Integrand size = 24, antiderivative size = 143 \[ \int \frac {x^3}{\sqrt {a x^2+b x^3+c x^4}} \, dx=\frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\left (3 b^2-4 a c\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2} \sqrt {a x^2+b x^3+c x^4}} \]
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Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1942, 1963, 12, 1928, 635, 212} \[ \int \frac {x^3}{\sqrt {a x^2+b x^3+c x^4}} \, dx=\frac {x \left (3 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2} \sqrt {a x^2+b x^3+c x^4}}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\sqrt {a x^2+b x^3+c x^4}}{2 c} \]
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Rule 12
Rule 212
Rule 635
Rule 1928
Rule 1942
Rule 1963
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\int \frac {x \left (a+\frac {3 b x}{2}\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{2 c} \\ & = \frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\int \frac {\left (3 b^2-4 a c\right ) x}{4 \sqrt {a x^2+b x^3+c x^4}} \, dx}{2 c^2} \\ & = \frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\left (3 b^2-4 a c\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 c^2} \\ & = \frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\left (\left (3 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^2 \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\left (\left (3 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 )}{4 c^2 \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {\sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {3 b \sqrt {a x^2+b x^3+c x^4}}{4 c^2 x}+\frac {\left (3 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^{5/2} \sqrt {a x^2+b x^3+c x^4}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.75 \[ \int \frac {x^3}{\sqrt {a x^2+b x^3+c x^4}} \, dx=\frac {x \left (2 \sqrt {c} (-3 b+2 c x) (a+x (b+c x))+\left (-3 b^2+4 a c\right ) \sqrt {a+x (b+c x)} \log \left (c^2 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )\right )}{8 c^{5/2} \sqrt {x^2 (a+x (b+c x))}} \]
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Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {4 \sqrt {c \,x^{2}+b x +a}\, c^{\frac {3}{2}} x -6 b \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}-4 \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) a c +3 \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) b^{2}}{8 c^{\frac {5}{2}}}\) | \(100\) |
risch | \(-\frac {\left (-2 c x +3 b \right ) \left (c \,x^{2}+b x +a \right ) x}{4 c^{2} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}-\frac {\left (4 a c -3 b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) x \sqrt {c \,x^{2}+b x +a}}{8 c^{\frac {5}{2}} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}\) | \(111\) |
default | \(\frac {x \sqrt {c \,x^{2}+b x +a}\, \left (4 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, x -6 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b -4 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,c^{2}+3 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) b^{2} c \right )}{8 \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, c^{\frac {7}{2}}}\) | \(144\) |
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Time = 0.29 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.58 \[ \int \frac {x^3}{\sqrt {a x^2+b x^3+c x^4}} \, dx=\left [-\frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x - 3 \, b c\right )}}{16 \, c^{3} x}, -\frac {{\left (3 \, b^{2} - 4 \, a c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x - 3 \, b c\right )}}{8 \, c^{3} x}\right ] \]
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\[ \int \frac {x^3}{\sqrt {a x^2+b x^3+c x^4}} \, dx=\int \frac {x^{3}}{\sqrt {x^{2} \left (a + b x + c x^{2}\right )}}\, dx \]
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\[ \int \frac {x^3}{\sqrt {a x^2+b x^3+c x^4}} \, dx=\int { \frac {x^{3}}{\sqrt {c x^{4} + b x^{3} + a x^{2}}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.94 \[ \int \frac {x^3}{\sqrt {a x^2+b x^3+c x^4}} \, dx=\frac {1}{4} \, \sqrt {c x^{2} + b x + a} {\left (\frac {2 \, x}{c \mathrm {sgn}\left (x\right )} - \frac {3 \, b}{c^{2} \mathrm {sgn}\left (x\right )}\right )} + \frac {{\left (3 \, b^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 4 \, a c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6 \, \sqrt {a} b \sqrt {c}\right )} \mathrm {sgn}\left (x\right )}{8 \, c^{\frac {5}{2}}} - \frac {{\left (3 \, b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x^3}{\sqrt {a x^2+b x^3+c x^4}} \, dx=\int \frac {x^3}{\sqrt {c\,x^4+b\,x^3+a\,x^2}} \,d x \]
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