Optimal. Leaf size=628 \[ \frac {1}{21} (3 x+2) \left (27 x^2-54 x+52\right )^{2/3}+\frac {25}{42} \left (27 x^2-54 x+52\right )^{2/3}+\frac {2700 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}+\frac {5\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{63\ 3^{3/4} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac {5\ 5^{5/6} \sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{126 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]
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Rubi [A] time = 0.54, antiderivative size = 628, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {742, 12, 640, 619, 235, 304, 219, 1879} \[ \frac {1}{21} (3 x+2) \left (27 x^2-54 x+52\right )^{2/3}+\frac {25}{42} \left (27 x^2-54 x+52\right )^{2/3}+\frac {2700 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}+\frac {5\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{63\ 3^{3/4} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac {5\ 5^{5/6} \sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{126 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 219
Rule 235
Rule 304
Rule 619
Rule 640
Rule 742
Rule 1879
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{\sqrt [3]{52-54 x+27 x^2}} \, dx &=\frac {1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{63} \int \frac {1350 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=\frac {1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {150}{7} \int \frac {x}{\sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=\frac {25}{42} \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {150}{7} \int \frac {1}{\sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=\frac {25}{42} \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{63} \left (5 \sqrt [3]{5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{2700}}} \, dx,x,-54+54 x\right )\\ &=\frac {25}{42} \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {\left (25 \sqrt {3} \sqrt [3]{5} \sqrt {(-54+54 x)^2}\right ) \operatorname {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 (-54+54 x)}\\ &=\frac {25}{42} \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}-\frac {\left (25 \sqrt {3} \sqrt [3]{5} \sqrt {(-54+54 x)^2}\right ) \operatorname {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 (-54+54 x)}+\frac {\left (25 \sqrt [3]{5} \sqrt {6 \left (2+\sqrt {3}\right )} \sqrt {(-54+54 x)^2}\right ) \operatorname {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 {25}{42} \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {2700 \sqrt [3]{5} (1-x)}{7 \left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )}-\frac {5\ 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 )}{126 \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 {5\ 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 )}{63\ 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] time = 0.02, size = 54, normalized size = 0.09 \[ \frac {1}{42} \left (180 \sqrt [3]{5} (x-1) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27}{25} (x-1)^2\right )+\left (27 x^2-54 x+52\right )^{2/3} (6 x+29)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {9 \, x^{2} + 12 \, x + 4}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (3 x +2\right )^{2}}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^2}{{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\left (3 x + 2\right )^{2}}{\sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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