3.8.1 \(\int \frac {(2+3 x)^2}{\sqrt [3]{4+27 x^2}} \, dx\) [701]

3.8.1.1 Optimal result

Integrand size = 19, antiderivative size = 551 \[ \int \frac {(2+3 x)^2}{\sqrt [3]{4+27 x^2}} \, dx=\frac {5}{21} \left (4+27 x^2\right )^{2/3}+\frac {1}{21} (2+3 x) \left (4+27 x^2\right )^{2/3}-\frac {72 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )}+\frac {4 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{7\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right ),-7+4 \sqrt {3}\right )}{21 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}} \]

output

5/21*(27*x^2+4)^(2/3)+1/21*(2+3*x)*(27*x^2+4)^(2/3)-72/7*x/(-(27*x^2+4)^(1 
/3)+2^(2/3)*(1-3^(1/2)))-8/63*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)+4/21*2^( 
1/3)*(2^(2/3)-(27*x^2+4)^(1/3))*EllipticE((-(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)
 

3.8.1.2 Mathematica [C] (verified)

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

Time = 16.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.09 \[ \int \frac {(2+3 x)^2}{\sqrt [3]{4+27 x^2}} \, dx=\frac {1}{21} \left ((7+3 x) \left (4+27 x^2\right )^{2/3}+36 \sqrt [3]{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27 x^2}{4}\right )\right ) \]

input

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

output

((7 + 3*x)*(4 + 27*x^2)^(2/3) + 36*2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3 
/2, (-27*x^2)/4])/21
 

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

Time = 0.49 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {497, 27, 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.

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

\(\Big \downarrow \) 497

\(\displaystyle \frac {1}{63} \int \frac {108 (5 x+2)}{\sqrt [3]{27 x^2+4}}dx+\frac {1}{21} \left (27 x^2+4\right )^{2/3} (3 x+2)\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {12}{7} \int \frac {5 x+2}{\sqrt [3]{27 x^2+4}}dx+\frac {1}{21} \left (27 x^2+4\right )^{2/3} (3 x+2)\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {12}{7} \left (2 \int \frac {1}{\sqrt [3]{27 x^2+4}}dx+\frac {5}{36} \left (27 x^2+4\right )^{2/3}\right )+\frac {1}{21} \left (27 x^2+4\right )^{2/3} (3 x+2)\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {12}{7} \left (\frac {\sqrt {x^2} \int \frac {\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}}{\sqrt {3} x}+\frac {5}{36} \left (27 x^2+4\right )^{2/3}\right )+\frac {1}{21} \left (27 x^2+4\right )^{2/3} (3 x+2)\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {12}{7} \left (\frac {\sqrt {x^2} \left (2^{2/3} \left (1+\sqrt {3}\right ) \int \frac {1}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}\right )}{\sqrt {3} x}+\frac {5}{36} \left (27 x^2+4\right )^{2/3}\right )+\frac {1}{21} \left (27 x^2+4\right )^{2/3} (3 x+2)\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {12}{7} \left (\frac {\sqrt {x^2} \left (-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}}}\right )}{\sqrt {3} x}+\frac {5}{36} \left (27 x^2+4\right )^{2/3}\right )+\frac {1}{21} \left (27 x^2+4\right )^{2/3} (3 x+2)\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {12}{7} \left (\frac {\sqrt {x^2} \left (-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}}}-\frac {6 \sqrt {3} \sqrt {x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )}{\sqrt {3} x}+\frac {5}{36} \left (27 x^2+4\right )^{2/3}\right )+\frac {1}{21} \left (27 x^2+4\right )^{2/3} (3 x+2)\)

input

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

output

((2 + 3*x)*(4 + 27*x^2)^(2/3))/21 + (12*((5*(4 + 27*x^2)^(2/3))/36 + (Sqrt 
[x^2]*((-6*Sqrt[3]*Sqrt[x^2])/(2^(2/3)*(1 - Sqrt[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))*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]*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)*S 
qrt[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)))/7
 

3.8.1.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 497

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b 
*(n + 2*p + 1))   Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 
 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n 
, p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p 
+ 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, 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 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]
 

3.8.1.4 Maple [C] (verified)

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

Time = 2.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.06

method	result	size
risch	\(\frac {\left (7+3 x \right ) \left (27 x^{2}+4\right )^{\frac {2}{3}}}{21}+\frac {12 \,2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^{2}}{4}\right )}{7}\)	\(35\)
meijerg	\(2 \,2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^{2}}{4}\right )+3 \,2^{\frac {1}{3}} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},1;2;-\frac {27 x^{2}}{4}\right )+\frac {3 \,2^{\frac {1}{3}} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {3}{2};\frac {5}{2};-\frac {27 x^{2}}{4}\right )}{2}\)	\(57\)

input

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

output

1/21*(7+3*x)*(27*x^2+4)^(2/3)+12/7*2^(1/3)*x*hypergeom([1/3,1/2],[3/2],-27 
/4*x^2)
 

3.8.1.5 Fricas [F]

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

input

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

output

integral((9*x^2 + 12*x + 4)/(27*x^2 + 4)^(1/3), x)
 

3.8.1.6 Sympy [A] (verification not implemented)

Time = 1.74 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.12 \[ \int \frac {(2+3 x)^2}{\sqrt [3]{4+27 x^2}} \, dx=\frac {3 \cdot \sqrt [3]{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{i \pi }}{4}} \right )}}{2} + 2 \cdot \sqrt [3]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {27 x^{2} e^{i \pi }}{4}} \right )} + \frac {\left (27 x^{2} + 4\right )^{\frac {2}{3}}}{3} \]

input

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

output

3*2**(1/3)*x**3*hyper((1/3, 3/2), (5/2,), 27*x**2*exp_polar(I*pi)/4)/2 + 2 
*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)/3
 

3.8.1.7 Maxima [F]

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

input

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

output

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

3.8.1.8 Giac [F]

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

input

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

output

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

3.8.1.9 Mupad [F(-1)]

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

input

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

output

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