Integrand size = 30, antiderivative size = 239 \[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=-\frac {\sqrt {b+\sqrt {b^2+4 a c}} \arctan \left (\frac {\sqrt {b+\sqrt {b^2+4 a c}} x \left (b-\sqrt {b^2+4 a c}-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}+\frac {\sqrt {-b+\sqrt {b^2+4 a c}} \text {arctanh}\left (\frac {\sqrt {-b+\sqrt {b^2+4 a c}} x \left (b+\sqrt {b^2+4 a c}-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d} \]
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Time = 0.12 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2097} \[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=\frac {\sqrt {\sqrt {4 a c+b^2}-b} \text {arctanh}\left (\frac {x \sqrt {\sqrt {4 a c+b^2}-b} \left (\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}-\frac {\sqrt {\sqrt {4 a c+b^2}+b} \arctan \left (\frac {x \sqrt {\sqrt {4 a c+b^2}+b} \left (-\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d} \]
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Rule 2097
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {b+\sqrt {b^2+4 a c}} \tan ^{-1}\left (\frac {\sqrt {b+\sqrt {b^2+4 a c}} x \left (b-\sqrt {b^2+4 a c}-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}+\frac {\sqrt {-b+\sqrt {b^2+4 a c}} \tanh ^{-1}\left (\frac {\sqrt {-b+\sqrt {b^2+4 a c}} x \left (b+\sqrt {b^2+4 a c}-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.64 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=\frac {i \left (\sqrt {-b-2 i \sqrt {a} \sqrt {c}} \arctan \left (\frac {\sqrt {-b-2 i \sqrt {a} \sqrt {c}} x}{\sqrt {a+b x^2-c x^4}}\right )-\sqrt {-b+2 i \sqrt {a} \sqrt {c}} \arctan \left (\frac {\sqrt {-b+2 i \sqrt {a} \sqrt {c}} x}{\sqrt {a+b x^2-c x^4}}\right )\right )}{4 \sqrt {a} \sqrt {c} d} \]
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Time = 2.49 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.47
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {2 \sqrt {4 a c +b^{2}}+2 b}\, \left (b -\sqrt {4 a c +b^{2}}\right ) \left (\ln \left (\frac {-c \,x^{4}+\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2 \sqrt {4 a c +b^{2}}+2 b}\, x +\sqrt {4 a c +b^{2}}\, x^{2}+b \,x^{2}+a}{x^{2}}\right )-\ln \left (\frac {c \,x^{4}+\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2 \sqrt {4 a c +b^{2}}+2 b}\, x -\sqrt {4 a c +b^{2}}\, x^{2}-b \,x^{2}-a}{x^{2}}\right )\right ) \sqrt {2 \sqrt {4 a c +b^{2}}-2 b}-16 a c \left (\arctan \left (\frac {\sqrt {2 \sqrt {4 a c +b^{2}}+2 b}\, x -2 \sqrt {-c \,x^{4}+b \,x^{2}+a}}{x \sqrt {2 \sqrt {4 a c +b^{2}}-2 b}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {4 a c +b^{2}}+2 b}\, x +2 \sqrt {-c \,x^{4}+b \,x^{2}+a}}{x \sqrt {2 \sqrt {4 a c +b^{2}}-2 b}}\right )\right )}{32 \sqrt {2 \sqrt {4 a c +b^{2}}-2 b}\, d a c}\) | \(351\) |
elliptic | \(\frac {\left (\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 d a c}+\frac {b^{2} \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}-2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 d a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}-\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 d a c}-\frac {\left (4 a c +b^{2}\right ) \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}-2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 d a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}-\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 d a c}+\frac {b^{2} \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 d a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}+\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 d a c}-\frac {\left (4 a c +b^{2}\right ) \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 d a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right ) \sqrt {2}}{2}\) | \(757\) |
default | \(\frac {\left (\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 a c}+\frac {\left (b +\sqrt {4 a c +b^{2}}\right ) b \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}-2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}-\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 a c}-\frac {\left (b +\sqrt {4 a c +b^{2}}\right ) \sqrt {4 a c +b^{2}}\, \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}-2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}-\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 a c}+\frac {\left (b +\sqrt {4 a c +b^{2}}\right ) b \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}+\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 a c}-\frac {\left (b +\sqrt {4 a c +b^{2}}\right ) \sqrt {4 a c +b^{2}}\, \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right ) \sqrt {2}}{2 d}\) | \(784\) |
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Leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (187) = 374\).
Time = 1.68 (sec) , antiderivative size = 669, normalized size of antiderivative = 2.80 \[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=-\frac {1}{8} \, \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} + \sqrt {-c x^{4} + b x^{2} + a} x^{2} + {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} - a d x\right )} \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} + a}\right ) + \frac {1}{8} \, \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} + \sqrt {-c x^{4} + b x^{2} + a} x^{2} - {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} - a d x\right )} \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} + a}\right ) - \frac {1}{8} \, \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} - \sqrt {-c x^{4} + b x^{2} + a} x^{2} + {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} + a d x\right )} \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} + a}\right ) + \frac {1}{8} \, \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} - \sqrt {-c x^{4} + b x^{2} + a} x^{2} - {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} + a d x\right )} \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} + a}\right ) \]
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\[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=\frac {\int \frac {\sqrt {a + b x^{2} - c x^{4}}}{a + c x^{4}}\, dx}{d} \]
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\[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=\int { \frac {\sqrt {-c x^{4} + b x^{2} + a}}{c d x^{4} + a d} \,d x } \]
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\[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=\int { \frac {\sqrt {-c x^{4} + b x^{2} + a}}{c d x^{4} + a d} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=\int \frac {\sqrt {-c\,x^4+b\,x^2+a}}{c\,d\,x^4+a\,d} \,d x \]
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