Optimal. Leaf size=445 \[ \frac {\sqrt {\sqrt {4 a c+b^2}+b} \left (-b \sqrt {4 a c+b^2}+a c+b^2\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} a^2 \sqrt {c} \sqrt {a+b x^2-c x^4}}-\frac {b \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} a^2 \sqrt {c} \sqrt {a+b x^2-c x^4}}+\frac {2 b \sqrt {a+b x^2-c x^4}}{3 a^2 x}-\frac {\sqrt {a+b x^2-c x^4}}{3 a x^3} \]
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Rubi [A] time = 0.47, antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1123, 1281, 1202, 524, 424, 419} \[ \frac {\sqrt {\sqrt {4 a c+b^2}+b} \left (-b \sqrt {4 a c+b^2}+a c+b^2\right ) \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} a^2 \sqrt {c} \sqrt {a+b x^2-c x^4}}-\frac {b \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} a^2 \sqrt {c} \sqrt {a+b x^2-c x^4}}+\frac {2 b \sqrt {a+b x^2-c x^4}}{3 a^2 x}-\frac {\sqrt {a+b x^2-c x^4}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 1123
Rule 1202
Rule 1281
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt {a+b x^2-c x^4}} \, dx &=-\frac {\sqrt {a+b x^2-c x^4}}{3 a x^3}+\frac {\int \frac {-2 b+c x^2}{x^2 \sqrt {a+b x^2-c x^4}} \, dx}{3 a}\\ &=-\frac {\sqrt {a+b x^2-c x^4}}{3 a x^3}+\frac {2 b \sqrt {a+b x^2-c x^4}}{3 a^2 x}-\frac {\int \frac {-a c-2 b c x^2}{\sqrt {a+b x^2-c x^4}} \, dx}{3 a^2}\\ &=-\frac {\sqrt {a+b x^2-c x^4}}{3 a x^3}+\frac {2 b \sqrt {a+b x^2-c x^4}}{3 a^2 x}-\frac {\left (\sqrt {1-\frac {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}}}\right ) \int \frac {-a c-2 b c x^2}{\sqrt {1-\frac {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}}}} \, dx}{3 a^2 \sqrt {a+b x^2-c x^4}}\\ &=-\frac {\sqrt {a+b x^2-c x^4}}{3 a x^3}+\frac {2 b \sqrt {a+b x^2-c x^4}}{3 a^2 x}-\frac {\left (b \left (b-\sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {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}}}\right ) \int \frac {\sqrt {1-\frac {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}}}} \, dx}{3 a^2 \sqrt {a+b x^2-c x^4}}+\frac {\left (\left (b^2+a c-b \sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {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}}}\right ) \int \frac {1}{\sqrt {1-\frac {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}}}} \, dx}{3 a^2 \sqrt {a+b x^2-c x^4}}\\ &=-\frac {\sqrt {a+b x^2-c x^4}}{3 a x^3}+\frac {2 b \sqrt {a+b x^2-c x^4}}{3 a^2 x}-\frac {b \left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {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 (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} a^2 \sqrt {c} \sqrt {a+b x^2-c x^4}}+\frac {\sqrt {b+\sqrt {b^2+4 a c}} \left (b^2+a c-b \sqrt {b^2+4 a c}\right ) \sqrt {1-\frac {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}}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{3 \sqrt {2} a^2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.76, size = 472, normalized size = 1.06 \[ \frac {-2 \sqrt {-\frac {c}{\sqrt {4 a c+b^2}+b}} \left (a-2 b x^2\right ) \left (a+b x^2-c x^4\right )+i \sqrt {2} x^3 \left (b \sqrt {4 a c+b^2}-a c-b^2\right ) \sqrt {\frac {\sqrt {4 a c+b^2}+b-2 c x^2}{\sqrt {4 a c+b^2}+b}} \sqrt {\frac {\sqrt {4 a c+b^2}-b+2 c x^2}{\sqrt {4 a c+b^2}-b}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )-i \sqrt {2} b x^3 \left (\sqrt {4 a c+b^2}-b\right ) \sqrt {\frac {\sqrt {4 a c+b^2}+b-2 c x^2}{\sqrt {4 a c+b^2}+b}} \sqrt {\frac {\sqrt {4 a c+b^2}-b+2 c x^2}{\sqrt {4 a c+b^2}-b}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{6 a^2 x^3 \sqrt {-\frac {c}{\sqrt {4 a c+b^2}+b}} \sqrt {a+b x^2-c x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c x^{4} + b x^{2} + a}}{c x^{8} - b x^{6} - a x^{4}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-c x^{4} + b x^{2} + a} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 417, normalized size = 0.94 \[ -\frac {\sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )+\EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )\right ) b c}{3 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {4 a c +b^{2}}\right ) a}+\frac {\sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) x^{2}}{a}+4}\, c \EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )}{12 \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}\, a}+\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, b}{3 a^{2} x}-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{3 a \,x^{3}} \]
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 \[ \int \frac {1}{\sqrt {-c x^{4} + b x^{2} + a} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,\sqrt {-c\,x^4+b\,x^2+a}} \,d x \]
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 \[ \int \frac {1}{x^{4} \sqrt {a + b x^{2} - c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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