3.8.2 \(\int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx\) [702]

3.8.2.1 Optimal result

Integrand size = 17, antiderivative size = 529 \[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\frac {1}{12} \left (4+27 x^2\right )^{2/3}-\frac {6 x}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {2\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right ),-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}} \]

output

1/12*(27*x^2+4)^(2/3)-6*x/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))-2/27*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)+1/9*2^(1/3)*(2^(2/3)-(27*x^2+4)^(1/3))*El 
lipticE((-(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)
 

3.8.2.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 9.92 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.08 \[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\frac {1}{12} \left (4+27 x^2\right )^{2/3}+\sqrt [3]{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27 x^2}{4}\right ) \]

input

Integrate[(2 + 3*x)/(4 + 27*x^2)^(1/3),x]
 

output

(4 + 27*x^2)^(2/3)/12 + 2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*x^ 
2)/4]
 

3.8.2.3 Rubi [A] (warning: unable to verify)

Time = 0.46 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {455, 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.

\(\displaystyle \int \frac {3 x+2}{\sqrt [3]{27 x^2+4}} \, dx\)

\(\Big \downarrow \) 455

\(\displaystyle 2 \int \frac {1}{\sqrt [3]{27 x^2+4}}dx+\frac {1}{12} \left (27 x^2+4\right )^{2/3}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {\sqrt {x^2} \int \frac {\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}}{\sqrt {3} x}+\frac {1}{12} \left (27 x^2+4\right )^{2/3}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {\sqrt {x^2} \left (2^{2/3} \left (1+\sqrt {3}\right ) \int \frac {1}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}\right )}{\sqrt {3} x}+\frac {1}{12} \left (27 x^2+4\right )^{2/3}\)

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2418

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

input

Int[(2 + 3*x)/(4 + 27*x^2)^(1/3),x]
 

output

(4 + 27*x^2)^(2/3)/12 + (Sqrt[x^2]*((-6*Sqrt[3]*Sqrt[x^2])/(2^(2/3)*(1 - S 
qrt[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))*Sq 
rt[(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)])))/(Sqrt[3]*x)
 

3.8.2.3.1 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 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

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 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]
 

3.8.2.4 Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 2.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.05

method	result	size
risch	\(\frac {\left (27 x^{2}+4\right )^{\frac {2}{3}}}{12}+2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^{2}}{4}\right )\)	\(29\)
meijerg	\(2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^{2}}{4}\right )+\frac {3 \,2^{\frac {1}{3}} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},1;2;-\frac {27 x^{2}}{4}\right )}{4}\)	\(37\)

input

int((2+3*x)/(27*x^2+4)^(1/3),x,method=_RETURNVERBOSE)
 

output

1/12*(27*x^2+4)^(2/3)+2^(1/3)*x*hypergeom([1/3,1/2],[3/2],-27/4*x^2)
 

3.8.2.5 Fricas [F]

\[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]

input

integrate((2+3*x)/(27*x^2+4)^(1/3),x, algorithm="fricas")
 

output

integral((3*x + 2)/(27*x^2 + 4)^(1/3), x)
 

3.8.2.6 Sympy [A] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.07 \[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{i \pi }}{4}} \right )} + \frac {\left (27 x^{2} + 4\right )^{\frac {2}{3}}}{12} \]

input

integrate((2+3*x)/(27*x**2+4)**(1/3),x)
 

output

2**(1/3)*x*hyper((1/3, 1/2), (3/2,), 27*x**2*exp_polar(I*pi)/4) + (27*x**2 
 + 4)**(2/3)/12
 

3.8.2.7 Maxima [F]

\[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]

input

integrate((2+3*x)/(27*x^2+4)^(1/3),x, algorithm="maxima")
 

output

integrate((3*x + 2)/(27*x^2 + 4)^(1/3), x)
 

3.8.2.8 Giac [F]

\[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}}} \,d x } \]

input

integrate((2+3*x)/(27*x^2+4)^(1/3),x, algorithm="giac")
 

output

integrate((3*x + 2)/(27*x^2 + 4)^(1/3), x)
 

3.8.2.9 Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.05 \[ \int \frac {2+3 x}{\sqrt [3]{4+27 x^2}} \, dx=\frac {{\left (27\,x^2+4\right )}^{2/3}}{12}+2^{1/3}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{2};\ \frac {3}{2};\ -\frac {27\,x^2}{4}\right ) \]

input

int((3*x + 2)/(27*x^2 + 4)^(1/3),x)
 

output

(27*x^2 + 4)^(2/3)/12 + 2^(1/3)*x*hypergeom([1/3, 1/2], 3/2, -(27*x^2)/4)