Optimal. Leaf size=50 \[ \frac {1}{4} x^2 \sqrt {a+c x^4}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}} \]
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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {281, 201, 223,
212} \begin {gather*} \frac {1}{4} x^2 \sqrt {a+c x^4}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 281
Rubi steps
\begin {align*} \int x \sqrt {a+c x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \sqrt {a+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{4} x^2 \sqrt {a+c x^4}+\frac {1}{4} a \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{4} x^2 \sqrt {a+c x^4}+\frac {1}{4} a \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {a+c x^4}}\right )\\ &=\frac {1}{4} x^2 \sqrt {a+c x^4}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{4} x^2 \sqrt {a+c x^4}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+c x^4}}{\sqrt {c} x^2}\right )}{4 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 40, normalized size = 0.80
method | result | size |
default | \(\frac {x^{2} \sqrt {x^{4} c +a}}{4}+\frac {a \ln \left (x^{2} \sqrt {c}+\sqrt {x^{4} c +a}\right )}{4 \sqrt {c}}\) | \(40\) |
risch | \(\frac {x^{2} \sqrt {x^{4} c +a}}{4}+\frac {a \ln \left (x^{2} \sqrt {c}+\sqrt {x^{4} c +a}\right )}{4 \sqrt {c}}\) | \(40\) |
elliptic | \(\frac {x^{2} \sqrt {x^{4} c +a}}{4}+\frac {a \ln \left (x^{2} \sqrt {c}+\sqrt {x^{4} c +a}\right )}{4 \sqrt {c}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs.
\(2 (38) = 76\).
time = 0.50, size = 78, normalized size = 1.56 \begin {gather*} -\frac {a \log \left (-\frac {\sqrt {c} - \frac {\sqrt {c x^{4} + a}}{x^{2}}}{\sqrt {c} + \frac {\sqrt {c x^{4} + a}}{x^{2}}}\right )}{8 \, \sqrt {c}} - \frac {\sqrt {c x^{4} + a} a}{4 \, {\left (c - \frac {c x^{4} + a}{x^{4}}\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 102, normalized size = 2.04 \begin {gather*} \left [\frac {2 \, \sqrt {c x^{4} + a} c x^{2} + a \sqrt {c} \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right )}{8 \, c}, \frac {\sqrt {c x^{4} + a} c x^{2} - a \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2}}{\sqrt {c x^{4} + a}}\right )}{4 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.88, size = 44, normalized size = 0.88 \begin {gather*} \frac {\sqrt {a} x^{2} \sqrt {1 + \frac {c x^{4}}{a}}}{4} + \frac {a \operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{4 \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 41, normalized size = 0.82 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{4} + a} x^{2} - \frac {a \log \left ({\left | -\sqrt {c} x^{2} + \sqrt {c x^{4} + a} \right |}\right )}{4 \, \sqrt {c}} \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 x\,\sqrt {c\,x^4+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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