Optimal. Leaf size=80 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {e} \sqrt {c d^2-b d e+a e^2}} \]
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Rubi [A]
time = 0.28, 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 = {2137, 211}
\begin {gather*} \frac {\text {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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 2137
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 \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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 10.56, size = 383, normalized size = 4.79 \begin {gather*} \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 (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 )-\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 )\right )}{\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d e \sqrt {a+b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.28, size = 555, normalized size = 6.94
method | result | size |
elliptic | \(-\frac {\arctan \left (\frac {d e \sqrt {c \,x^{4}+b \,x^{2}+a}}{x \sqrt {\left (a \,e^{2}-d e b +c \,d^{2}\right ) d e}}\right )}{\sqrt {\left (a \,e^{2}-d e b +c \,d^{2}\right ) d e}}\) | \(66\) |
default | \(-\frac {\sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 e d \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\sqrt {2}\, \sqrt {1+\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \EllipticPi \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a e}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{e d \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\sqrt {2}\, \sqrt {1+\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {-4 a c +b^{2}}}{2 a}}\, \EllipticPi \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 c d}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) e}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{d e \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\) | \(555\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 29.13, size = 477, normalized size = 5.96 \begin {gather*} \left [-\frac {\sqrt {-c d^{3} e + b d^{2} e^{2} - a d e^{3}} \log \left (-\frac {c^{2} d^{4} x^{4} + a^{2} x^{4} e^{4} - 4 \, {\left (c d^{2} x^{3} + a x^{3} e^{2} - {\left (c d x^{5} + 2 \, b d x^{3} + a d x\right )} e\right )} \sqrt {c x^{4} + b x^{2} + a} \sqrt {-c d^{3} e + b d^{2} e^{2} - a d e^{3}} - 2 \, {\left (3 \, a c d x^{6} + 4 \, a b d x^{4} + 3 \, a^{2} d x^{2}\right )} e^{3} + {\left (c^{2} d^{2} x^{8} + 8 \, b c d^{2} x^{6} + 4 \, {\left (2 \, b^{2} + a c\right )} d^{2} x^{4} + 8 \, a b d^{2} x^{2} + a^{2} d^{2}\right )} e^{2} - 2 \, {\left (3 \, c^{2} d^{3} x^{6} + 4 \, b c d^{3} x^{4} + 3 \, a c d^{3} x^{2}\right )} e}{c^{2} d^{4} x^{4} + a^{2} x^{4} e^{4} + 2 \, {\left (a c d x^{6} + a^{2} d x^{2}\right )} e^{3} + {\left (c^{2} d^{2} x^{8} + 4 \, a c d^{2} x^{4} + a^{2} d^{2}\right )} e^{2} + 2 \, {\left (c^{2} d^{3} x^{6} + a c d^{3} x^{2}\right )} e}\right )}{4 \, {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )}}, -\frac {\arctan \left (\frac {2 \, \sqrt {c x^{4} + b x^{2} + a} \sqrt {c d^{3} e - b d^{2} e^{2} + a d e^{3}} x}{c d^{2} x^{2} + a x^{2} e^{2} - {\left (c d x^{4} + 2 \, b d x^{2} + a d\right )} e}\right )}{2 \, \sqrt {c d^{3} e - b d^{2} e^{2} + a d e^{3}}}\right ] \end {gather*}
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 \begin {gather*} - \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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