Integrand size = 20, antiderivative size = 603 \[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\frac {90 \sqrt [3]{5} (1-x)}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}-\frac {5^{5/6} \sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{108 \sqrt {2} \sqrt [4]{3} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac {5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{54\ 3^{3/4} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}} \]
1/12*(27*x^2-54*x+52)^(2/3)+90*5^(1/3)*(1-x)/(-10^(1/3)*(2700+(-54+54*x)^2 )^(1/3)+30-30*3^(1/2))+1/162*5^(5/6)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3 ))*EllipticF((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30+30*3^(1/2))/(-10^(1/3 )*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2)),2*I-I*3^(1/2))*((900+30*10^(1/3 )*(2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3) *(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2))^2)^(1/2)*3^(1/4)/(1-x)/((-30+10^ (1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-3 0*3^(1/2))^2)^(1/2)-1/324*5^(5/6)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))* EllipticE((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30+30*3^(1/2))/(-10^(1/3)*( 2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2)),2*I-I*3^(1/2))*((900+30*10^(1/3)*( 2700+(-54+54*x)^2)^(1/3)+10^(2/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2 700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2))^2)^(1/2)*(1/2+1/2*3^(1/2))*3^(3/4)/ (1-x)/((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x )^2)^(1/3)+30-30*3^(1/2))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.08 \[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\sqrt [3]{5} (-1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27}{25} (-1+x)^2\right ) \]
(52 - 54*x + 27*x^2)^(2/3)/12 + 5^(1/3)*(-1 + x)*Hypergeometric2F1[1/3, 1/ 2, 3/2, (-27*(-1 + x)^2)/25]
Time = 0.45 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.61, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1160, 1090, 233, 833, 760, 2418}
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.
| \(\displaystyle \int \frac {3 x+2}{\sqrt [3]{27 x^2-54 x+52}} \, dx\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle 5 \int \frac {1}{\sqrt [3]{27 x^2-54 x+52}}dx+\frac {1}{12} \left (27 x^2-54 x+52\right )^{2/3}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{54} \sqrt [3]{5} \int \frac {1}{\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}d(54 x-54)+\frac {1}{12} \left (27 x^2-54 x+52\right )^{2/3}\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {5 \sqrt [3]{5} \sqrt {(54 x-54)^2} \int \frac {30 \sqrt {3} \sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}{2 \sqrt {3} (54 x-54)}+\frac {1}{12} \left (27 x^2-54 x+52\right )^{2/3}\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {5 \sqrt [3]{5} \sqrt {(54 x-54)^2} \left (\left (1+\sqrt {3}\right ) \int \frac {30 \sqrt {3}}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}-\int \frac {30 \sqrt {3} \left (-54 x+\sqrt {3}+55\right )}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}\right )}{2 \sqrt {3} (54 x-54)}+\frac {1}{12} \left (27 x^2-54 x+52\right )^{2/3}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {5 \sqrt [3]{5} \sqrt {(54 x-54)^2} \left (-\int \frac {30 \sqrt {3} \left (-54 x+\sqrt {3}+55\right )}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}-\frac {60 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}\right )}{2 \sqrt {3} (54 x-54)}+\frac {1}{12} \left (27 x^2-54 x+52\right )^{2/3}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {5 \sqrt [3]{5} \sqrt {(54 x-54)^2} \left (-\frac {60 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}+\frac {30\ 3^{3/4} \sqrt {2+\sqrt {3}} (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} E\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}-\frac {\sqrt {(54 x-54)^2}}{15 \sqrt {3} \left (-54 x-\sqrt {3}+55\right )}\right )}{2 \sqrt {3} (54 x-54)}+\frac {1}{12} \left (27 x^2-54 x+52\right )^{2/3}\) |
(52 - 54*x + 27*x^2)^(2/3)/12 + (5*5^(1/3)*Sqrt[(-54 + 54*x)^2]*(-1/15*Sqr t[(-54 + 54*x)^2]/(Sqrt[3]*(55 - Sqrt[3] - 54*x)) + (30*3^(3/4)*Sqrt[2 + S qrt[3]]*(55 - 54*x)*Sqrt[(-53 + 54*x + (1 + (-54 + 54*x)^2/2700)^(2/3))/(5 5 - Sqrt[3] - 54*x)^2]*EllipticE[ArcSin[(55 + Sqrt[3] - 54*x)/(55 - Sqrt[3 ] - 54*x)], -7 + 4*Sqrt[3]])/(Sqrt[-((55 - 54*x)/(55 - Sqrt[3] - 54*x)^2)] *Sqrt[(-54 + 54*x)^2]) - (60*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*(55 - 54*x)*Sqrt[(-53 + 54*x + (1 + (-54 + 54*x)^2/2700)^(2/3))/(55 - Sqrt[3] - 54*x)^2]*EllipticF[ArcSin[(55 + Sqrt[3] - 54*x)/(55 - Sqrt[3] - 54*x)], - 7 + 4*Sqrt[3]])/(Sqrt[-((55 - 54*x)/(55 - Sqrt[3] - 54*x)^2)]*Sqrt[(-54 + 54*x)^2])))/(2*Sqrt[3]*(-54 + 54*x))
3.25.99.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b }, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) *s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
\[\int \frac {2+3 x}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x\]
\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {3 x + 2}{\sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]
\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {3\,x+2}{{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \]