Optimal. Leaf size=59 \[ \frac {2 \sqrt {c+d x^3}}{3 d^2}-\frac {8 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{3 \sqrt {3} d^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 59, 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 = {457, 81, 65,
209} \begin {gather*} \frac {2 \sqrt {c+d x^3}}{3 d^2}-\frac {8 \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{3 \sqrt {3} d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 81
Rule 209
Rule 457
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x}{\sqrt {c+d x} (4 c+d x)} \, dx,x,x^3\right )\\ &=\frac {2 \sqrt {c+d x^3}}{3 d^2}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{\sqrt {c+d x} (4 c+d x)} \, dx,x,x^3\right )}{3 d}\\ &=\frac {2 \sqrt {c+d x^3}}{3 d^2}-\frac {(8 c) \text {Subst}\left (\int \frac {1}{3 c+x^2} \, dx,x,\sqrt {c+d x^3}\right )}{3 d^2}\\ &=\frac {2 \sqrt {c+d x^3}}{3 d^2}-\frac {8 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{3 \sqrt {3} d^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 56, normalized size = 0.95 \begin {gather*} \frac {6 \sqrt {c+d x^3}-8 \sqrt {3} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {3} \sqrt {c}}\right )}{9 d^2} \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 = 0.39, size = 425, normalized size = 7.20 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 43, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (4 \, \sqrt {3} \sqrt {c} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right ) - 3 \, \sqrt {d x^{3} + c}\right )}}{9 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.55, size = 112, normalized size = 1.90 \begin {gather*} \left [\frac {2 \, {\left (2 \, \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 ) + 3 \, \sqrt {d x^{3} + c}\right )}}{9 \, d^{2}}, -\frac {2 \, {\left (4 \, \sqrt {3} \sqrt {c} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right ) - 3 \, \sqrt {d x^{3} + c}\right )}}{9 \, d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.00, size = 65, normalized size = 1.10 \begin {gather*} \begin {cases} \frac {2 \left (- \frac {4 \sqrt {3} \sqrt {c} \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {c + d x^{3}}}{3 \sqrt {c}} \right )}}{9 d} + \frac {\sqrt {c + d x^{3}}}{3 d}\right )}{d} & \text {for}\: d \neq 0 \\\frac {x^{6}}{24 c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.74, size = 49, normalized size = 0.83 \begin {gather*} -\frac {2 \, {\left (\frac {4 \, \sqrt {3} \sqrt {c} \arctan \left (\frac {\sqrt {3} \sqrt {d x^{3} + c}}{3 \, \sqrt {c}}\right )}{d} - \frac {3 \, \sqrt {d x^{3} + c}}{d}\right )}}{9 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.86, size = 71, normalized size = 1.20 \begin {gather*} \frac {2\,\sqrt {d\,x^3+c}}{3\,d^2}+\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 )\,4{}\mathrm {i}}{9\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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