Integrand size = 22, antiderivative size = 109 \[ \int \frac {(2+3 x)^4}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {11}{39} (2+3 x)^2 \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{39} (2+3 x)^3 \left (52-54 x+27 x^2\right )^{2/3}+\frac {20}{91} (76+29 x) \left (52-54 x+27 x^2\right )^{2/3}-\frac {2000}{91} \sqrt [3]{5} (1-x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27}{25} (1-x)^2\right ) \] Output:
11/39*(2+3*x)^2*(27*x^2-54*x+52)^(2/3)+1/39*(2+3*x)^3*(27*x^2-54*x+52)^(2/ 3)+20/91*(76+29*x)*(27*x^2-54*x+52)^(2/3)-2000/91*5^(1/3)*(1-x)*hypergeom( [1/3, 1/2],[3/2],-27/25*(1-x)^2)
Time = 10.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.59 \[ \int \frac {(2+3 x)^4}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {1}{273} \left (\left (52-54 x+27 x^2\right )^{2/3} \left (4924+2916 x+1071 x^2+189 x^3\right )+6000 \sqrt [3]{5} (-1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27}{25} (-1+x)^2\right )\right ) \] Input:
Integrate[(2 + 3*x)^4/(52 - 54*x + 27*x^2)^(1/3),x]
Output:
((52 - 54*x + 27*x^2)^(2/3)*(4924 + 2916*x + 1071*x^2 + 189*x^3) + 6000*5^ (1/3)*(-1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*(-1 + x)^2)/25])/273
Leaf count is larger than twice the leaf count of optimal. \(430\) vs. \(2(109)=218\).
Time = 0.54 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.94, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1166, 27, 1236, 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.
\(\displaystyle \int \frac {(3 x+2)^4}{\sqrt [3]{27 x^2-54 x+52}} \, dx\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle \frac {1}{117} \int -\frac {90 (8-33 x) (3 x+2)^2}{\sqrt [3]{27 x^2-54 x+52}}dx+\frac {1}{39} \left (27 x^2-54 x+52\right )^{2/3} (3 x+2)^3\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2-54 x+52\right )^{2/3}-\frac {10}{13} \int \frac {(8-33 x) (3 x+2)^2}{\sqrt [3]{27 x^2-54 x+52}}dx\) |
\(\Big \downarrow \) 1236 |
\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2-54 x+52\right )^{2/3}-\frac {10}{13} \left (\frac {1}{90} \int \frac {180 (52-87 x) (3 x+2)}{\sqrt [3]{27 x^2-54 x+52}}dx-\frac {11}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2-54 x+52\right )^{2/3}-\frac {10}{13} \left (2 \int \frac {(52-87 x) (3 x+2)}{\sqrt [3]{27 x^2-54 x+52}}dx-\frac {11}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}\right )\) |
\(\Big \downarrow \) 1225 |
\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2-54 x+52\right )^{2/3}-\frac {10}{13} \left (2 \left (-\frac {500}{7} \int \frac {1}{\sqrt [3]{27 x^2-54 x+52}}dx-\frac {1}{7} \left (27 x^2-54 x+52\right )^{2/3} (29 x+76)\right )-\frac {11}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}\right )\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2-54 x+52\right )^{2/3}-\frac {10}{13} \left (2 \left (-\frac {50}{189} \sqrt [3]{5} \int \frac {1}{\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}d(54 x-54)-\frac {1}{7} \left (27 x^2-54 x+52\right )^{2/3} (29 x+76)\right )-\frac {11}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}\right )\) |
\(\Big \downarrow \) 233 |
\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2-54 x+52\right )^{2/3}-\frac {10}{13} \left (2 \left (-\frac {250 \sqrt [3]{5} \sqrt {(54 x-54)^2} \int \frac {30 \sqrt {3} \sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}}{7 \sqrt {3} (54 x-54)}-\frac {1}{7} \left (27 x^2-54 x+52\right )^{2/3} (29 x+76)\right )-\frac {11}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}\right )\) |
\(\Big \downarrow \) 833 |
\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2-54 x+52\right )^{2/3}-\frac {10}{13} \left (2 \left (-\frac {250 \sqrt [3]{5} \sqrt {(54 x-54)^2} \left (\left (1+\sqrt {3}\right ) \int \frac {30 \sqrt {3}}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}-\int \frac {30 \sqrt {3} \left (-54 x+\sqrt {3}+55\right )}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}\right )}{7 \sqrt {3} (54 x-54)}-\frac {1}{7} \left (27 x^2-54 x+52\right )^{2/3} (29 x+76)\right )-\frac {11}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}\right )\) |
\(\Big \downarrow \) 760 |
\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2-54 x+52\right )^{2/3}-\frac {10}{13} \left (2 \left (-\frac {250 \sqrt [3]{5} \sqrt {(54 x-54)^2} \left (-\int \frac {30 \sqrt {3} \left (-54 x+\sqrt {3}+55\right )}{\sqrt {(54 x-54)^2}}d\sqrt [3]{\frac {(54 x-54)^2}{2700}+1}-\frac {60 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}\right )}{7 \sqrt {3} (54 x-54)}-\frac {1}{7} \left (27 x^2-54 x+52\right )^{2/3} (29 x+76)\right )-\frac {11}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}\right )\) |
\(\Big \downarrow \) 2418 |
\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2-54 x+52\right )^{2/3}-\frac {10}{13} \left (2 \left (-\frac {250 \sqrt [3]{5} \sqrt {(54 x-54)^2} \left (-\frac {60 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}+\frac {30\ 3^{3/4} \sqrt {2+\sqrt {3}} (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} E\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}-\frac {\sqrt {(54 x-54)^2}}{15 \sqrt {3} \left (-54 x-\sqrt {3}+55\right )}\right )}{7 \sqrt {3} (54 x-54)}-\frac {1}{7} \left (27 x^2-54 x+52\right )^{2/3} (29 x+76)\right )-\frac {11}{30} (3 x+2)^2 \left (27 x^2-54 x+52\right )^{2/3}\right )\) |
Input:
Int[(2 + 3*x)^4/(52 - 54*x + 27*x^2)^(1/3),x]
Output:
((2 + 3*x)^3*(52 - 54*x + 27*x^2)^(2/3))/39 - (10*((-11*(2 + 3*x)^2*(52 - 54*x + 27*x^2)^(2/3))/30 + 2*(-1/7*((76 + 29*x)*(52 - 54*x + 27*x^2)^(2/3) ) - (250*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 - Sqrt[3] - 54*x)^2]*Ell ipticE[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[-((5 5 - 54*x)/(55 - Sqrt[3] - 54*x)^2)]*Sqrt[(-54 + 54*x)^2])))/(7*Sqrt[3]*(-5 4 + 54*x)))))/13
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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
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]
\[\int \frac {\left (3 x +2\right )^{4}}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x\]
Input:
int((3*x+2)^4/(27*x^2-54*x+52)^(1/3),x)
Output:
int((3*x+2)^4/(27*x^2-54*x+52)^(1/3),x)
\[ \int \frac {(2+3 x)^4}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{4}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((2+3*x)^4/(27*x^2-54*x+52)^(1/3),x, algorithm="fricas")
Output:
integral((81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)/(27*x^2 - 54*x + 52)^(1/ 3), x)
\[ \int \frac {(2+3 x)^4}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {\left (3 x + 2\right )^{4}}{\sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \] Input:
integrate((2+3*x)**4/(27*x**2-54*x+52)**(1/3),x)
Output:
Integral((3*x + 2)**4/(27*x**2 - 54*x + 52)**(1/3), x)
\[ \int \frac {(2+3 x)^4}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{4}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((2+3*x)^4/(27*x^2-54*x+52)^(1/3),x, algorithm="maxima")
Output:
integrate((3*x + 2)^4/(27*x^2 - 54*x + 52)^(1/3), x)
\[ \int \frac {(2+3 x)^4}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{4}}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((2+3*x)^4/(27*x^2-54*x+52)^(1/3),x, algorithm="giac")
Output:
integrate((3*x + 2)^4/(27*x^2 - 54*x + 52)^(1/3), x)
Timed out. \[ \int \frac {(2+3 x)^4}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {{\left (3\,x+2\right )}^4}{{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \] Input:
int((3*x + 2)^4/(27*x^2 - 54*x + 52)^(1/3),x)
Output:
int((3*x + 2)^4/(27*x^2 - 54*x + 52)^(1/3), x)
\[ \int \frac {(2+3 x)^4}{\sqrt [3]{52-54 x+27 x^2}} \, dx=81 \left (\int \frac {x^{4}}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x \right )+216 \left (\int \frac {x^{3}}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x \right )+216 \left (\int \frac {x^{2}}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x \right )+96 \left (\int \frac {x}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x \right )+16 \left (\int \frac {1}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x \right ) \] Input:
int((2+3*x)^4/(27*x^2-54*x+52)^(1/3),x)
Output:
81*int(x**4/(27*x**2 - 54*x + 52)**(1/3),x) + 216*int(x**3/(27*x**2 - 54*x + 52)**(1/3),x) + 216*int(x**2/(27*x**2 - 54*x + 52)**(1/3),x) + 96*int(x /(27*x**2 - 54*x + 52)**(1/3),x) + 16*int(1/(27*x**2 - 54*x + 52)**(1/3),x )