Integrand size = 19, antiderivative size = 558 \[ \int \frac {(2+3 x)^3}{\sqrt [3]{4+27 x^2}} \, dx=\frac {1}{30} (2+3 x)^2 \left (4+27 x^2\right )^{2/3}+\frac {4}{35} (7+4 x) \left (4+27 x^2\right )^{2/3}-\frac {96 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )}+\frac {16 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{21\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {32\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right ),-7+4 \sqrt {3}\right )}{63 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}} \]
1/30*(2+3*x)^2*(27*x^2+4)^(2/3)+4/35*(7+4*x)*(27*x^2+4)^(2/3)-96/7*x/(-(27 *x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))-32/189*2^(5/6)*(2^(2/3)-(27*x^2+4)^(1/3 ))*EllipticF((-(27*x^2+4)^(1/3)+2^(2/3)*(1+3^(1/2)))/(-(27*x^2+4)^(1/3)+2^ (2/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((2*2^(1/3)+2^(2/3)*(27*x^2+4)^(1/3)+(27 *x^2+4)^(2/3))/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))^2)^(1/2)*3^(3/4)/x/ ((-2^(2/3)+(27*x^2+4)^(1/3))/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))^2)^(1 /2)+16/63*2^(1/3)*(2^(2/3)-(27*x^2+4)^(1/3))*EllipticE((-(27*x^2+4)^(1/3)+ 2^(2/3)*(1+3^(1/2)))/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2))),2*I-I*3^(1/2) )*((2*2^(1/3)+2^(2/3)*(27*x^2+4)^(1/3)+(27*x^2+4)^(2/3))/(-(27*x^2+4)^(1/3 )+2^(2/3)*(1-3^(1/2)))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*3^(1/4)/x/((-2^( 2/3)+(27*x^2+4)^(1/3))/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 16.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.09 \[ \int \frac {(2+3 x)^3}{\sqrt [3]{4+27 x^2}} \, dx=\frac {1}{210} \left (4+27 x^2\right )^{2/3} \left (196+180 x+63 x^2\right )+\frac {16}{7} \sqrt [3]{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27 x^2}{4}\right ) \]
((4 + 27*x^2)^(2/3)*(196 + 180*x + 63*x^2))/210 + (16*2^(1/3)*x*Hypergeome tric2F1[1/3, 1/2, 3/2, (-27*x^2)/4])/7
Time = 0.53 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {497, 27, 676, 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.
| \(\displaystyle \int \frac {(3 x+2)^3}{\sqrt [3]{27 x^2+4}} \, dx\) |
| \(\Big \downarrow \) 497 |
| \(\displaystyle \frac {1}{90} \int \frac {288 (3 x+1) (3 x+2)}{\sqrt [3]{27 x^2+4}}dx+\frac {1}{30} \left (27 x^2+4\right )^{2/3} (3 x+2)^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {16}{5} \int \frac {(3 x+1) (3 x+2)}{\sqrt [3]{27 x^2+4}}dx+\frac {1}{30} \left (27 x^2+4\right )^{2/3} (3 x+2)^2\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {16}{5} \left (\frac {10}{7} \int \frac {1}{\sqrt [3]{27 x^2+4}}dx+\frac {1}{7} \left (27 x^2+4\right )^{2/3} x+\frac {1}{4} \left (27 x^2+4\right )^{2/3}\right )+\frac {1}{30} \left (27 x^2+4\right )^{2/3} (3 x+2)^2\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {16}{5} \left (\frac {5 \sqrt {x^2} \int \frac {\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}}{7 \sqrt {3} x}+\frac {1}{7} \left (27 x^2+4\right )^{2/3} x+\frac {1}{4} \left (27 x^2+4\right )^{2/3}\right )+\frac {1}{30} \left (27 x^2+4\right )^{2/3} (3 x+2)^2\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {16}{5} \left (\frac {5 \sqrt {x^2} \left (2^{2/3} \left (1+\sqrt {3}\right ) \int \frac {1}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}\right )}{7 \sqrt {3} x}+\frac {1}{7} \left (27 x^2+4\right )^{2/3} x+\frac {1}{4} \left (27 x^2+4\right )^{2/3}\right )+\frac {1}{30} \left (27 x^2+4\right )^{2/3} (3 x+2)^2\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {16}{5} \left (\frac {5 \sqrt {x^2} \left (-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}}}\right )}{7 \sqrt {3} x}+\frac {1}{7} \left (27 x^2+4\right )^{2/3} x+\frac {1}{4} \left (27 x^2+4\right )^{2/3}\right )+\frac {1}{30} \left (27 x^2+4\right )^{2/3} (3 x+2)^2\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {16}{5} \left (\frac {5 \sqrt {x^2} \left (-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}}}-\frac {6 \sqrt {3} \sqrt {x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )}{7 \sqrt {3} x}+\frac {1}{7} \left (27 x^2+4\right )^{2/3} x+\frac {1}{4} \left (27 x^2+4\right )^{2/3}\right )+\frac {1}{30} \left (27 x^2+4\right )^{2/3} (3 x+2)^2\) |
((2 + 3*x)^2*(4 + 27*x^2)^(2/3))/30 + (16*((4 + 27*x^2)^(2/3)/4 + (x*(4 + 27*x^2)^(2/3))/7 + (5*Sqrt[x^2]*((-6*Sqrt[3]*Sqrt[x^2])/(2^(2/3)*(1 - Sqrt [3]) - (4 + 27*x^2)^(1/3)) + (2^(1/3)*Sqrt[2 + Sqrt[3]]*(2^(2/3) - (4 + 27 *x^2)^(1/3))*Sqrt[(2*2^(1/3) + 2^(2/3)*(4 + 27*x^2)^(1/3) + (4 + 27*x^2)^( 2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2]*EllipticE[ArcSin[(2^ (2/3)*(1 + Sqrt[3]) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27 *x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[x^2]*Sqrt[-((2^(2/3) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2)]) - (2*2^ (1/3)*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*(2^(2/3) - (4 + 27*x^2)^(1/3))*Sqrt[ (2*2^(1/3) + 2^(2/3)*(4 + 27*x^2)^(1/3) + (4 + 27*x^2)^(2/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2]*EllipticF[ArcSin[(2^(2/3)*(1 + Sqrt[3] ) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3*3^(3/4)*Sqrt[x^2]*Sqrt[-((2^(2/3) - (4 + 27*x^2)^(1/3))/( 2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2)])))/(7*Sqrt[3]*x)))/5
3.7.100.3.1 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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
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[((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]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 2.63 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.07
| method | result | size |
| risch | \(\frac {\left (63 x^{2}+180 x +196\right ) \left (27 x^{2}+4\right )^{\frac {2}{3}}}{210}+\frac {16 \,2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^{2}}{4}\right )}{7}\) | \(40\) |
| meijerg | \(4 \,2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^{2}}{4}\right )+9 \,2^{\frac {1}{3}} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},1;2;-\frac {27 x^{2}}{4}\right )+9 \,2^{\frac {1}{3}} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {3}{2};\frac {5}{2};-\frac {27 x^{2}}{4}\right )+\frac {27 \,2^{\frac {1}{3}} x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},2;3;-\frac {27 x^{2}}{4}\right )}{8}\) | \(76\) |
1/210*(63*x^2+180*x+196)*(27*x^2+4)^(2/3)+16/7*2^(1/3)*x*hypergeom([1/3,1/ 2],[3/2],-27/4*x^2)
\[ \int \frac {(2+3 x)^3}{\sqrt [3]{4+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{3}}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
Time = 2.77 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.15 \[ \int \frac {(2+3 x)^3}{\sqrt [3]{4+27 x^2}} \, dx=9 \cdot \sqrt [3]{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{i \pi }}{4}} \right )} + \frac {3 x^{2} \left (27 x^{2} + 4\right )^{\frac {2}{3}}}{10} + 4 \cdot \sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{i \pi }}{4}} \right )} + \frac {14 \left (27 x^{2} + 4\right )^{\frac {2}{3}}}{15} \]
9*2**(1/3)*x**3*hyper((1/3, 3/2), (5/2,), 27*x**2*exp_polar(I*pi)/4) + 3*x **2*(27*x**2 + 4)**(2/3)/10 + 4*2**(1/3)*x*hyper((1/3, 1/2), (3/2,), 27*x* *2*exp_polar(I*pi)/4) + 14*(27*x**2 + 4)**(2/3)/15
\[ \int \frac {(2+3 x)^3}{\sqrt [3]{4+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{3}}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {(2+3 x)^3}{\sqrt [3]{4+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{3}}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^3}{\sqrt [3]{4+27 x^2}} \, dx=\int \frac {{\left (3\,x+2\right )}^3}{{\left (27\,x^2+4\right )}^{1/3}} \,d x \]