Optimal. Leaf size=47 \[ -\frac {\sqrt {a+c x^4}}{4 x^4}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]
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Rubi [A] time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ -\frac {\sqrt {a+c x^4}}{4 x^4}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {a+c x^4}}{x^5} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {a+c x^4}}{4 x^4}+\frac {1}{8} c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {a+c x^4}}{4 x^4}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^4}\right )\\ &=-\frac {\sqrt {a+c x^4}}{4 x^4}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{4 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 1.26 \[ -\frac {c x^4 \sqrt {\frac {c x^4}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {c x^4}{a}+1}\right )+a+c x^4}{4 x^4 \sqrt {a+c x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 108, normalized size = 2.30 \[ \left [\frac {\sqrt {a} c x^{4} \log \left (\frac {c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 2 \, \sqrt {c x^{4} + a} a}{8 \, a x^{4}}, \frac {\sqrt {-a} c x^{4} \arctan \left (\frac {\sqrt {c x^{4} + a} \sqrt {-a}}{a}\right ) - \sqrt {c x^{4} + a} a}{4 \, a x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 0.98 \[ \frac {\frac {c^{2} \arctan \left (\frac {\sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {c x^{4} + a} c}{x^{4}}}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 1.34 \[ -\frac {c \ln \left (\frac {2 a +2 \sqrt {c \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{4 \sqrt {a}}+\frac {\sqrt {c \,x^{4}+a}\, c}{4 a}-\frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 53, normalized size = 1.13 \[ \frac {c \log \left (\frac {\sqrt {c x^{4} + a} - \sqrt {a}}{\sqrt {c x^{4} + a} + \sqrt {a}}\right )}{8 \, \sqrt {a}} - \frac {\sqrt {c x^{4} + a}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 35, normalized size = 0.74 \[ -\frac {\sqrt {c\,x^4+a}}{4\,x^4}-\frac {c\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^4+a}}{\sqrt {a}}\right )}{4\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.06, size = 46, normalized size = 0.98 \[ - \frac {\sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{4 x^{2}} - \frac {c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x^{2}} \right )}}{4 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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