Integrand size = 39, antiderivative size = 296 \[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {2 \sqrt {c} x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}-\frac {2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt {a+b x^2+c x^4}}+\frac {\left (b+2 \sqrt {a} \sqrt {c}-\sqrt {b^2-4 a c}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}} \]
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Time = 0.07 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1211, 1117, 1209} \[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\left (-\sqrt {b^2-4 a c}+2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt {a+b x^2+c x^4}}+\frac {2 \sqrt {c} x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2} \]
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Rule 1117
Rule 1209
Rule 1211
Rubi steps \begin{align*} \text {integral}& = -\left (\left (2 \sqrt {a} \sqrt {c}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx\right )+\left (b+2 \sqrt {a} \sqrt {c}-\sqrt {b^2-4 a c}\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx \\ & = \frac {2 \sqrt {c} x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}-\frac {2 \sqrt [4]{a} \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 )}{\sqrt {a+b x^2+c x^4}}+\frac {\left (b+2 \sqrt {a} \sqrt {c}-\sqrt {b^2-4 a c}\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}} 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 )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.23 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.63 \[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {a+b x^2+c x^4}} \, dx=-\frac {2 i \sqrt {2} a \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {a+b x^2+c x^4}} \]
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Time = 7.02 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.74
method | result | size |
default | \(\frac {b \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {c a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}-\frac {\sqrt {-4 a c +b^{2}}\, \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\) | \(515\) |
elliptic | \(-\frac {\left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right ) \sqrt {-\left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}\, \left (\frac {\left (4 a c -b^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) x^{2}}{a \left (4 a c -b^{2}\right )}}\, \sqrt {4+\frac {2 \left (\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) x^{2}}{a \left (4 a c -b^{2}\right )}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}}{a \left (4 a c -b^{2}\right )}}}{2}, \frac {\sqrt {-4+\frac {2 \left (-4 a b c +b^{3}\right ) \left (\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right )}{a \left (4 a c -b^{2}\right ) \left (-4 a \,c^{2}+b^{2} c \right )}}}{2}\right )}{4 \sqrt {\frac {\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}}{a \left (4 a c -b^{2}\right )}}\, \sqrt {-4 a \,c^{2} x^{4}+b^{2} c \,x^{4}-4 a b c \,x^{2}+b^{3} x^{2}-4 c \,a^{2}+b^{2} a}}+\frac {b \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 )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {c 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 )}{\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 )}{2 \sqrt {-\left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}\, c \,x^{2}+4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a c -\sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2}+\sqrt {-\left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}\, b}\) | \(807\) |
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Time = 0.11 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.09 \[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} {\left (a c x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - a b x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left (\sqrt {b^{2} - 4 \, a c} b x - {\left (2 \, a b + b^{2}\right )} x + {\left ({\left (2 \, a - b\right )} c x + \sqrt {b^{2} - 4 \, a c} c x\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 4 \, \sqrt {c x^{4} + b x^{2} + a} a c}{2 \, a c x} \]
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\[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {b + 2 c x^{2} - \sqrt {- 4 a c + b^{2}}}{\sqrt {a + b x^{2} + c x^{4}}}\, dx \]
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\[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {2 \, c x^{2} + b - \sqrt {b^{2} - 4 \, a c}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \]
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\[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {2 \, c x^{2} + b - \sqrt {b^{2} - 4 \, a c}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {b+2\,c\,x^2-\sqrt {b^2-4\,a\,c}}{\sqrt {c\,x^4+b\,x^2+a}} \,d x \]
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