Optimal. Leaf size=57 \[ \frac {2 \sqrt {c+d x^3}}{3 d}-\frac {2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} d} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {455, 52, 65,
209} \begin {gather*} \frac {2 \sqrt {c+d x^3}}{3 d}-\frac {2 \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 455
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {c+d x^3}}{4 c+d x^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {c+d x}}{4 c+d x} \, dx,x,x^3\right )\\ &=\frac {2 \sqrt {c+d x^3}}{3 d}-c \text {Subst}\left (\int \frac {1}{\sqrt {c+d x} (4 c+d x)} \, dx,x,x^3\right )\\ &=\frac {2 \sqrt {c+d x^3}}{3 d}-\frac {(2 c) \text {Subst}\left (\int \frac {1}{3 c+x^2} \, dx,x,\sqrt {c+d x^3}\right )}{d}\\ &=\frac {2 \sqrt {c+d x^3}}{3 d}-\frac {2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{\sqrt {3} d}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 54, normalized size = 0.95 \begin {gather*} \frac {2 \left (\sqrt {c+d x^3}-\sqrt {3} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.37, size = 425, normalized size = 7.46 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 42, normalized size = 0.74 \begin {gather*} -\frac {2 \, {\left (\sqrt {3} \sqrt {c} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right ) - \sqrt {d x^{3} + c}\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.87, size = 110, normalized size = 1.93 \begin {gather*} \left [\frac {\sqrt {3} \sqrt {-c} \log \left (\frac {d x^{3} - 2 \, \sqrt {3} \sqrt {d x^{3} + c} \sqrt {-c} - 2 \, c}{d x^{3} + 4 \, c}\right ) + 2 \, \sqrt {d x^{3} + c}}{3 \, d}, -\frac {2 \, {\left (\sqrt {3} \sqrt {c} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right ) - \sqrt {d x^{3} + c}\right )}}{3 \, d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.77, size = 51, normalized size = 0.89 \begin {gather*} \frac {2 \left (- \frac {\sqrt {3} \sqrt {c} \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {c + d x^{3}}}{3 \sqrt {c}} \right )}}{3} + \frac {\sqrt {c + d x^{3}}}{3}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.60, size = 44, normalized size = 0.77 \begin {gather*} -\frac {2 \, \sqrt {3} \sqrt {c} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right )}{3 \, d} + \frac {2 \, \sqrt {d x^{3} + c}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.87, size = 71, normalized size = 1.25 \begin {gather*} \frac {2\,\sqrt {d\,x^3+c}}{3\,d}+\frac {\sqrt {3}\,\sqrt {c}\,\ln \left (\frac {2\,\sqrt {3}\,c-\sqrt {3}\,d\,x^3+\sqrt {c}\,\sqrt {d\,x^3+c}\,6{}\mathrm {i}}{d\,x^3+4\,c}\right )\,1{}\mathrm {i}}{3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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