Optimal. Leaf size=56 \[ -\frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 \sqrt {b}}-\frac {\sqrt {b x^2+c x^4}}{2 x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2020, 2008, 206} \[ -\frac {\sqrt {b x^2+c x^4}}{2 x^3}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^4} \, dx &=-\frac {\sqrt {b x^2+c x^4}}{2 x^3}+\frac {1}{2} c \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx\\ &=-\frac {\sqrt {b x^2+c x^4}}{2 x^3}-\frac {1}{2} c \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )\\ &=-\frac {\sqrt {b x^2+c x^4}}{2 x^3}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 1.12 \[ -\frac {c x^2 \sqrt {\frac {c x^2}{b}+1} \tanh ^{-1}\left (\sqrt {\frac {c x^2}{b}+1}\right )+b+c x^2}{2 x \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 134, normalized size = 2.39 \[ \left [\frac {\sqrt {b} c x^{3} \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) - 2 \, \sqrt {c x^{4} + b x^{2}} b}{4 \, b x^{3}}, \frac {\sqrt {-b} c x^{3} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) - \sqrt {c x^{4} + b x^{2}} b}{2 \, b x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 50, normalized size = 0.89 \[ \frac {\frac {c^{2} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b}} - \frac {\sqrt {c x^{2} + b} c \mathrm {sgn}\relax (x)}{x^{2}}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 85, normalized size = 1.52 \[ -\frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (\sqrt {b}\, c \,x^{2} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-\sqrt {c \,x^{2}+b}\, c \,x^{2}+\left (c \,x^{2}+b \right )^{\frac {3}{2}}\right )}{2 \sqrt {c \,x^{2}+b}\, b \,x^{3}} \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {c\,x^4+b\,x^2}}{x^4} \,d x \]
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 \[ \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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