Optimal. Leaf size=257 \[ -\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{1920 c^4 x}+\frac {b x \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}-\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c} \]
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Rubi [A] time = 0.59, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1919, 1949, 12, 1914, 621, 206} \begin {gather*} -\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}+\frac {b x \left (7 b^2-12 a c\right ) \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 )}{256 c^{9/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 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^2 \sqrt {a x^2+b x^3+c x^4} \, dx &=\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}+\frac {\int \frac {x^3 \left (-3 a b-\frac {1}{2} \left (7 b^2-16 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{40 c}\\ &=-\frac {\left (7 b^2-16 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\int \frac {x^2 \left (-a \left (7 b^2-16 a c\right )-\frac {1}{4} b \left (35 b^2-116 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{120 c^2}\\ &=\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}-\frac {\left (7 b^2-16 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}+\frac {\int \frac {x \left (-\frac {1}{4} a b \left (35 b^2-116 a c\right )-\frac {1}{8} \left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{240 c^3}\\ &=\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac {\left (7 b^2-16 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\int -\frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) x}{16 \sqrt {a x^2+b x^3+c x^4}} \, dx}{240 c^4}\\ &=\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac {\left (7 b^2-16 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}+\frac {\left (b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{256 c^4}\\ &=\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac {\left (7 b^2-16 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}+\frac {\left (b \left (7 b^2-12 a c\right ) \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}{256 c^4 \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac {\left (7 b^2-16 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}+\frac {\left (b \left (7 b^2-12 a c\right ) \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 )}{128 c^4 \sqrt {a x^2+b x^3+c x^4}}\\ &=\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac {\left (7 b^2-16 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}+\frac {b \left (7 b^2-12 a c\right ) \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 )}{256 c^{9/2} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 180, normalized size = 0.70 \begin {gather*} \frac {15 x \left (48 a^2 b c^2-40 a b^3 c+7 b^5\right ) \sqrt {a+x (b+c x)} \log \left (2 \sqrt {c} \sqrt {a+x (b+c x)}+b+2 c x\right )+2 \sqrt {c} x (a+x (b+c x)) \left (128 c^2 \left (-2 a^2+a c x^2+3 c^2 x^4\right )+4 b^2 c \left (115 a-14 c x^2\right )+8 b c^2 x \left (6 c x^2-29 a\right )-105 b^4+70 b^3 c x\right )}{3840 c^{9/2} \sqrt {x^2 (a+x (b+c x))}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.84, size = 211, normalized size = 0.82 \begin {gather*} \frac {\log (x) \left (48 a^2 b c^2-40 a b^3 c+7 b^5\right )}{256 c^{9/2}}+\frac {\left (-48 a^2 b c^2+40 a b^3 c-7 b^5\right ) \log \left (-2 c^{9/2} \sqrt {a x^2+b x^3+c x^4}+b c^4 x+2 c^5 x^2\right )}{256 c^{9/2}}+\frac {\sqrt {a x^2+b x^3+c x^4} \left (-256 a^2 c^2+460 a b^2 c-232 a b c^2 x+128 a c^3 x^2-105 b^4+70 b^3 c x-56 b^2 c^2 x^2+48 b c^3 x^3+384 c^4 x^4\right )}{1920 c^4 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.23, size = 390, normalized size = 1.52 \begin {gather*} \left [\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b 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 (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \, {\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{7680 \, c^{5} x}, -\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b 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 (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \, {\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{3840 \, c^{5} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 283, normalized size = 1.10 \begin {gather*} \frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x \mathrm {sgn}\relax (x) + \frac {b \mathrm {sgn}\relax (x)}{c}\right )} x - \frac {7 \, b^{2} c^{2} \mathrm {sgn}\relax (x) - 16 \, a c^{3} \mathrm {sgn}\relax (x)}{c^{4}}\right )} x + \frac {35 \, b^{3} c \mathrm {sgn}\relax (x) - 116 \, a b c^{2} \mathrm {sgn}\relax (x)}{c^{4}}\right )} x - \frac {105 \, b^{4} \mathrm {sgn}\relax (x) - 460 \, a b^{2} c \mathrm {sgn}\relax (x) + 256 \, a^{2} c^{2} \mathrm {sgn}\relax (x)}{c^{4}}\right )} - \frac {{\left (7 \, b^{5} \mathrm {sgn}\relax (x) - 40 \, a b^{3} c \mathrm {sgn}\relax (x) + 48 \, a^{2} b 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 )}{256 \, c^{\frac {9}{2}}} + \frac {{\left (105 \, b^{5} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 600 \, a b^{3} c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 720 \, a^{2} b c^{2} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 210 \, \sqrt {a} b^{4} \sqrt {c} - 920 \, a^{\frac {3}{2}} b^{2} c^{\frac {3}{2}} + 512 \, a^{\frac {5}{2}} c^{\frac {5}{2}}\right )} \mathrm {sgn}\relax (x)}{3840 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 310, normalized size = 1.21 \begin {gather*} \frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (720 a^{2} b \,c^{3} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-600 a \,b^{3} c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+105 b^{5} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+720 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{\frac {7}{2}} x -420 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{\frac {5}{2}} x +768 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {9}{2}} x^{2}+360 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{\frac {5}{2}}-210 \sqrt {c \,x^{2}+b x +a}\, b^{4} c^{\frac {3}{2}}-672 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b \,c^{\frac {7}{2}} x -512 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,c^{\frac {7}{2}}+560 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} c^{\frac {5}{2}}\right )}{3840 \sqrt {c \,x^{2}+b x +a}\, c^{\frac {11}{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^{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 = -0.00 \begin {gather*} \int 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 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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