Integrand size = 22, antiderivative size = 585 \[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {5}{42} \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (28+54 x+27 x^2\right )^{2/3}-\frac {108 (1+x)}{7 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )}+\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{21 \sqrt {2} \sqrt [4]{3} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}-\frac {2 \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{21\ 3^{3/4} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}} \]
-5/42*(27*x^2+54*x+28)^(2/3)+1/21*(2+3*x)*(27*x^2+54*x+28)^(2/3)-108/7*(1+ x)/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2))-2/63*(6-2^(1/3)*(108+(54 +54*x)^2)^(1/3))*EllipticF((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6+6*3^(1/2))/ (-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2)),2*I-I*3^(1/2))*((1+(27*x^2+ 54*x+28)^(1/3)+(27*x^2+54*x+28)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6 -6*3^(1/2))^2)^(1/2)*3^(1/4)/(1+x)/((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/( -2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2))^2)^(1/2)+1/126*(6-2^(1/3)*(1 08+(54+54*x)^2)^(1/3))*EllipticE((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6+6*3^( 1/2))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2)),2*I-I*3^(1/2))*((1+(2 7*x^2+54*x+28)^(1/3)+(27*x^2+54*x+28)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^( 1/3)+6-6*3^(1/2))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*3^(3/4)/(1+x)*2^(1/2) /((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6 -6*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 \[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {1}{42} \left ((-1+6 x) \left (28+54 x+27 x^2\right )^{2/3}+36 (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-27 (1+x)^2\right )\right ) \]
((-1 + 6*x)*(28 + 54*x + 27*x^2)^(2/3) + 36*(1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, -27*(1 + x)^2])/42
Time = 0.45 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.67, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1166, 27, 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.
\(\displaystyle \int \frac {(3 x+2)^2}{\sqrt [3]{27 x^2+54 x+28}} \, dx\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle \frac {1}{63} \int -\frac {54 (5 x+4)}{\sqrt [3]{27 x^2+54 x+28}}dx+\frac {1}{21} \left (27 x^2+54 x+28\right )^{2/3} (3 x+2)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac {6}{7} \int \frac {5 x+4}{\sqrt [3]{27 x^2+54 x+28}}dx\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac {6}{7} \left (\frac {5}{36} \left (27 x^2+54 x+28\right )^{2/3}-\int \frac {1}{\sqrt [3]{27 x^2+54 x+28}}dx\right )\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac {6}{7} \left (\frac {5}{36} \left (27 x^2+54 x+28\right )^{2/3}-\frac {1}{54} \int \frac {1}{\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}d(54 x+54)\right )\) |
\(\Big \downarrow \) 233 |
\(\displaystyle \frac {1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac {6}{7} \left (\frac {5}{36} \left (27 x^2+54 x+28\right )^{2/3}-\frac {\sqrt {(54 x+54)^2} \int \frac {6 \sqrt {3} \sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}{2 \sqrt {3} (54 x+54)}\right )\) |
\(\Big \downarrow \) 833 |
\(\displaystyle \frac {1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac {6}{7} \left (\frac {5}{36} \left (27 x^2+54 x+28\right )^{2/3}-\frac {\sqrt {(54 x+54)^2} \left (\left (1+\sqrt {3}\right ) \int \frac {6 \sqrt {3}}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}-\int \frac {6 \sqrt {3} \left (-54 x+\sqrt {3}-53\right )}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}\right )}{2 \sqrt {3} (54 x+54)}\right )\) |
\(\Big \downarrow \) 760 |
\(\displaystyle \frac {1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac {6}{7} \left (\frac {5}{36} \left (27 x^2+54 x+28\right )^{2/3}-\frac {\sqrt {(54 x+54)^2} \left (-\int \frac {6 \sqrt {3} \left (-54 x+\sqrt {3}-53\right )}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}\right )}{2 \sqrt {3} (54 x+54)}\right )\) |
\(\Big \downarrow \) 2418 |
\(\displaystyle \frac {1}{21} (3 x+2) \left (27 x^2+54 x+28\right )^{2/3}-\frac {6}{7} \left (\frac {5}{36} \left (27 x^2+54 x+28\right )^{2/3}-\frac {\sqrt {(54 x+54)^2} \left (-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}+\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} E\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}-\frac {\sqrt {(54 x+54)^2}}{3 \sqrt {3} \left (-54 x-\sqrt {3}-53\right )}\right )}{2 \sqrt {3} (54 x+54)}\right )\) |
((2 + 3*x)*(28 + 54*x + 27*x^2)^(2/3))/21 - (6*((5*(28 + 54*x + 27*x^2)^(2 /3))/36 - (Sqrt[(54 + 54*x)^2]*(-1/3*Sqrt[(54 + 54*x)^2]/(Sqrt[3]*(-53 - S qrt[3] - 54*x)) + (6*3^(3/4)*Sqrt[2 + Sqrt[3]]*(-53 - 54*x)*Sqrt[(55 + 54* x + (1 + (54 + 54*x)^2/108)^(2/3))/(-53 - Sqrt[3] - 54*x)^2]*EllipticE[Arc Sin[(-53 + Sqrt[3] - 54*x)/(-53 - Sqrt[3] - 54*x)], -7 + 4*Sqrt[3]])/(Sqrt [-((-53 - 54*x)/(-53 - Sqrt[3] - 54*x)^2)]*Sqrt[(54 + 54*x)^2]) - (12*3^(1 /4)*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*(-53 - 54*x)*Sqrt[(55 + 54*x + (1 + (5 4 + 54*x)^2/108)^(2/3))/(-53 - Sqrt[3] - 54*x)^2]*EllipticF[ArcSin[(-53 + Sqrt[3] - 54*x)/(-53 - Sqrt[3] - 54*x)], -7 + 4*Sqrt[3]])/(Sqrt[-((-53 - 5 4*x)/(-53 - Sqrt[3] - 54*x)^2)]*Sqrt[(54 + 54*x)^2])))/(2*Sqrt[3]*(54 + 54 *x))))/7
3.26.4.3.1 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_))*((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[((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[((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]
\[\int \frac {\left (2+3 x \right )^{2}}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x\]
\[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {\left (3 x + 2\right )^{2}}{\sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \]
\[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{2}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^2}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {{\left (3\,x+2\right )}^2}{{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \]