Integrand size = 22, antiderivative size = 108 \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{3 \sqrt {3} 10^{2/3}}-\frac {\log (2+3 x)}{6\ 10^{2/3}}+\frac {\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{6\ 10^{2/3}} \]
-1/60*ln(2+3*x)*10^(1/3)+1/60*ln(216-81*x-27*10^(1/3)*(27*x^2-54*x+52)^(1/ 3))*10^(1/3)+1/90*arctan(-1/3*3^(1/2)-1/15*2^(2/3)*(8-3*x)*5^(2/3)/(27*x^2 -54*x+52)^(1/3)*3^(1/2))*10^(1/3)*3^(1/2)
Time = 0.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {8\ 10^{2/3}-3\ 10^{2/3} x+5 \sqrt [3]{52-54 x+27 x^2}}{5 \sqrt {3} \sqrt [3]{52-54 x+27 x^2}}\right )-2 \log \left (-8 10^{2/3}+3\ 10^{2/3} x+10 \sqrt [3]{52-54 x+27 x^2}\right )+\log \left (-64 \sqrt [3]{10}+48 \sqrt [3]{10} x-9 \sqrt [3]{10} x^2+10^{2/3} (-8+3 x) \sqrt [3]{52-54 x+27 x^2}-10 \left (52-54 x+27 x^2\right )^{2/3}\right )}{18\ 10^{2/3}} \]
-1/18*(2*Sqrt[3]*ArcTan[(8*10^(2/3) - 3*10^(2/3)*x + 5*(52 - 54*x + 27*x^2 )^(1/3))/(5*Sqrt[3]*(52 - 54*x + 27*x^2)^(1/3))] - 2*Log[-8*10^(2/3) + 3*1 0^(2/3)*x + 10*(52 - 54*x + 27*x^2)^(1/3)] + Log[-64*10^(1/3) + 48*10^(1/3 )*x - 9*10^(1/3)*x^2 + 10^(2/3)*(-8 + 3*x)*(52 - 54*x + 27*x^2)^(1/3) - 10 *(52 - 54*x + 27*x^2)^(2/3)])/10^(2/3)
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1175}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2-54 x+52}} \, dx\) |
| \(\Big \downarrow \) 1175 |
| \(\displaystyle -\frac {\arctan \left (\frac {2^{2/3} (8-3 x)}{\sqrt {3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} 10^{2/3}}+\frac {\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac {\log (3 x+2)}{6\ 10^{2/3}}\) |
-1/3*ArcTan[1/Sqrt[3] + (2^(2/3)*(8 - 3*x))/(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1/3))]/(Sqrt[3]*10^(2/3)) - Log[2 + 3*x]/(6*10^(2/3)) + Log[216 - 81*x - 27*10^(1/3)*(52 - 54*x + 27*x^2)^(1/3)]/(6*10^(2/3))
3.25.100.3.1 Defintions of rubi rules used
Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Sy mbol] :> With[{q = Rt[3*c*e^2*(2*c*d - b*e), 3]}, Simp[(-Sqrt[3])*c*e*(ArcT an[1/Sqrt[3] + 2*((c*d - b*e - c*e*x)/(Sqrt[3]*q*(a + b*x + c*x^2)^(1/3)))] /q^2), x] + (-Simp[3*c*e*(Log[d + e*x]/(2*q^2)), x] + Simp[3*c*e*(Log[c*d - b*e - c*e*x - q*(a + b*x + c*x^2)^(1/3)]/(2*q^2)), x])] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d^2 - b*c*d*e + b^2*e^2 - 3*a*c*e^2, 0] && PosQ[c*e^2 *(2*c*d - b*e)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.76 (sec) , antiderivative size = 2182, normalized size of antiderivative = 20.20
1/90*RootOf(_Z^3-10)*ln((10710567117700515*RootOf(RootOf(_Z^3-10)^2+30*_Z* RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^3+36039316478811750*RootOf(R ootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x^3+3 5356252380606720*(27*x^2-54*x+52)^(2/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*Roo tOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^2*x+19773354678831720*RootOf(RootOf (_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x^2+66534122 730114000*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootO f(_Z^3-10)^2*x^2-94283339681617920*(27*x^2-54*x+52)^(2/3)*RootOf(RootOf(_Z ^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*RootOf(_Z^3-10)^2-380074665718431 *(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x^2+94666517170267230*(27*x^2-54 *x+52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*Root Of(_Z^3-10)*x^2+17576315270072640*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z ^3-10)+900*_Z^2)*RootOf(_Z^3-10)^3*x+59141442426768000*RootOf(RootOf(_Z^3- 10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)^2*RootOf(_Z^3-10)^2*x+20270648838316 32*(27*x^2-54*x+52)^(1/3)*RootOf(_Z^3-10)^2*x-504888091574758560*(27*x^2-5 4*x+52)^(1/3)*RootOf(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*Roo tOf(_Z^3-10)*x-3515263054014528*RootOf(_Z^3-10)*x^3-11828288485353600*Root Of(RootOf(_Z^3-10)^2+30*_Z*RootOf(_Z^3-10)+900*_Z^2)*x^3+10518501907807470 *(27*x^2-54*x+52)^(2/3)*x-2702753178442176*(27*x^2-54*x+52)^(1/3)*RootOf(_ Z^3-10)^2+673184122099678080*(27*x^2-54*x+52)^(1/3)*RootOf(RootOf(_Z^3-...
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (83) = 166\).
Time = 2.41 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.98 \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=-\frac {1}{90} \cdot 100^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {100^{\frac {1}{6}} {\left (2 \cdot 100^{\frac {2}{3}} \sqrt {3} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {2}{3}} {\left (3 \, x - 8\right )} + 100^{\frac {1}{3}} \sqrt {3} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} + 20 \, \sqrt {3} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (9 \, x^{2} - 48 \, x + 64\right )}\right )}}{90 \, {\left (9 \, x^{3} - 162 \, x^{2} + 372 \, x - 344\right )}}\right ) - \frac {1}{1800} \cdot 100^{\frac {2}{3}} \log \left (\frac {100^{\frac {2}{3}} {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {2}{3}} + 100^{\frac {1}{3}} {\left (9 \, x^{2} - 48 \, x + 64\right )} - 10 \, {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x - 8\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac {1}{900} \cdot 100^{\frac {2}{3}} \log \left (\frac {100^{\frac {1}{3}} {\left (3 \, x - 8\right )} + 10 \, {\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}}{3 \, x + 2}\right ) \]
-1/90*100^(1/6)*sqrt(3)*arctan(1/90*100^(1/6)*(2*100^(2/3)*sqrt(3)*(27*x^2 - 54*x + 52)^(2/3)*(3*x - 8) + 100^(1/3)*sqrt(3)*(27*x^3 + 54*x^2 + 36*x + 8) + 20*sqrt(3)*(27*x^2 - 54*x + 52)^(1/3)*(9*x^2 - 48*x + 64))/(9*x^3 - 162*x^2 + 372*x - 344)) - 1/1800*100^(2/3)*log((100^(2/3)*(27*x^2 - 54*x + 52)^(2/3) + 100^(1/3)*(9*x^2 - 48*x + 64) - 10*(27*x^2 - 54*x + 52)^(1/3 )*(3*x - 8))/(9*x^2 + 12*x + 4)) + 1/900*100^(2/3)*log((100^(1/3)*(3*x - 8 ) + 10*(27*x^2 - 54*x + 52)^(1/3))/(3*x + 2))
\[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]
\[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}} \,d x } \]
\[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {1}{\left (3\,x+2\right )\,{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \]