Optimal. Leaf size=80 \[ \frac {\tan ^{-1}\left (\frac {x \sqrt {a e^2-b d e+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {e} \sqrt {a e^2-b d e+c d^2}} \]
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Rubi [A] time = 0.45, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2112, 205} \[ \frac {\tan ^{-1}\left (\frac {x \sqrt {a e^2-b d e+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {e} \sqrt {a e^2-b d e+c d^2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2112
Rubi steps
\begin {align*} \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 &=a \operatorname {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*}
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Mathematica [C] time = 0.91, size = 383, normalized size = 4.79 \[ \frac {i \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \left (-\Pi \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) d}{2 a e};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\Pi \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {2} d e \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 67.60, size = 472, normalized size = 5.90 \[ \left [-\frac {\sqrt {-c d^{3} e + b d^{2} e^{2} - a d e^{3}} \log \left (-\frac {c^{2} d^{2} e^{2} x^{8} - 2 \, {\left (3 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2} + 3 \, a c d e^{3}\right )} x^{6} + a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} - 8 \, b c d^{3} e - 8 \, a b d e^{3} + a^{2} e^{4} + 4 \, {\left (2 \, b^{2} + a c\right )} d^{2} e^{2}\right )} x^{4} - 2 \, {\left (3 \, a c d^{3} e - 4 \, a b d^{2} e^{2} + 3 \, a^{2} d e^{3}\right )} x^{2} + 4 \, {\left (c d e x^{5} + a d e x - {\left (c d^{2} - 2 \, b d e + a e^{2}\right )} x^{3}\right )} \sqrt {-c d^{3} e + b d^{2} e^{2} - a d e^{3}} \sqrt {c x^{4} + b x^{2} + a}}{c^{2} d^{2} e^{2} x^{8} + 2 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{6} + a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 4 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{4} + 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x^{2}}\right )}{4 \, {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )}}, \frac {\arctan \left (\frac {2 \, \sqrt {c d^{3} e - b d^{2} e^{2} + a d e^{3}} \sqrt {c x^{4} + b x^{2} + a} x}{c d e x^{4} + a d e - {\left (c d^{2} - 2 \, b d e + a e^{2}\right )} x^{2}}\right )}{2 \, \sqrt {c d^{3} e - b d^{2} e^{2} + a d e^{3}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {c x^{4} - a}{\sqrt {c x^{4} + b x^{2} + a} {\left (c d x^{2} + a e\right )} {\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 555, normalized size = 6.94 \[ -\frac {\sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, d e}+\frac {\sqrt {2}\, \sqrt {\frac {b \,x^{2}}{2 a}-\frac {\sqrt {-4 a c +b^{2}}\, x^{2}}{2 a}+1}\, \sqrt {\frac {b \,x^{2}}{2 a}+\frac {\sqrt {-4 a c +b^{2}}\, x^{2}}{2 a}+1}\, \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, -\frac {2 a e}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, d e}+\frac {\sqrt {2}\, \sqrt {\frac {b \,x^{2}}{2 a}-\frac {\sqrt {-4 a c +b^{2}}\, x^{2}}{2 a}+1}\, \sqrt {\frac {b \,x^{2}}{2 a}+\frac {\sqrt {-4 a c +b^{2}}\, x^{2}}{2 a}+1}\, \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, -\frac {2 c d}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) e}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {c x^{4} - a}{\sqrt {c x^{4} + b x^{2} + a} {\left (c d x^{2} + a e\right )} {\left (e x^{2} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a-c\,x^4}{\left (e\,x^2+d\right )\,\left (c\,d\,x^2+a\,e\right )\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {a}{a d e \sqrt {a + b x^{2} + c x^{4}} + a e^{2} x^{2} \sqrt {a + b x^{2} + c x^{4}} + c d^{2} x^{2} \sqrt {a + b x^{2} + c x^{4}} + c d e x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \frac {c x^{4}}{a d e \sqrt {a + b x^{2} + c x^{4}} + a e^{2} x^{2} \sqrt {a + b x^{2} + c x^{4}} + c d^{2} x^{2} \sqrt {a + b x^{2} + c x^{4}} + c d e x^{4} \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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