\(\int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx\) [705]

Optimal result

Integrand size = 20, antiderivative size = 52 \[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {1}{12} \left (52-54 x+27 x^2\right )^{2/3}-\sqrt [3]{5} (1-x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27}{25} (1-x)^2\right ) \] Output:

1/12*(27*x^2-54*x+52)^(2/3)-5^(1/3)*(1-x)*hypergeom([1/3, 1/2],[3/2],-27/2 
5*(1-x)^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 10.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.90 \[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\sqrt [3]{5} (-1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27}{25} (-1+x)^2\right ) \] Input:

Integrate[(2 + 3*x)/(52 - 54*x + 27*x^2)^(1/3),x]
 

Output:

(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]
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(367\) vs. \(2(52)=104\).

Time = 0.44 (sec) , antiderivative size = 367, normalized size of antiderivative = 7.06, 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

\(\displaystyle \frac {5 \sqrt [3]{5} \sqrt {(54 x-54)^2} \left (-\frac {60 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}+\frac {30\ 3^{3/4} \sqrt {2+\sqrt {3}} (55-54 x) \sqrt {\frac {54 x+\left (\frac {(54 x-54)^2}{2700}+1\right )^{2/3}-53}{\left (-54 x-\sqrt {3}+55\right )^2}} E\left (\arcsin \left (\frac {-54 x+\sqrt {3}+55}{-54 x-\sqrt {3}+55}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {55-54 x}{\left (-54 x-\sqrt {3}+55\right )^2}} \sqrt {(54 x-54)^2}}-\frac {\sqrt {(54 x-54)^2}}{15 \sqrt {3} \left (-54 x-\sqrt {3}+55\right )}\right )}{2 \sqrt {3} (54 x-54)}+\frac {1}{12} \left (27 x^2-54 x+52\right )^{2/3}\)

Input:

Int[(2 + 3*x)/(52 - 54*x + 27*x^2)^(1/3),x]
 

Output:

(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))
 

Defintions of rubi rules used

rule 233

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]
 

rule 760

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]
 

rule 833

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]
 

rule 1090

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]
 

rule 1160

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]
 

rule 2418

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]
 

Maple [F]

\[\int \frac {3 x +2}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x\]

Input:

int((3*x+2)/(27*x^2-54*x+52)^(1/3),x)
 

Output:

int((3*x+2)/(27*x^2-54*x+52)^(1/3),x)
 

Fricas [F]

\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)/(27*x^2-54*x+52)^(1/3),x, algorithm="fricas")
 

Output:

integral((3*x + 2)/(27*x^2 - 54*x + 52)^(1/3), x)
 

Sympy [F]

\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {3 x + 2}{\sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \] Input:

integrate((2+3*x)/(27*x**2-54*x+52)**(1/3),x)
 

Output:

Integral((3*x + 2)/(27*x**2 - 54*x + 52)**(1/3), x)
 

Maxima [F]

\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)/(27*x^2-54*x+52)^(1/3),x, algorithm="maxima")
 

Output:

integrate((3*x + 2)/(27*x^2 - 54*x + 52)^(1/3), x)
 

Giac [F]

\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)/(27*x^2-54*x+52)^(1/3),x, algorithm="giac")
 

Output:

integrate((3*x + 2)/(27*x^2 - 54*x + 52)^(1/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {3\,x+2}{{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \] Input:

int((3*x + 2)/(27*x^2 - 54*x + 52)^(1/3),x)
 

Output:

int((3*x + 2)/(27*x^2 - 54*x + 52)^(1/3), x)
 

Reduce [F]

\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=3 \left (\int \frac {x}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x \right )+2 \left (\int \frac {1}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x \right ) \] Input:

int((2+3*x)/(27*x^2-54*x+52)^(1/3),x)
                                                                                    
                                                                                    
 

Output:

3*int(x/(27*x**2 - 54*x + 52)**(1/3),x) + 2*int(1/(27*x**2 - 54*x + 52)**( 
1/3),x)