Optimal. Leaf size=635 \[ \frac {1}{30} (2+3 x)^2 \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{7} (27+8 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {9000 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )}-\frac {25\ 5^{5/6} \sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{189 \sqrt {2} \sqrt [4]{3} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac {50\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{189\ 3^{3/4} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}} \]
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Rubi [A]
time = 0.50, antiderivative size = 635, 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 {50\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\text {ArcSin}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{189\ 3^{3/4} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}}}-\frac {25\ 5^{5/6} \sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\text {ArcSin}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{189 \sqrt {2} \sqrt [4]{3} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}}}+\frac {1}{30} \left (27 x^2-54 x+52\right )^{2/3} (3 x+2)^2+\frac {1}{7} (8 x+27) \left (27 x^2-54 x+52\right )^{2/3}+\frac {9000 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\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]{52-54 x+27 x^2}} \, dx &=\frac {1}{30} (2+3 x)^2 \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{90} \int \frac {(2+3 x) (-360+2160 x)}{\sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=\frac {1}{30} (2+3 x)^2 \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{7} (27+8 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {500}{7} \int \frac {1}{\sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=\frac {1}{30} (2+3 x)^2 \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{7} (27+8 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{189} \left (50 \sqrt [3]{5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{2700}}} \, dx,x,-54+54 x\right )\\ &=\frac {1}{30} (2+3 x)^2 \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{7} (27+8 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {\left (250 \sqrt [3]{5} \sqrt {(-54+54 x)^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\frac {\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{7 \sqrt {3} (-54+54 x)}\\ &=\frac {1}{30} (2+3 x)^2 \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{7} (27+8 x) \left (52-54 x+27 x^2\right )^{2/3}-\frac {\left (250 \sqrt [3]{5} \sqrt {(-54+54 x)^2}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\frac {\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{7 \sqrt {3} (-54+54 x)}+\frac {\left (250 \sqrt [3]{5} \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,\frac {\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{7 (-54+54 x)}\\ &=\frac {1}{30} (2+3 x)^2 \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{7} (27+8 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {9000 \sqrt [3]{5} (1-x)}{7 \left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )}-\frac {25\ 5^{5/6} \sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {30+30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{189 \sqrt {2} \sqrt [4]{3} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac {50\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {30+30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{189\ 3^{3/4} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 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 = 10.03, size = 59, normalized size = 0.09 \begin {gather*} \frac {1}{210} \left (\left (52-54 x+27 x^2\right )^{2/3} \left (838+324 x+63 x^2\right )+3000 \sqrt [3]{5} (-1+x) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27}{25} (-1+x)^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (2+3 x \right )^{3}}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}\, dx\]
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 + 52}}\, 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+52\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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