Integrand size = 24, antiderivative size = 114 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^4} \, dx=-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1934, 1965, 12, 1918, 212} \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^4} \, dx=\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{3/2}}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}-\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3} \]
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Rule 12
Rule 212
Rule 1918
Rule 1934
Rule 1965
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}+\frac {1}{4} \int \frac {b+2 c x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}-\frac {\int \frac {b^2-4 a c}{2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{4 a} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 a} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}+\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x (2 a+b x)}{\sqrt {a x^2+b x^3+c x^4}}\right )}{4 a} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{2 x^3}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{4 a x^2}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{3/2}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^4} \, dx=-\frac {\sqrt {x^2 (a+x (b+c x))} \left (\sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)}+\left (b^2-4 a c\right ) x^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )\right )}{4 a^{3/2} x^3 \sqrt {a+x (b+c x)}} \]
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Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (b x +2 a \right )}{4 x^{2} a}+\frac {\left (4 a c -b^{2}\right ) \left (\ln \left (2\right )-\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )\right )}{8 a^{\frac {3}{2}}}\) | \(81\) |
risch | \(-\frac {\left (b x +2 a \right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{4 x^{3} a}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{8 a^{\frac {3}{2}} x \sqrt {c \,x^{2}+b x +a}}\) | \(108\) |
default | \(-\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (4 c \,a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) x^{2}+2 c \sqrt {c \,x^{2}+b x +a}\, b \,x^{3}-4 c \sqrt {c \,x^{2}+b x +a}\, a \,x^{2}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{2} x^{2}-2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b x +2 \sqrt {c \,x^{2}+b x +a}\, b^{2} x^{2}+4 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \right )}{8 x^{3} \sqrt {c \,x^{2}+b x +a}\, a^{2}}\) | \(207\) |
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Time = 0.30 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^4} \, dx=\left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {a} x^{3} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (a b x + 2 \, a^{2}\right )}}{16 \, a^{2} x^{3}}, -\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (a b x + 2 \, a^{2}\right )}}{8 \, a^{2} x^{3}}\right ] \]
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\[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^4} \, dx=\int \frac {\sqrt {x^{2} \left (a + b x + c x^{2}\right )}}{x^{4}}\, dx \]
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\[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^4} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{3} + a x^{2}}}{x^{4}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^4} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^3+a\,x^2}}{x^4} \,d x \]
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