Integrand size = 46, antiderivative size = 80 \[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {e} \sqrt {c d^2-b d e+a e^2}} \]
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Time = 0.36 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2137, 211} \[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\frac {\arctan \left (\frac {x \sqrt {a e^2-b d e+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {e} \sqrt {a e^2-b d e+c d^2}} \]
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Rule 211
Rule 2137
Rubi steps \begin{align*} \text {integral}& = a \text {Subst}\left (\int \frac {1}{a d e-\left (a b d e-a \left (c d^2+a e^2\right )\right ) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2+c x^4}}\right ) \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {e} \sqrt {c d^2-b d e+a e^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 10.57 (sec) , antiderivative size = 383, normalized size of antiderivative = 4.79 \[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\frac {i \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) d}{2 a e},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d e \sqrt {a+b x^2+c x^4}} \]
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Time = 3.40 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {\arctan \left (\frac {d e \sqrt {c \,x^{4}+b \,x^{2}+a}}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{\sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\) | \(66\) |
elliptic | \(-\frac {\arctan \left (\frac {d e \sqrt {c \,x^{4}+b \,x^{2}+a}}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{\sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\) | \(66\) |
pseudoelliptic | \(-\frac {\arctan \left (\frac {d e \sqrt {c \,x^{4}+b \,x^{2}+a}}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{\sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\) | \(66\) |
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Time = 28.97 (sec) , antiderivative size = 472, normalized size of antiderivative = 5.90 \[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\left [-\frac {\sqrt {-c d^{3} e + b d^{2} e^{2} - a d e^{3}} \log \left (-\frac {c^{2} d^{2} e^{2} x^{8} - 2 \, {\left (3 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2} + 3 \, a c d e^{3}\right )} x^{6} + a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} - 8 \, b c d^{3} e - 8 \, a b d e^{3} + a^{2} e^{4} + 4 \, {\left (2 \, b^{2} + a c\right )} d^{2} e^{2}\right )} x^{4} - 2 \, {\left (3 \, a c d^{3} e - 4 \, a b d^{2} e^{2} + 3 \, a^{2} d e^{3}\right )} x^{2} + 4 \, {\left (c d e x^{5} + a d e x - {\left (c d^{2} - 2 \, b d e + a e^{2}\right )} x^{3}\right )} \sqrt {-c d^{3} e + b d^{2} e^{2} - a d e^{3}} \sqrt {c x^{4} + b x^{2} + a}}{c^{2} d^{2} e^{2} x^{8} + 2 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{6} + a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 4 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{4} + 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x^{2}}\right )}{4 \, {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )}}, \frac {\arctan \left (\frac {2 \, \sqrt {c d^{3} e - b d^{2} e^{2} + a d e^{3}} \sqrt {c x^{4} + b x^{2} + a} x}{c d e x^{4} + a d e - {\left (c d^{2} - 2 \, b d e + a e^{2}\right )} x^{2}}\right )}{2 \, \sqrt {c d^{3} e - b d^{2} e^{2} + a d e^{3}}}\right ] \]
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\[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=- \int \left (- \frac {a}{a d e \sqrt {a + b x^{2} + c x^{4}} + a e^{2} x^{2} \sqrt {a + b x^{2} + c x^{4}} + c d^{2} x^{2} \sqrt {a + b x^{2} + c x^{4}} + c d e x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \frac {c x^{4}}{a d e \sqrt {a + b x^{2} + c x^{4}} + a e^{2} x^{2} \sqrt {a + b x^{2} + c x^{4}} + c d^{2} x^{2} \sqrt {a + b x^{2} + c x^{4}} + c d e x^{4} \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
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\[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\int { -\frac {c x^{4} - a}{\sqrt {c x^{4} + b x^{2} + a} {\left (c d x^{2} + a e\right )} {\left (e x^{2} + d\right )}} \,d x } \]
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\[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\int { -\frac {c x^{4} - a}{\sqrt {c x^{4} + b x^{2} + a} {\left (c d x^{2} + a e\right )} {\left (e x^{2} + d\right )}} \,d x } \]
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Timed out. \[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {a-c\,x^4}{\left (e\,x^2+d\right )\,\left (c\,d\,x^2+a\,e\right )\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \]
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