Optimal. Leaf size=52 \[ \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )-\frac {\sqrt {b x^2+c x^4}}{x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2018, 662, 620, 206} \[ \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )-\frac {\sqrt {b x^2+c x^4}}{x^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 662
Rule 2018
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {b x^2+c x^4}}{x^2}+\frac {1}{2} c \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {b x^2+c x^4}}{x^2}+c \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )\\ &=-\frac {\sqrt {b x^2+c x^4}}{x^2}+\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 60, normalized size = 1.15 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\frac {\sqrt {c} x \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {\frac {c x^2}{b}+1}}-1\right )}{x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 115, normalized size = 2.21 \[ \left [\frac {\sqrt {c} x^{2} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, \sqrt {c x^{4} + b x^{2}}}{2 \, x^{2}}, -\frac {\sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + \sqrt {c x^{4} + b x^{2}}}{x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 61, normalized size = 1.17 \[ -\frac {1}{2} \, \sqrt {c} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\relax (x) + \frac {2 \, b \sqrt {c} \mathrm {sgn}\relax (x)}{{\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 1.62 \[ \frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (b c x \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+\sqrt {c \,x^{2}+b}\, c^{\frac {3}{2}} x^{2}-\left (c \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {c}\right )}{\sqrt {c \,x^{2}+b}\, b \sqrt {c}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 51, normalized size = 0.98 \[ \frac {1}{2} \, \sqrt {c} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - \frac {\sqrt {c x^{4} + b x^{2}}}{x^{2}} \]
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 \[ \int \frac {\sqrt {c\,x^4+b\,x^2}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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