Optimal. Leaf size=744 \[ -\frac {\left (27 x^2-54 x+52\right )^{2/3}}{1500 (3 x+2)}-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}+\frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{600\ 10^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{300 \sqrt {3} 10^{2/3}}+\frac {9 (1-x)}{50\ 5^{2/3} \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac {\log (3 x+2)}{600\ 10^{2/3}}+\frac {\left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{27000\ 3^{3/4} \sqrt [6]{5} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac {\sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{54000 \sqrt {2} \sqrt [4]{3} \sqrt [6]{5} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]
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Rubi [A] time = 0.67, antiderivative size = 744, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {744, 834, 843, 619, 235, 304, 219, 1879, 750} \[ -\frac {\left (27 x^2-54 x+52\right )^{2/3}}{1500 (3 x+2)}-\frac {\left (27 x^2-54 x+52\right )^{2/3}}{600 (3 x+2)^2}+\frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{600\ 10^{2/3}}-\frac {\tan ^{-1}\left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{300 \sqrt {3} 10^{2/3}}+\frac {9 (1-x)}{50\ 5^{2/3} \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )}-\frac {\log (3 x+2)}{600\ 10^{2/3}}+\frac {\left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} F\left (\sin ^{-1}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{27000\ 3^{3/4} \sqrt [6]{5} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac {\sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\sin ^{-1}\left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{54000 \sqrt {2} \sqrt [4]{3} \sqrt [6]{5} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)} \]
Antiderivative was successfully verified.
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Rule 219
Rule 235
Rule 304
Rule 619
Rule 744
Rule 750
Rule 834
Rule 843
Rule 1879
Rubi steps
\begin {align*} \int \frac {1}{(2+3 x)^3 \sqrt [3]{52-54 x+27 x^2}} \, dx &=-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac {\int \frac {-324+54 x}{(2+3 x)^2 \sqrt [3]{52-54 x+27 x^2}} \, dx}{1800}\\ &=-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}+\frac {\int \frac {22680+9720 x}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx}{1620000}\\ &=-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}+\frac {1}{500} \int \frac {1}{\sqrt [3]{52-54 x+27 x^2}} \, dx+\frac {1}{100} \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx\\ &=-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{300 \sqrt {3} 10^{2/3}}-\frac {\log (2+3 x)}{600\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{600\ 10^{2/3}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{2700}}} \, dx,x,-54+54 x\right )}{27000\ 5^{2/3}}\\ &=-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{300 \sqrt {3} 10^{2/3}}-\frac {\log (2+3 x)}{600\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{600\ 10^{2/3}}+\frac {\sqrt {(-54+54 x)^2} \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 )}{200 \sqrt {3} 5^{2/3} (-54+54 x)}\\ &=-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{300 \sqrt {3} 10^{2/3}}-\frac {\log (2+3 x)}{600\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{600\ 10^{2/3}}-\frac {\sqrt {(-54+54 x)^2} \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 )}{200 \sqrt {3} 5^{2/3} (-54+54 x)}+\frac {\left (\sqrt {\frac {1}{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 )}{100\ 5^{2/3} (-54+54 x)}\\ &=-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{600 (2+3 x)^2}-\frac {\left (52-54 x+27 x^2\right )^{2/3}}{1500 (2+3 x)}+\frac {9 (1-x)}{50\ 5^{2/3} \left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{300 \sqrt {3} 10^{2/3}}-\frac {\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 )}{54000 \sqrt {2} \sqrt [4]{3} \sqrt [6]{5} (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 {\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 )}{27000\ 3^{3/4} \sqrt [6]{5} (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 {\log (2+3 x)}{600\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{600\ 10^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 233, normalized size = 0.31 \[ \frac {-150 \sqrt [3]{3} \sqrt [3]{\frac {9 x-5 i \sqrt {3}-9}{3 x+2}} \sqrt [3]{\frac {9 x+5 i \sqrt {3}-9}{3 x+2}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {15-5 i \sqrt {3}}{9 x+6},\frac {15+5 i \sqrt {3}}{9 x+6}\right )+3^{5/6} 10^{2/3} \sqrt [3]{-9 i x+5 \sqrt {3}+9 i} \left (9 x-5 i \sqrt {3}-9\right ) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {9 i x+5 \sqrt {3}-9 i}{10 \sqrt {3}}\right )-\frac {90 (2 x+3) \left (27 x^2-54 x+52\right )}{(3 x+2)^2}}{90000 \sqrt [3]{27 x^2-54 x+52}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 7.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {2}{3}}}{729 \, x^{5} - 540 \, x^{3} + 1080 \, x^{2} + 1440 \, x + 416}, x\right ) \]
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 \[ \int \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (3 x +2\right )^{3} \left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}^{3}}\,{d x} \]
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 \[ \int \frac {1}{{\left (3\,x+2\right )}^3\,{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{\left (3 x + 2\right )^{3} \sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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