Integrand size = 20, antiderivative size = 341 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^4} \, dx=-\frac {\sqrt {a+b x^2+c x^4}}{3 x^3}-\frac {b \sqrt {a+b x^2+c x^4}}{3 a x}+\frac {b \sqrt {c} x \sqrt {a+b x^2+c x^4}}{3 a \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {b \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {\left (b+2 \sqrt {a} \sqrt {c}\right ) \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{3/4} \sqrt {a+b x^2+c x^4}} \]
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Time = 0.10 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1131, 1295, 1211, 1117, 1209} \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^4} \, dx=\frac {\sqrt [4]{c} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{3 a x}+\frac {b \sqrt {c} x \sqrt {a+b x^2+c x^4}}{3 a \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {a+b x^2+c x^4}}{3 x^3} \]
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Rule 1117
Rule 1131
Rule 1209
Rule 1211
Rule 1295
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x^2+c x^4}}{3 x^3}+\frac {1}{3} \int \frac {b+2 c x^2}{x^2 \sqrt {a+b x^2+c x^4}} \, dx \\ & = -\frac {\sqrt {a+b x^2+c x^4}}{3 x^3}-\frac {b \sqrt {a+b x^2+c x^4}}{3 a x}-\frac {\int \frac {-2 a c-b c x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{3 a} \\ & = -\frac {\sqrt {a+b x^2+c x^4}}{3 x^3}-\frac {b \sqrt {a+b x^2+c x^4}}{3 a x}-\frac {\left (b \sqrt {c}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{3 \sqrt {a}}+\frac {1}{3} \left (\left (\frac {b}{\sqrt {a}}+2 \sqrt {c}\right ) \sqrt {c}\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx \\ & = -\frac {\sqrt {a+b x^2+c x^4}}{3 x^3}-\frac {b \sqrt {a+b x^2+c x^4}}{3 a x}+\frac {b \sqrt {c} x \sqrt {a+b x^2+c x^4}}{3 a \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {b \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {\left (b+2 \sqrt {a} \sqrt {c}\right ) \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{3/4} \sqrt {a+b x^2+c x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.56 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^4} \, dx=\frac {-4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (a+b x^2\right ) \left (a+b x^2+c x^4\right )+i b \left (-b+\sqrt {b^2-4 a c}\right ) x^3 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right ) x^3 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{12 a \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x^3 \sqrt {a+b x^2+c x^4}} \]
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Time = 1.08 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.17
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b \,x^{2}+a \right )}{3 x^{3} a}+\frac {c \left (\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )}{3 a}\) | \(398\) |
default | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x^{3}}-\frac {b \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 a x}+\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(404\) |
elliptic | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x^{3}}-\frac {b \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 a x}+\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(404\) |
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Time = 0.08 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^4} \, dx=-\frac {\sqrt {\frac {1}{2}} {\left (a b x^{3} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b^{2} x^{3}\right )} \sqrt {a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + \sqrt {\frac {1}{2}} {\left ({\left (2 \, a^{2} - a b\right )} x^{3} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + {\left (2 \, a b + b^{2}\right )} x^{3}\right )} \sqrt {a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (a b x^{2} + a^{2}\right )}}{6 \, a^{2} x^{3}} \]
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\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^4} \, dx=\int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{4}}\, dx \]
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\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^4} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a}}{x^{4}} \,d x } \]
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\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^4} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^4} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^4} \,d x \]
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