Optimal. Leaf size=49 \[ -\frac {x^2}{4 c \left (a+c x^4\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 \sqrt {a} c^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 294, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 \sqrt {a} c^{3/2}}-\frac {x^2}{4 c \left (a+c x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 281
Rule 294
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {x^2}{4 c \left (a+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {x^2}{4 c \left (a+c x^4\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 \sqrt {a} c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 1.00 \begin {gather*} -\frac {x^2}{4 c \left (a+c x^4\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 \sqrt {a} c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 40, normalized size = 0.82
method | result | size |
default | \(-\frac {x^{2}}{4 c \left (x^{4} c +a \right )}+\frac {\arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{4 c \sqrt {a c}}\) | \(40\) |
risch | \(-\frac {x^{2}}{4 c \left (x^{4} c +a \right )}-\frac {\ln \left (x^{2} \sqrt {-a c}-a \right )}{8 \sqrt {-a c}\, c}+\frac {\ln \left (x^{2} \sqrt {-a c}+a \right )}{8 \sqrt {-a c}\, c}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 40, normalized size = 0.82 \begin {gather*} -\frac {x^{2}}{4 \, {\left (c^{2} x^{4} + a c\right )}} + \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 127, normalized size = 2.59 \begin {gather*} \left [-\frac {2 \, a c x^{2} + {\left (c x^{4} + a\right )} \sqrt {-a c} \log \left (\frac {c x^{4} - 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right )}{8 \, {\left (a c^{3} x^{4} + a^{2} c^{2}\right )}}, -\frac {a c x^{2} + {\left (c x^{4} + a\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right )}{4 \, {\left (a c^{3} x^{4} + a^{2} c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs.
\(2 (39) = 78\).
time = 0.14, size = 83, normalized size = 1.69 \begin {gather*} - \frac {x^{2}}{4 a c + 4 c^{2} x^{4}} - \frac {\sqrt {- \frac {1}{a c^{3}}} \log {\left (- a c \sqrt {- \frac {1}{a c^{3}}} + x^{2} \right )}}{8} + \frac {\sqrt {- \frac {1}{a c^{3}}} \log {\left (a c \sqrt {- \frac {1}{a c^{3}}} + x^{2} \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 39, normalized size = 0.80 \begin {gather*} -\frac {x^{2}}{4 \, {\left (c x^{4} + a\right )} c} + \frac {\arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, \sqrt {a c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 37, normalized size = 0.76 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{4\,\sqrt {a}\,c^{3/2}}-\frac {x^2}{4\,c\,\left (c\,x^4+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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