{4} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-48 \sqrt {c \,x^{2}+b x +a}\, a \,c^{\frac {7}{2}} x +60 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{\frac {5}{2}} x -24 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{\frac {5}{2}}+30 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{\frac {3}{2}}+96 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {7}{2}} x -80 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b \,c^{\frac {5}{2}}\right )}{384 \sqrt {c \,x^{2}+b x +a}\, c^{\frac {9}{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 \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 = -0.00 \begin {gather*} \int 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 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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