Optimal. Leaf size=59 \[ \frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac {\sqrt {b x^2+c x^4}}{2 b x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 59, 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 = {2025, 2008, 206} \[ \frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac {\sqrt {b x^2+c x^4}}{2 b x^3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2025
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx &=-\frac {\sqrt {b x^2+c x^4}}{2 b x^3}-\frac {c \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{2 b}\\ &=-\frac {\sqrt {b x^2+c x^4}}{2 b x^3}+\frac {c \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )}{2 b}\\ &=-\frac {\sqrt {b x^2+c x^4}}{2 b x^3}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{2 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 68, normalized size = 1.15 \[ \frac {c \sqrt {x^2 \left (b+c x^2\right )} \left (\frac {\tanh ^{-1}\left (\sqrt {\frac {c x^2}{b}+1}\right )}{2 \sqrt {\frac {c x^2}{b}+1}}-\frac {b}{2 c x^2}\right )}{b^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 133, normalized size = 2.25 \[ \left [\frac {\sqrt {b} c x^{3} \log \left (-\frac {c x^{3} + 2 \, b x + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) - 2 \, \sqrt {c x^{4} + b x^{2}} b}{4 \, b^{2} x^{3}}, -\frac {\sqrt {-b} c x^{3} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + \sqrt {c x^{4} + b x^{2}} b}{2 \, b^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 73, normalized size = 1.24 \[ -\frac {\sqrt {c \,x^{2}+b}\, \left (-b c \,x^{2} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )+\sqrt {c \,x^{2}+b}\, b^{\frac {3}{2}}\right )}{2 \sqrt {c \,x^{4}+b \,x^{2}}\, b^{\frac {5}{2}} x} \]
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 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.47, size = 76, normalized size = 1.29 \[ -\frac {\left (\frac {\sqrt {c}\,x^2\,\sqrt {c+\frac {b}{x^2}}}{2\,b}+\frac {c^{3/2}\,x^3\,\mathrm {asin}\left (\frac {\sqrt {b}\,1{}\mathrm {i}}{\sqrt {c}\,x}\right )\,1{}\mathrm {i}}{2\,b^{3/2}}\right )\,\sqrt {\frac {b}{c\,x^2}+1}}{x\,\sqrt {c\,x^4+b\,x^2}} \]
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 \[ \int \frac {1}{x^{2} \sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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