Optimal. Leaf size=551 \[ \frac {5}{21} \left (4+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (4+27 x^2\right )^{2/3}-\frac {72 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )}+\frac {4 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{7\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{21 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}} \]
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Rubi [A]
time = 0.22, antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {757, 655, 241,
310, 225, 1893} \begin {gather*} -\frac {8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} F\left (\text {ArcSin}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{21 \sqrt [4]{3} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac {4 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\text {ArcSin}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{7\ 3^{3/4} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac {72 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )}+\frac {1}{21} (3 x+2) \left (27 x^2+4\right )^{2/3}+\frac {5}{21} \left (27 x^2+4\right )^{2/3} \end {gather*}
Antiderivative was successfully verified.
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Rule 225
Rule 241
Rule 310
Rule 655
Rule 757
Rule 1893
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{\sqrt [3]{4+27 x^2}} \, dx &=\frac {1}{21} (2+3 x) \left (4+27 x^2\right )^{2/3}+\frac {1}{63} \int \frac {216+540 x}{\sqrt [3]{4+27 x^2}} \, dx\\ &=\frac {5}{21} \left (4+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (4+27 x^2\right )^{2/3}+\frac {24}{7} \int \frac {1}{\sqrt [3]{4+27 x^2}} \, dx\\ &=\frac {5}{21} \left (4+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (4+27 x^2\right )^{2/3}+\frac {\left (4 \sqrt {3} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{7 x}\\ &=\frac {5}{21} \left (4+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (4+27 x^2\right )^{2/3}-\frac {\left (4 \sqrt {3} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{7 x}+\frac {\left (8 \sqrt [6]{2} \sqrt {3} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{7 \sqrt {2-\sqrt {3}} x}\\ &=\frac {5}{21} \left (4+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (4+27 x^2\right )^{2/3}-\frac {72 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )}+\frac {4 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{7\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{21 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 14.95, size = 47, normalized size = 0.09 \begin {gather*} \frac {1}{21} \left ((7+3 x) \left (4+27 x^2\right )^{2/3}+36 \sqrt [3]{2} x \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^2}{4}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.48, size = 35, normalized size = 0.06
method | result | size |
risch | \(\frac {\left (7+3 x \right ) \left (27 x^{2}+4\right )^{\frac {2}{3}}}{21}+\frac {12 \,2^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {27 x^{2}}{4}\right )}{7}\) | \(35\) |
meijerg | \(2 \,2^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {27 x^{2}}{4}\right )+3 \,2^{\frac {1}{3}} x^{2} \hypergeom \left (\left [\frac {1}{3}, 1\right ], \left [2\right ], -\frac {27 x^{2}}{4}\right )+\frac {3 \,2^{\frac {1}{3}} x^{3} \hypergeom \left (\left [\frac {1}{3}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], -\frac {27 x^{2}}{4}\right )}{2}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.99, size = 68, normalized size = 0.12 \begin {gather*} \frac {3 \cdot \sqrt [3]{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{i \pi }}{4}} \right )}}{2} + 2 \cdot \sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{i \pi }}{4}} \right )} + \frac {\left (27 x^{2} + 4\right )^{\frac {2}{3}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^2}{{\left (27\,x^2+4\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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