Integrand size = 24, antiderivative size = 173 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=-\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}-\frac {b x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {\sqrt {c} x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}} \]
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Time = 0.08 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1934, 1947, 857, 635, 212, 738} \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=-\frac {b x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {\sqrt {c} x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}}-\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2} \]
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Rule 212
Rule 635
Rule 738
Rule 857
Rule 1934
Rule 1947
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}+\frac {1}{2} \int \frac {b+2 c x}{\sqrt {a x^2+b x^3+c x^4}} \, dx \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}+\frac {\left (x \sqrt {a+b x+c x^2}\right ) \int \frac {b+2 c x}{x \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {a x^2+b x^3+c x^4}} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}+\frac {\left (b x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (c x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{\sqrt {a x^2+b x^3+c x^4}} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}-\frac {\left (b x \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}}+\frac {\left (2 c 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 )}{\sqrt {a x^2+b x^3+c x^4}} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{x^2}-\frac {b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}+\frac {\sqrt {c} 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 )}{\sqrt {a x^2+b x^3+c x^4}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=\frac {\sqrt {a+x (b+c x)} \left (b x \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-\sqrt {a} \left (\sqrt {a+x (b+c x)}+\sqrt {c} x \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )\right )}{\sqrt {a} \sqrt {x^2 (a+x (b+c x))}} \]
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Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.58
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {c}\, \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) x \sqrt {a}+b x \ln \left (2\right )-b x \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{2 x \sqrt {a}}\) | \(101\) |
risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{x^{2}}+\frac {\left (\sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 \sqrt {a}}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{x \sqrt {c \,x^{2}+b x +a}}\) | \(120\) |
default | \(\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (2 x^{2} \sqrt {c \,x^{2}+b x +a}\, c^{\frac {5}{2}}-c^{\frac {3}{2}} \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b x -2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {3}{2}}+2 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b x +2 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,c^{2} x \right )}{2 x^{2} \sqrt {c \,x^{2}+b x +a}\, a \,c^{\frac {3}{2}}}\) | \(174\) |
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Time = 0.32 (sec) , antiderivative size = 653, normalized size of antiderivative = 3.77 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=\left [\frac {2 \, a \sqrt {c} x^{2} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + \sqrt {a} b x^{2} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{4 \, a x^{2}}, -\frac {4 \, a \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - \sqrt {a} b x^{2} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{4 \, a x^{2}}, \frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + a \sqrt {c} x^{2} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{2 \, a x^{2}}, \frac {\sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 2 \, a \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} a}{2 \, a x^{2}}\right ] \]
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\[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=\int \frac {\sqrt {x^{2} \left (a + b x + c x^{2}\right )}}{x^{3}}\, dx \]
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Timed out. \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{3} + a x^{2}}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^3} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^3+a\,x^2}}{x^3} \,d x \]
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