Optimal. Leaf size=397 \[ -\frac {\sqrt [4]{c} \left (\sqrt {a} b \sqrt {c}-6 a c+2 b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 a^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {2 \sqrt [4]{c} \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 a^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {2 \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{15 a^2 x}-\frac {2 \sqrt {c} x \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{15 a^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {a+b x^2+c x^4}}{5 x^5}-\frac {b \sqrt {a+b x^2+c x^4}}{15 a x^3} \]
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Rubi [A] time = 0.26, antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1117, 1281, 1197, 1103, 1195} \[ \frac {2 \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{15 a^2 x}-\frac {2 \sqrt {c} x \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{15 a^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{c} \left (\sqrt {a} b \sqrt {c}-6 a c+2 b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 a^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {2 \sqrt [4]{c} \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 a^{7/4} \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{15 a x^3}-\frac {\sqrt {a+b x^2+c x^4}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1117
Rule 1195
Rule 1197
Rule 1281
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2+c x^4}}{x^6} \, dx &=-\frac {\sqrt {a+b x^2+c x^4}}{5 x^5}+\frac {1}{5} \int \frac {b+2 c x^2}{x^4 \sqrt {a+b x^2+c x^4}} \, dx\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{5 x^5}-\frac {b \sqrt {a+b x^2+c x^4}}{15 a x^3}-\frac {\int \frac {2 \left (b^2-3 a c\right )+b c x^2}{x^2 \sqrt {a+b x^2+c x^4}} \, dx}{15 a}\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{5 x^5}-\frac {b \sqrt {a+b x^2+c x^4}}{15 a x^3}+\frac {2 \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{15 a^2 x}+\frac {\int \frac {-a b c-2 c \left (b^2-3 a c\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{15 a^2}\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{5 x^5}-\frac {b \sqrt {a+b x^2+c x^4}}{15 a x^3}+\frac {2 \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{15 a^2 x}+\frac {\left (2 \sqrt {c} \left (b^2-3 a c\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{15 a^{3/2}}--\frac {\left (-\sqrt {a} b c^{3/2}-2 c \left (b^2-3 a c\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{15 a^{3/2} \sqrt {c}}\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{5 x^5}-\frac {b \sqrt {a+b x^2+c x^4}}{15 a x^3}+\frac {2 \left (b^2-3 a c\right ) \sqrt {a+b x^2+c x^4}}{15 a^2 x}-\frac {2 \sqrt {c} \left (b^2-3 a c\right ) x \sqrt {a+b x^2+c x^4}}{15 a^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {2 \sqrt [4]{c} \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 a^{7/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (2 b^2+\sqrt {a} b \sqrt {c}-6 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 a^{7/4} \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 1.34, size = 530, normalized size = 1.34 \[ \frac {-2 \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \left (3 a^3+a^2 \left (4 b x^2+9 c x^4\right )+a \left (-b^2 x^4+7 b c x^6+6 c^2 x^8\right )-2 b^2 x^6 \left (b+c x^2\right )\right )-i x^5 \left (b^2-3 a c\right ) \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i x^5 \left (b^2 \sqrt {b^2-4 a c}-3 a c \sqrt {b^2-4 a c}+4 a b c-b^3\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{30 a^2 x^5 \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2} + a}}{x^{6}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2} + a}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 452, normalized size = 1.14 \[ -\frac {\sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, b c \EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )}{60 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, a}-\frac {\left (3 a c -b^{2}\right ) \sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )+\EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )\right ) c}{15 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right ) a}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, b}{15 a \,x^{3}}-\frac {2 \left (3 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 a^{2} x}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{5 x^{5}} \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2} + a}}{x^{6}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^6} \,d x \]
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 \[ \int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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