Optimal. Leaf size=35 \[ -\frac {3}{2} \left (1+\sqrt {-3+x}\right )^{4/3}+\frac {6}{7} \left (1+\sqrt {-3+x}\right )^{7/3} \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 196, 45}
\begin {gather*} \frac {6}{7} \left (\sqrt {x-3}+1\right )^{7/3}-\frac {3}{2} \left (\sqrt {x-3}+1\right )^{4/3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 196
Rule 253
Rubi steps
\begin {align*} \int \sqrt [3]{1+\sqrt {-3+x}} \, dx &=\text {Subst}\left (\int \sqrt [3]{1+\sqrt {x}} \, dx,x,-3+x\right )\\ &=2 \text {Subst}\left (\int x \sqrt [3]{1+x} \, dx,x,\sqrt {-3+x}\right )\\ &=2 \text {Subst}\left (\int \left (-\sqrt [3]{1+x}+(1+x)^{4/3}\right ) \, dx,x,\sqrt {-3+x}\right )\\ &=-\frac {3}{2} \left (1+\sqrt {-3+x}\right )^{4/3}+\frac {6}{7} \left (1+\sqrt {-3+x}\right )^{7/3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.80 \begin {gather*} \frac {3}{14} \left (1+\sqrt {-3+x}\right )^{4/3} \left (-3+4 \sqrt {-3+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 24, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {3 \left (1+\sqrt {x -3}\right )^{\frac {4}{3}}}{2}+\frac {6 \left (1+\sqrt {x -3}\right )^{\frac {7}{3}}}{7}\) | \(24\) |
default | \(-\frac {3 \left (1+\sqrt {x -3}\right )^{\frac {4}{3}}}{2}+\frac {6 \left (1+\sqrt {x -3}\right )^{\frac {7}{3}}}{7}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 23, normalized size = 0.66 \begin {gather*} \frac {6}{7} \, {\left (\sqrt {x - 3} + 1\right )}^{\frac {7}{3}} - \frac {3}{2} \, {\left (\sqrt {x - 3} + 1\right )}^{\frac {4}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 21, normalized size = 0.60 \begin {gather*} \frac {3}{14} \, {\left (4 \, x + \sqrt {x - 3} - 15\right )} {\left (\sqrt {x - 3} + 1\right )}^{\frac {1}{3}} \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. 184 vs.
\(2 (29) = 58\).
time = 0.61, size = 184, normalized size = 5.26 \begin {gather*} \frac {12 \left (x - 3\right )^{\frac {7}{2}} \sqrt [3]{\sqrt {x - 3} + 1}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} - \frac {6 \left (x - 3\right )^{\frac {5}{2}} \sqrt [3]{\sqrt {x - 3} + 1}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} + \frac {9 \left (x - 3\right )^{\frac {5}{2}}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} + \frac {15 \left (x - 3\right )^{3} \sqrt [3]{\sqrt {x - 3} + 1}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} - \frac {9 \left (x - 3\right )^{2} \sqrt [3]{\sqrt {x - 3} + 1}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} + \frac {9 \left (x - 3\right )^{2}}{14 \left (x - 3\right )^{\frac {5}{2}} + 14 \left (x - 3\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.73, size = 23, normalized size = 0.66 \begin {gather*} \frac {6}{7} \, {\left (\sqrt {x - 3} + 1\right )}^{\frac {7}{3}} - \frac {3}{2} \, {\left (\sqrt {x - 3} + 1\right )}^{\frac {4}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.51, size = 16, normalized size = 0.46 \begin {gather*} \left (x-3\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},2;\ 3;\ -\sqrt {x-3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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