Integrand size = 24, antiderivative size = 257 \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac {\left (7 b^2-16 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}+\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2} \sqrt {a x^2+b x^3+c x^4}} \]
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Time = 0.37 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1933, 1963, 12, 1928, 635, 212} \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{1920 c^4 x}+\frac {b x \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}-\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c} \]
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Rule 12
Rule 212
Rule 635
Rule 1928
Rule 1933
Rule 1963
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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*}
Time = 0.42 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.72 \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\frac {2 \sqrt {c} x (a+x (b+c x)) \left (-105 b^4+70 b^3 c x+4 b^2 c \left (115 a-14 c x^2\right )+8 b c^2 x \left (-29 a+6 c x^2\right )+128 c^2 \left (-2 a^2+a c x^2+3 c^2 x^4\right )\right )-15 \left (7 b^5-40 a b^3 c+48 a^2 b c^2\right ) x \sqrt {a+x (b+c x)} \log \left (c^4 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{3840 c^{9/2} \sqrt {x^2 (a+x (b+c x))}} \]
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Time = 0.23 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.52
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (-\frac {45}{32} a^{2} b \,c^{2}+\frac {75}{64} a \,b^{3} c -\frac {105}{512} b^{5}\right ) \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right )+\sqrt {c \,x^{2}+b x +a}\, \left (\left (\frac {7}{32} b^{2} x^{2}+\frac {29}{32} a b x +a^{2}\right ) c^{\frac {5}{2}}-\frac {115 \left (\frac {7 b x}{46}+a \right ) b^{2} c^{\frac {3}{2}}}{64}-\frac {\left (\frac {3 b x}{8}+a \right ) x^{2} c^{\frac {7}{2}}}{2}-\frac {3 c^{\frac {9}{2}} x^{4}}{2}+\frac {105 \sqrt {c}\, b^{4}}{256}\right )\right )}{15 c^{\frac {9}{2}}}\) | \(133\) |
risch | \(-\frac {\left (-384 c^{4} x^{4}-48 b \,c^{3} x^{3}-128 a \,c^{3} x^{2}+56 b^{2} c^{2} x^{2}+232 a b \,c^{2} x -70 b^{3} c x +256 a^{2} c^{2}-460 a \,b^{2} c +105 b^{4}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{1920 c^{4} x}+\frac {b \left (48 a^{2} c^{2}-40 a \,b^{2} c +7 b^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{256 c^{\frac {9}{2}} x \sqrt {c \,x^{2}+b x +a}}\) | \(182\) |
default | \(\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (768 x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {9}{2}}-672 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b x -512 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a +720 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a b x +560 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2}-420 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{3} x +360 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{2}-210 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{4}+720 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{2} b \,c^{3}-600 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,b^{3} c^{2}+105 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) b^{5} c \right )}{3840 x \sqrt {c \,x^{2}+b x +a}\, c^{\frac {11}{2}}}\) | \(310\) |
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Time = 0.29 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.52 \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\left [\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, {\left (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \, {\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{7680 \, c^{5} x}, -\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, {\left (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \, {\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{3840 \, c^{5} x}\right ] \]
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\[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\int x^{2} \sqrt {x^{2} \left (a + b x + c x^{2}\right )}\, dx \]
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\[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{3} + a x^{2}} x^{2} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.07 \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x \mathrm {sgn}\left (x\right ) + \frac {b \mathrm {sgn}\left (x\right )}{c}\right )} x - \frac {7 \, b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 16 \, a c^{3} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} x + \frac {35 \, b^{3} c \mathrm {sgn}\left (x\right ) - 116 \, a b c^{2} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} x - \frac {105 \, b^{4} \mathrm {sgn}\left (x\right ) - 460 \, a b^{2} c \mathrm {sgn}\left (x\right ) + 256 \, a^{2} c^{2} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} - \frac {{\left (7 \, b^{5} \mathrm {sgn}\left (x\right ) - 40 \, a b^{3} c \mathrm {sgn}\left (x\right ) + 48 \, a^{2} b c^{2} \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {9}{2}}} + \frac {{\left (105 \, b^{5} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 600 \, a b^{3} c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 720 \, a^{2} b c^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 210 \, \sqrt {a} b^{4} \sqrt {c} - 920 \, a^{\frac {3}{2}} b^{2} c^{\frac {3}{2}} + 512 \, a^{\frac {5}{2}} c^{\frac {5}{2}}\right )} \mathrm {sgn}\left (x\right )}{3840 \, c^{\frac {9}{2}}} \]
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Timed out. \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\int x^2\,\sqrt {c\,x^4+b\,x^3+a\,x^2} \,d x \]
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