Integrand size = 24, antiderivative size = 155 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^5} \, dx=-\frac {\sqrt {a x^2+b x^3+c x^4}}{3 x^4}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{12 a x^3}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{16 a^{5/2}} \]
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Time = 0.15 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1934, 1965, 12, 1918, 212} \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^5} \, dx=-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{16 a^{5/2}}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{12 a x^3}-\frac {\sqrt {a x^2+b x^3+c x^4}}{3 x^4} \]
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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}}{3 x^4}+\frac {1}{6} \int \frac {b+2 c x}{x^2 \sqrt {a x^2+b x^3+c x^4}} \, dx \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{3 x^4}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{12 a x^3}-\frac {\int \frac {\frac {1}{2} \left (3 b^2-8 a c\right )+b c x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{12 a} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{3 x^4}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{12 a x^3}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^2 x^2}+\frac {\int \frac {3 b \left (b^2-4 a c\right )}{4 \sqrt {a x^2+b x^3+c x^4}} \, dx}{12 a^2} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{3 x^4}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{12 a x^3}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^2 x^2}+\frac {\left (b \left (b^2-4 a c\right )\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{16 a^2} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{3 x^4}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{12 a x^3}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac {\left (b \left (b^2-4 a c\right )\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 )}{8 a^2} \\ & = -\frac {\sqrt {a x^2+b x^3+c x^4}}{3 x^4}-\frac {b \sqrt {a x^2+b x^3+c x^4}}{12 a x^3}+\frac {\left (3 b^2-8 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^2 x^2}-\frac {b \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 )}{16 a^{5/2}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^5} \, dx=\frac {\sqrt {x^2 (a+x (b+c x))} \left (\sqrt {a} \sqrt {a+x (b+c x)} \left (-8 a^2+3 b^2 x^2-2 a x (b+4 c x)\right )+3 b \left (b^2-4 a c\right ) x^3 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )\right )}{24 a^{5/2} x^4 \sqrt {a+x (b+c x)}} \]
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Time = 0.15 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.74
method | result | size |
pseudoelliptic | \(\frac {b \,x^{3} \left (a c -\frac {b^{2}}{4}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )+\left (-\frac {x \left (4 c x +b \right ) a^{\frac {3}{2}}}{3}+\frac {\sqrt {a}\, b^{2} x^{2}}{2}-\frac {4 a^{\frac {5}{2}}}{3}\right ) \sqrt {c \,x^{2}+b x +a}-\ln \left (2\right ) x^{3} b \left (a c -\frac {b^{2}}{4}\right )}{4 a^{\frac {5}{2}} x^{3}}\) | \(115\) |
risch | \(-\frac {\left (8 a c \,x^{2}-3 b^{2} x^{2}+2 a b x +8 a^{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{24 x^{4} a^{2}}+\frac {\left (4 a c -b^{2}\right ) b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{16 a^{\frac {5}{2}} x \sqrt {c \,x^{2}+b x +a}}\) | \(128\) |
default | \(\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (12 c \,a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b \,x^{3}+6 c \sqrt {c \,x^{2}+b x +a}\, b^{2} x^{4}-12 c \sqrt {c \,x^{2}+b x +a}\, a b \,x^{3}-3 \sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{3} x^{3}-6 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} x^{2}+6 \sqrt {c \,x^{2}+b x +a}\, b^{3} x^{3}+12 a \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b x -16 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2}\right )}{48 x^{4} \sqrt {c \,x^{2}+b x +a}\, a^{3}}\) | \(234\) |
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Time = 0.33 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^5} \, dx=\left [-\frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {a} x^{4} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, a^{2} b x + 8 \, a^{3} - {\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{2}\right )}}{96 \, a^{3} x^{4}}, \frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, a^{2} b x + 8 \, a^{3} - {\left (3 \, a b^{2} - 8 \, a^{2} c\right )} x^{2}\right )}}{48 \, a^{3} x^{4}}\right ] \]
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\[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^5} \, dx=\int \frac {\sqrt {x^{2} \left (a + b x + c x^{2}\right )}}{x^{5}}\, dx \]
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\[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^5} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{3} + a x^{2}}}{x^{5}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^5} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {a x^2+b x^3+c x^4}}{x^5} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^3+a\,x^2}}{x^5} \,d x \]
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