Optimal. Leaf size=31 \[ \frac {x}{c}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 327, 211}
\begin {gather*} \frac {x}{c}-\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 327
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^4}{b x^2+c x^4} \, dx &=\int \frac {x^2}{b+c x^2} \, dx\\ &=\frac {x}{c}-\frac {b \int \frac {1}{b+c x^2} \, dx}{c}\\ &=\frac {x}{c}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} \frac {x}{c}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 27, normalized size = 0.87
method | result | size |
default | \(\frac {x}{c}-\frac {b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{c \sqrt {b c}}\) | \(27\) |
risch | \(\frac {x}{c}+\frac {\sqrt {-b c}\, \ln \left (-\sqrt {-b c}\, x -b \right )}{2 c^{2}}-\frac {\sqrt {-b c}\, \ln \left (\sqrt {-b c}\, x -b \right )}{2 c^{2}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 26, normalized size = 0.84 \begin {gather*} -\frac {b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c} + \frac {x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 82, normalized size = 2.65 \begin {gather*} \left [\frac {\sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) + 2 \, x}{2 \, c}, -\frac {\sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) - x}{c}\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. 56 vs.
\(2 (26) = 52\).
time = 0.06, size = 56, normalized size = 1.81 \begin {gather*} \frac {\sqrt {- \frac {b}{c^{3}}} \log {\left (- c \sqrt {- \frac {b}{c^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{c^{3}}} \log {\left (c \sqrt {- \frac {b}{c^{3}}} + x \right )}}{2} + \frac {x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.25, size = 26, normalized size = 0.84 \begin {gather*} -\frac {b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c} + \frac {x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 23, normalized size = 0.74 \begin {gather*} \frac {x}{c}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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