Integrand size = 22, antiderivative size = 77 \[ \int \frac {1}{x^3 \sqrt {-a+b x^2+c x^4}} \, dx=\frac {\sqrt {-a+b x^2+c x^4}}{2 a x^2}-\frac {b \arctan \left (\frac {2 a-b x^2}{2 \sqrt {a} \sqrt {-a+b x^2+c x^4}}\right )}{4 a^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1128, 744, 738, 210} \[ \int \frac {1}{x^3 \sqrt {-a+b x^2+c x^4}} \, dx=\frac {\sqrt {-a+b x^2+c x^4}}{2 a x^2}-\frac {b \arctan \left (\frac {2 a-b x^2}{2 \sqrt {a} \sqrt {-a+b x^2+c x^4}}\right )}{4 a^{3/2}} \]
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Rule 210
Rule 738
Rule 744
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {-a+b x+c x^2}} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {-a+b x^2+c x^4}}{2 a x^2}+\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {-a+b x+c x^2}} \, dx,x,x^2\right )}{4 a} \\ & = \frac {\sqrt {-a+b x^2+c x^4}}{2 a x^2}-\frac {b \text {Subst}\left (\int \frac {1}{-4 a-x^2} \, dx,x,\frac {-2 a+b x^2}{\sqrt {-a+b x^2+c x^4}}\right )}{2 a} \\ & = \frac {\sqrt {-a+b x^2+c x^4}}{2 a x^2}-\frac {b \tan ^{-1}\left (\frac {2 a-b x^2}{2 \sqrt {a} \sqrt {-a+b x^2+c x^4}}\right )}{4 a^{3/2}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^3 \sqrt {-a+b x^2+c x^4}} \, dx=\frac {\sqrt {-a+b x^2+c x^4}}{2 a x^2}-\frac {b \arctan \left (\frac {\sqrt {c} x^2-\sqrt {-a+b x^2+c x^4}}{\sqrt {a}}\right )}{2 a^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}-a}}{2 a \,x^{2}}-\frac {b \ln \left (\frac {-2 a +b \,x^{2}+2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{4 a \sqrt {-a}}\) | \(74\) |
elliptic | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}-a}}{2 a \,x^{2}}-\frac {b \ln \left (\frac {-2 a +b \,x^{2}+2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{4 a \sqrt {-a}}\) | \(74\) |
pseudoelliptic | \(\frac {b \ln \left (\frac {-2 a +b \,x^{2}+2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right ) x^{2}-2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{4 x^{2} \left (-a \right )^{\frac {3}{2}}}\) | \(77\) |
risch | \(-\frac {-c \,x^{4}-b \,x^{2}+a}{2 a \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}-a}}-\frac {b \ln \left (\frac {-2 a +b \,x^{2}+2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{4 a \sqrt {-a}}\) | \(88\) |
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none
Time = 0.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.44 \[ \int \frac {1}{x^3 \sqrt {-a+b x^2+c x^4}} \, dx=\left [-\frac {\sqrt {-a} b x^{2} \log \left (\frac {{\left (b^{2} - 4 \, a c\right )} x^{4} - 8 \, a b x^{2} - 4 \, \sqrt {c x^{4} + b x^{2} - a} {\left (b x^{2} - 2 \, a\right )} \sqrt {-a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, \sqrt {c x^{4} + b x^{2} - a} a}{8 \, a^{2} x^{2}}, \frac {\sqrt {a} b x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} - a} {\left (b x^{2} - 2 \, a\right )} \sqrt {a}}{2 \, {\left (a c x^{4} + a b x^{2} - a^{2}\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} - a} a}{4 \, a^{2} x^{2}}\right ] \]
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\[ \int \frac {1}{x^3 \sqrt {-a+b x^2+c x^4}} \, dx=\int \frac {1}{x^{3} \sqrt {- a + b x^{2} + c x^{4}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^3 \sqrt {-a+b x^2+c x^4}} \, dx=-\frac {b \arcsin \left (-\frac {b}{\sqrt {b^{2} + 4 \, a c}} + \frac {2 \, a}{\sqrt {b^{2} + 4 \, a c} x^{2}}\right )}{4 \, a^{\frac {3}{2}}} + \frac {\sqrt {c x^{4} + b x^{2} - a}}{2 \, a x^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x^3 \sqrt {-a+b x^2+c x^4}} \, dx=\frac {b \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}}{\sqrt {a}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )} b - 2 \, a \sqrt {c}}{2 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{2} + a\right )} a} \]
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Time = 13.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^3 \sqrt {-a+b x^2+c x^4}} \, dx=\frac {\sqrt {c\,x^4+b\,x^2-a}}{2\,a\,x^2}-\frac {b\,\mathrm {atanh}\left (\frac {a-\frac {b\,x^2}{2}}{\sqrt {-a}\,\sqrt {c\,x^4+b\,x^2-a}}\right )}{4\,{\left (-a\right )}^{3/2}} \]
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