\(\int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx\) [712]

Optimal result

Integrand size = 20, antiderivative size = 41 \[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {1}{12} \left (28+54 x+27 x^2\right )^{2/3}-(1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-27 (1+x)^2\right ) \] Output:

1/12*(27*x^2+54*x+28)^(2/3)-(1+x)*hypergeom([1/3, 1/2],[3/2],-27*(1+x)^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 10.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {1}{12} \left (28+54 x+27 x^2\right )^{2/3}-(1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-27 (1+x)^2\right ) \] Input:

Integrate[(2 + 3*x)/(28 + 54*x + 27*x^2)^(1/3),x]
 

Output:

(28 + 54*x + 27*x^2)^(2/3)/12 - (1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, - 
27*(1 + x)^2]
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(362\) vs. \(2(41)=82\).

Time = 0.43 (sec) , antiderivative size = 362, normalized size of antiderivative = 8.83, 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.

\(\displaystyle \int \frac {3 x+2}{\sqrt [3]{27 x^2+54 x+28}} \, dx\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{12} \left (27 x^2+54 x+28\right )^{2/3}-\int \frac {1}{\sqrt [3]{27 x^2+54 x+28}}dx\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{12} \left (27 x^2+54 x+28\right )^{2/3}-\frac {1}{54} \int \frac {1}{\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}d(54 x+54)\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {1}{12} \left (27 x^2+54 x+28\right )^{2/3}-\frac {\sqrt {(54 x+54)^2} \int \frac {6 \sqrt {3} \sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}{2 \sqrt {3} (54 x+54)}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {1}{12} \left (27 x^2+54 x+28\right )^{2/3}-\frac {\sqrt {(54 x+54)^2} \left (\left (1+\sqrt {3}\right ) \int \frac {6 \sqrt {3}}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}-\int \frac {6 \sqrt {3} \left (-54 x+\sqrt {3}-53\right )}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}\right )}{2 \sqrt {3} (54 x+54)}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {1}{12} \left (27 x^2+54 x+28\right )^{2/3}-\frac {\sqrt {(54 x+54)^2} \left (-\int \frac {6 \sqrt {3} \left (-54 x+\sqrt {3}-53\right )}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}\right )}{2 \sqrt {3} (54 x+54)}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {1}{12} \left (27 x^2+54 x+28\right )^{2/3}-\frac {\sqrt {(54 x+54)^2} \left (-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}+\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} E\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}-\frac {\sqrt {(54 x+54)^2}}{3 \sqrt {3} \left (-54 x-\sqrt {3}-53\right )}\right )}{2 \sqrt {3} (54 x+54)}\)

Input:

Int[(2 + 3*x)/(28 + 54*x + 27*x^2)^(1/3),x]
 

Output:

(28 + 54*x + 27*x^2)^(2/3)/12 - (Sqrt[(54 + 54*x)^2]*(-1/3*Sqrt[(54 + 54*x 
)^2]/(Sqrt[3]*(-53 - Sqrt[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[ArcSin[(-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)*Sqr 
t[(55 + 54*x + (1 + (54 + 54*x)^2/108)^(2/3))/(-53 - Sqrt[3] - 54*x)^2]*El 
lipticF[ArcSin[(-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]) 
))/(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]

\[\int \frac {3 x +2}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x\]

Input:

int((3*x+2)/(27*x^2+54*x+28)^(1/3),x)
 

Output:

int((3*x+2)/(27*x^2+54*x+28)^(1/3),x)
 

Fricas [F]

\[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)/(27*x^2+54*x+28)^(1/3),x, algorithm="fricas")
 

Output:

integral((3*x + 2)/(27*x^2 + 54*x + 28)^(1/3), x)
 

Sympy [F]

\[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {3 x + 2}{\sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \] Input:

integrate((2+3*x)/(27*x**2+54*x+28)**(1/3),x)
 

Output:

Integral((3*x + 2)/(27*x**2 + 54*x + 28)**(1/3), x)
 

Maxima [F]

\[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)/(27*x^2+54*x+28)^(1/3),x, algorithm="maxima")
 

Output:

integrate((3*x + 2)/(27*x^2 + 54*x + 28)^(1/3), x)
 

Giac [F]

\[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)/(27*x^2+54*x+28)^(1/3),x, algorithm="giac")
 

Output:

integrate((3*x + 2)/(27*x^2 + 54*x + 28)^(1/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {3\,x+2}{{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \] Input:

int((3*x + 2)/(54*x + 27*x^2 + 28)^(1/3),x)
 

Output:

int((3*x + 2)/(54*x + 27*x^2 + 28)^(1/3), x)
 

Reduce [F]

\[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=3 \left (\int \frac {x}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x \right )+2 \left (\int \frac {1}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x \right ) \] Input:

int((2+3*x)/(27*x^2+54*x+28)^(1/3),x)
                                                                                    
                                                                                    
 

Output:

3*int(x/(27*x**2 + 54*x + 28)**(1/3),x) + 2*int(1/(27*x**2 + 54*x + 28)**( 
1/3),x)