Optimal. Leaf size=341 \[ \frac {\sqrt [4]{c} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{3 a x}+\frac {b \sqrt {c} x \sqrt {a+b x^2+c x^4}}{3 a \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {a+b x^2+c x^4}}{3 x^3} \]
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Rubi [A] time = 0.16, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {\sqrt [4]{c} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{3 a x}+\frac {b \sqrt {c} x \sqrt {a+b x^2+c x^4}}{3 a \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {a+b x^2+c x^4}}{3 x^3} \]
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^4} \, dx &=-\frac {\sqrt {a+b x^2+c x^4}}{3 x^3}+\frac {1}{3} \int \frac {b+2 c x^2}{x^2 \sqrt {a+b x^2+c x^4}} \, dx\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{3 x^3}-\frac {b \sqrt {a+b x^2+c x^4}}{3 a x}-\frac {\int \frac {-2 a c-b c x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{3 a}\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{3 x^3}-\frac {b \sqrt {a+b x^2+c x^4}}{3 a x}-\frac {\left (b \sqrt {c}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{3 \sqrt {a}}+\frac {1}{3} \left (\left (\frac {b}{\sqrt {a}}+2 \sqrt {c}\right ) \sqrt {c}\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx\\ &=-\frac {\sqrt {a+b x^2+c x^4}}{3 x^3}-\frac {b \sqrt {a+b x^2+c x^4}}{3 a x}+\frac {b \sqrt {c} x \sqrt {a+b x^2+c x^4}}{3 a \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {b \sqrt [4]{c} \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 )}{3 a^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {\left (b+2 \sqrt {a} \sqrt {c}\right ) \sqrt [4]{c} \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 )}{6 a^{3/4} \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.90, size = 459, normalized size = 1.35 \[ \frac {-4 \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \left (a+b x^2\right ) \left (a+b x^2+c x^4\right )-i x^3 \left (b \sqrt {b^2-4 a c}+4 a c-b^2\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i b x^3 \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{12 a x^3 \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \sqrt {a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2} + a}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2} + a}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 404, normalized size = 1.18 \[ -\frac {\sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )+\EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )\right ) b c}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {\sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, c \EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, b}{3 a x}-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x^{3}} \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2} + a}}{x^{4}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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