Integrand size = 22, antiderivative size = 635 \[ \int \frac {(2+3 x)^3}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {1}{30} (2+3 x)^2 \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{7} (27+8 x) \left (52-54 x+27 x^2\right )^{2/3}+\frac {9000 \sqrt [3]{5} (1-x)}{7 \left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )}-\frac {25\ 5^{5/6} \sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{189 \sqrt {2} \sqrt [4]{3} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac {50\ 5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{189\ 3^{3/4} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}} \]
1/30*(2+3*x)^2*(27*x^2-54*x+52)^(2/3)+1/7*(27+8*x)*(27*x^2-54*x+52)^(2/3)+ 9000/7*5^(1/3)*(1-x)/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2))+5 0/567*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)+30-30*3^(1/2))^2)^(1/2)-25/ 567*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)*(2700+(-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 = 10.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.09 \[ \int \frac {(2+3 x)^3}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {1}{210} \left (\left (52-54 x+27 x^2\right )^{2/3} \left (838+324 x+63 x^2\right )+3000 \sqrt [3]{5} (-1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27}{25} (-1+x)^2\right )\right ) \]
((52 - 54*x + 27*x^2)^(2/3)*(838 + 324*x + 63*x^2) + 3000*5^(1/3)*(-1 + x) *Hypergeometric2F1[1/3, 1/2, 3/2, (-27*(-1 + x)^2)/25])/210
Time = 0.45 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.63, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1166, 27, 1225, 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.
\(\displaystyle \int \frac {(3 x+2)^3}{\sqrt [3]{27 x^2-54 x+52}} \, dx\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle \frac {1}{90} \int -\frac {360 (1-6 x) (3 x+2)}{\sqrt [3]{27 x^2-54 x+52}}dx+\frac {1}{30} \left (27 x^2-54 x+52\right )^{2/3} (3 x+2)^2\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}-4 \int \frac {(1-6 x) (3 x+2)}{\sqrt [3]{27 x^2-54 x+52}}dx\) |
\(\Big \downarrow \) 1225 |
\(\displaystyle \frac {1}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}-4 \left (-\frac {125}{7} \int \frac {1}{\sqrt [3]{27 x^2-54 x+52}}dx-\frac {1}{28} \left (27 x^2-54 x+52\right )^{2/3} (8 x+27)\right )\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}-4 \left (-\frac {25}{378} \sqrt [3]{5} \int \frac {1}{\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}d(54 x-54)-\frac {1}{28} \left (27 x^2-54 x+52\right )^{2/3} (8 x+27)\right )\) |
\(\Big \downarrow \) 233 |
\(\displaystyle \frac {1}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}-4 \left (-\frac {125 \sqrt [3]{5} \sqrt {(54 x-54)^2} \int \frac {30 \sqrt {3} \sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}{14 \sqrt {3} (54 x-54)}-\frac {1}{28} \left (27 x^2-54 x+52\right )^{2/3} (8 x+27)\right )\) |
\(\Big \downarrow \) 833 |
\(\displaystyle \frac {1}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}-4 \left (-\frac {125 \sqrt [3]{5} \sqrt {(54 x-54)^2} \left (\left (1+\sqrt {3}\right ) \int \frac {30 \sqrt {3}}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}-\int \frac {30 \sqrt {3} \left (-54 x+\sqrt {3}+55\right )}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}\right )}{14 \sqrt {3} (54 x-54)}-\frac {1}{28} \left (27 x^2-54 x+52\right )^{2/3} (8 x+27)\right )\) |
\(\Big \downarrow \) 760 |
\(\displaystyle \frac {1}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}-4 \left (-\frac {125 \sqrt [3]{5} \sqrt {(54 x-54)^2} \left (-\int \frac {30 \sqrt {3} \left (-54 x+\sqrt {3}+55\right )}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}-\frac {60 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}\right )}{14 \sqrt {3} (54 x-54)}-\frac {1}{28} \left (27 x^2-54 x+52\right )^{2/3} (8 x+27)\right )\) |
\(\Big \downarrow \) 2418 |
\(\displaystyle \frac {1}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}-4 \left (-\frac {125 \sqrt [3]{5} \sqrt {(54 x-54)^2} \left (-\frac {60 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}+\frac {30\ 3^{3/4} \sqrt {2+\sqrt {3}} (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} E\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}-\frac {\sqrt {(54 x-54)^2}}{15 \sqrt {3} \left (-54 x-\sqrt {3}+55\right )}\right )}{14 \sqrt {3} (54 x-54)}-\frac {1}{28} \left (27 x^2-54 x+52\right )^{2/3} (8 x+27)\right )\) |
((2 + 3*x)^2*(52 - 54*x + 27*x^2)^(2/3))/30 - 4*(-1/28*((27 + 8*x)*(52 - 5 4*x + 27*x^2)^(2/3)) - (125*5^(1/3)*Sqrt[(-54 + 54*x)^2]*(-1/15*Sqrt[(-54 + 54*x)^2]/(Sqrt[3]*(55 - Sqrt[3] - 54*x)) + (30*3^(3/4)*Sqrt[2 + Sqrt[3]] *(55 - 54*x)*Sqrt[(-53 + 54*x + (1 + (-54 + 54*x)^2/2700)^(2/3))/(55 - Sqr t[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*S qrt[3]])/(Sqrt[-((55 - 54*x)/(55 - Sqrt[3] - 54*x)^2)]*Sqrt[(-54 + 54*x)^2 ])))/(14*Sqrt[3]*(-54 + 54*x)))
3.25.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[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]
\[\int \frac {\left (2+3 x \right )^{3}}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x\]
\[ \int \frac {(2+3 x)^3}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{3}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {(2+3 x)^3}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {\left (3 x + 2\right )^{3}}{\sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]
\[ \int \frac {(2+3 x)^3}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{3}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {(2+3 x)^3}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{3}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^3}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {{\left (3\,x+2\right )}^3}{{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \]