Optimal. Leaf size=52 \[ -\frac {\sqrt {b x^2+c x^4}}{x^2}+\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right ) \]
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Rubi [A]
time = 0.05, 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 = {2043, 676, 634,
212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 676
Rule 2043
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^3} \, dx &=\frac {1}{2} \text {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 \text {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 \text {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 = 0.05, size = 68, normalized size = 1.31 \begin {gather*} -\frac {\sqrt {b+c x^2} \left (\sqrt {b+c x^2}+\sqrt {c} x \log \left (-\sqrt {c} x+\sqrt {b+c x^2}\right )\right )}{\sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 84, normalized size = 1.62
method | result | size |
risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}}{x^{2}}+\frac {\sqrt {c}\, \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{x \sqrt {c \,x^{2}+b}}\) | \(65\) |
default | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (c^{\frac {3}{2}} \sqrt {c \,x^{2}+b}\, x^{2}-\left (c \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {c}+\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) b c x \right )}{x^{2} \sqrt {c \,x^{2}+b}\, b \sqrt {c}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 51, normalized size = 0.98 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 115, normalized size = 2.21 \begin {gather*} \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 ] \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.38, size = 61, normalized size = 1.17 \begin {gather*} -\frac {1}{2} \, \sqrt {c} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2}\right ) \mathrm {sgn}\left (x\right ) + \frac {2 \, b \sqrt {c} \mathrm {sgn}\left (x\right )}{{\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c\,x^4+b\,x^2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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