Optimal. Leaf size=589 \[ \frac {1}{30} (2+3 x)^2 \left (28+54 x+27 x^2\right )^{2/3}-\frac {1}{35} (1+8 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac {72 (1+x)}{7 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )}-\frac {\sqrt {2 \left (2+\sqrt {3}\right )} \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{63 \sqrt [4]{3} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}+\frac {4 \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{63\ 3^{3/4} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}} \]
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Rubi [A]
time = 0.39, antiderivative size = 589, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {756, 793, 633,
241, 310, 225, 1893} \begin {gather*} \frac {4 \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} F\left (\text {ArcSin}\left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt {3}\right )}{63\ 3^{3/4} (x+1) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}}}-\frac {\sqrt {2 \left (2+\sqrt {3}\right )} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\text {ArcSin}\left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt {3}\right )}{63 \sqrt [4]{3} (x+1) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}}}+\frac {1}{30} \left (27 x^2+54 x+28\right )^{2/3} (3 x+2)^2-\frac {1}{35} (8 x+1) \left (27 x^2+54 x+28\right )^{2/3}+\frac {72 (x+1)}{7 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 225
Rule 241
Rule 310
Rule 633
Rule 756
Rule 793
Rule 1893
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{\sqrt [3]{28+54 x+27 x^2}} \, dx &=\frac {1}{30} (2+3 x)^2 \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{90} \int \frac {(-360-432 x) (2+3 x)}{\sqrt [3]{28+54 x+27 x^2}} \, dx\\ &=\frac {1}{30} (2+3 x)^2 \left (28+54 x+27 x^2\right )^{2/3}-\frac {1}{35} (1+8 x) \left (28+54 x+27 x^2\right )^{2/3}-\frac {4}{7} \int \frac {1}{\sqrt [3]{28+54 x+27 x^2}} \, dx\\ &=\frac {1}{30} (2+3 x)^2 \left (28+54 x+27 x^2\right )^{2/3}-\frac {1}{35} (1+8 x) \left (28+54 x+27 x^2\right )^{2/3}-\frac {2}{189} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{108}}} \, dx,x,54+54 x\right )\\ &=\frac {1}{30} (2+3 x)^2 \left (28+54 x+27 x^2\right )^{2/3}-\frac {1}{35} (1+8 x) \left (28+54 x+27 x^2\right )^{2/3}-\frac {\left (2 \sqrt {(54+54 x)^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{7 \sqrt {3} (54+54 x)}\\ &=\frac {1}{30} (2+3 x)^2 \left (28+54 x+27 x^2\right )^{2/3}-\frac {1}{35} (1+8 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac {\left (2 \sqrt {(54+54 x)^2}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{7 \sqrt {3} (54+54 x)}-\frac {\left (2 \sqrt {\frac {2}{3} \left (2+\sqrt {3}\right )} \sqrt {(54+54 x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{7 (54+54 x)}\\ &=\frac {1}{30} (2+3 x)^2 \left (28+54 x+27 x^2\right )^{2/3}-\frac {1}{35} (1+8 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac {12 (1+x)}{7 \left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )}-\frac {2 \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{28+54 x+27 x^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}{1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{21\ 3^{3/4} (1+x) \sqrt {-\frac {1-\sqrt [3]{28+54 x+27 x^2}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}}}+\frac {4 \sqrt {2} \left (1-\sqrt [3]{28+54 x+27 x^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}{1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{63 \sqrt [4]{3} (1+x) \sqrt {-\frac {1-\sqrt [3]{28+54 x+27 x^2}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+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 = 10.02, size = 53, normalized size = 0.09 \begin {gather*} \frac {1}{210} \left (28+54 x+27 x^2\right )^{2/3} \left (22+36 x+63 x^2\right )-\frac {4}{7} (1+x) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-27 (1+x)^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (2+3 x \right )^{3}}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{3}}{\sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \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 )}^3}{{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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