Integrand size = 21, antiderivative size = 408 \[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=-\frac {\sqrt {a+b x^2-c x^4}}{a x}+\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} a \sqrt {c} \sqrt {a+b x^2-c x^4}}-\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} a \sqrt {c} \sqrt {a+b x^2-c x^4}} \]
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Time = 0.26 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1137, 12, 1154, 507, 435, 430} \[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=-\frac {\left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} a \sqrt {c} \sqrt {a+b x^2-c x^4}}+\frac {\left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} a \sqrt {c} \sqrt {a+b x^2-c x^4}}-\frac {\sqrt {a+b x^2-c x^4}}{a x} \]
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Rule 12
Rule 430
Rule 435
Rule 507
Rule 1137
Rule 1154
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x^2-c x^4}}{a x}-\frac {\int \frac {c x^2}{\sqrt {a+b x^2-c x^4}} \, dx}{a} \\ & = -\frac {\sqrt {a+b x^2-c x^4}}{a x}-\frac {c \int \frac {x^2}{\sqrt {a+b x^2-c x^4}} \, dx}{a} \\ & = -\frac {\sqrt {a+b x^2-c x^4}}{a x}-\frac {\left (c \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}\right ) \int \frac {x^2}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx}{a \sqrt {a+b x^2-c x^4}} \\ & = -\frac {\sqrt {a+b x^2-c x^4}}{a x}-\frac {\left (\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}\right ) \int \frac {1}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx}{2 a \sqrt {a+b x^2-c x^4}}+\frac {\left (\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}\right ) \int \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}}}{\sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}}} \, dx}{2 a \sqrt {a+b x^2-c x^4}} \\ & = -\frac {\sqrt {a+b x^2-c x^4}}{a x}+\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} a \sqrt {c} \sqrt {a+b x^2-c x^4}}-\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{2 \sqrt {2} a \sqrt {c} \sqrt {a+b x^2-c x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.29 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\frac {-\frac {4 a}{x}-4 b x+4 c x^3+\frac {i \left (-b+\sqrt {b^2+4 a c}\right ) \sqrt {2+\frac {4 c x^2}{-b+\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \left (E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )\right )}{\sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}}}}{4 a \sqrt {a+b x^2-c x^4}} \]
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Time = 0.68 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.59
method | result | size |
default | \(-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{a x}+\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\) | \(241\) |
risch | \(-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{a x}+\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\) | \(241\) |
elliptic | \(-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{a x}+\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4-\frac {2 b \left (b +\sqrt {4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right )}\) | \(241\) |
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Time = 0.09 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (a^{\frac {3}{2}} x \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - \sqrt {a} b x\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b^{2} + 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left (a^{\frac {3}{2}} x \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - \sqrt {a} b x\right )} \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,-\frac {a b \sqrt {\frac {b^{2} + 4 \, a c}{a^{2}}} + b^{2} + 2 \, a c}{2 \, a c}) + 2 \, \sqrt {-c x^{4} + b x^{2} + a} a}{2 \, a^{2} x} \]
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\[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\int \frac {1}{x^{2} \sqrt {a + b x^{2} - c x^{4}}}\, dx \]
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\[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + b x^{2} + a} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + b x^{2} + a} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 \sqrt {a+b x^2-c x^4}} \, dx=\int \frac {1}{x^2\,\sqrt {-c\,x^4+b\,x^2+a}} \,d x \]
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