3.8.5 \(\int \frac {1}{(2+3 x)^3 \sqrt [3]{4+27 x^2}} \, dx\) [705]

3.8.5.1 Optimal result

Integrand size = 19, antiderivative size = 656 \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{4+27 x^2}} \, dx=-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)^2}-\frac {\left (4+27 x^2\right )^{2/3}}{96 (2+3 x)}-\frac {3 x}{32 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )}-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{4+27 x^2}}\right )}{96 \sqrt [3]{2} \sqrt {3}}+\frac {\sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{96\ 2^{2/3} 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {\left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right ),-7+4 \sqrt {3}\right )}{144 \sqrt [6]{2} \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {\log (2+3 x)}{192 \sqrt [3]{2}}+\frac {\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{192 \sqrt [3]{2}} \]

output

-1/96*(27*x^2+4)^(2/3)/(2+3*x)^2-1/96*(27*x^2+4)^(2/3)/(2+3*x)-1/384*ln(2+ 
3*x)*2^(2/3)+1/384*ln(54-81*x-27*2^(2/3)*(27*x^2+4)^(1/3))*2^(2/3)-3/32*x/ 
(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))+1/576*arctan(-1/3*3^(1/2)-1/3*2^(1 
/3)*(2-3*x)/(27*x^2+4)^(1/3)*3^(1/2))*2^(2/3)*3^(1/2)-1/864*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)*(2 
7*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)+1/576*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)
 

3.8.5.2 Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 18.83 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.34 \[ \int \frac {1}{(2+3 x)^3 \sqrt [3]{4+27 x^2}} \, dx=\frac {-12 \left (4+4 x+27 x^2+27 x^3\right )-4 \sqrt [3]{3} (2+3 x)^2 \sqrt [3]{\frac {-2 i \sqrt {3}+9 x}{2+3 x}} \sqrt [3]{\frac {2 i \sqrt {3}+9 x}{2+3 x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {6-2 i \sqrt {3}}{6+9 x},\frac {6+2 i \sqrt {3}}{6+9 x}\right )+\sqrt [3]{6} \sqrt [3]{2 \sqrt {3}-9 i x} (2+3 x)^2 \left (-2 i+3 \sqrt {3} x\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {1}{2}+\frac {3}{4} i \sqrt {3} x\right )}{384 (2+3 x)^2 \sqrt [3]{4+27 x^2}} \]

input

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

output

(-12*(4 + 4*x + 27*x^2 + 27*x^3) - 4*3^(1/3)*(2 + 3*x)^2*(((-2*I)*Sqrt[3] 
+ 9*x)/(2 + 3*x))^(1/3)*(((2*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2 
/3, 1/3, 1/3, 5/3, (6 - (2*I)*Sqrt[3])/(6 + 9*x), (6 + (2*I)*Sqrt[3])/(6 + 
 9*x)] + 6^(1/3)*(2*Sqrt[3] - (9*I)*x)^(1/3)*(2 + 3*x)^2*(-2*I + 3*Sqrt[3] 
*x)*Hypergeometric2F1[1/3, 2/3, 5/3, 1/2 + ((3*I)/4)*Sqrt[3]*x])/(384*(2 + 
 3*x)^2*(4 + 27*x^2)^(1/3))
 

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

Time = 0.61 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {498, 27, 688, 27, 719, 233, 501, 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.

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

\(\Big \downarrow \) 498

\(\displaystyle -\frac {3}{32} \int -\frac {2 (2-x)}{(3 x+2)^2 \sqrt [3]{27 x^2+4}}dx-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{16} \int \frac {2-x}{(3 x+2)^2 \sqrt [3]{27 x^2+4}}dx-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {3}{16} \left (-\frac {1}{144} \int -\frac {24 (3 x+4)}{(3 x+2) \sqrt [3]{27 x^2+4}}dx-\frac {\left (27 x^2+4\right )^{2/3}}{18 (3 x+2)}\right )-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{16} \left (\frac {1}{6} \int \frac {3 x+4}{(3 x+2) \sqrt [3]{27 x^2+4}}dx-\frac {\left (27 x^2+4\right )^{2/3}}{18 (3 x+2)}\right )-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {3}{16} \left (\frac {1}{6} \left (\int \frac {1}{\sqrt [3]{27 x^2+4}}dx+2 \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2+4}}dx\right )-\frac {\left (27 x^2+4\right )^{2/3}}{18 (3 x+2)}\right )-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3}{16} \left (\frac {1}{6} \left (2 \int \frac {1}{(3 x+2) \sqrt [3]{27 x^2+4}}dx+\frac {\sqrt {x^2} \int \frac {\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}}{2 \sqrt {3} x}\right )-\frac {\left (27 x^2+4\right )^{2/3}}{18 (3 x+2)}\right )-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}\)

\(\Big \downarrow \) 501

\(\displaystyle \frac {3}{16} \left (\frac {1}{6} \left (\frac {\sqrt {x^2} \int \frac {\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}}{2 \sqrt {3} x}+2 \left (-\frac {\arctan \left (\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{27 x^2+4}}+\frac {1}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{12 \sqrt [3]{2}}-\frac {\log (3 x+2)}{12 \sqrt [3]{2}}\right )\right )-\frac {\left (27 x^2+4\right )^{2/3}}{18 (3 x+2)}\right )-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3}{16} \left (\frac {1}{6} \left (\frac {\sqrt {x^2} \left (2^{2/3} \left (1+\sqrt {3}\right ) \int \frac {1}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}\right )}{2 \sqrt {3} x}+2 \left (-\frac {\arctan \left (\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{27 x^2+4}}+\frac {1}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{12 \sqrt [3]{2}}-\frac {\log (3 x+2)}{12 \sqrt [3]{2}}\right )\right )-\frac {\left (27 x^2+4\right )^{2/3}}{18 (3 x+2)}\right )-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3}{16} \left (\frac {1}{6} \left (\frac {\sqrt {x^2} \left (-\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{3 \sqrt {3} \sqrt {x^2}}d\sqrt [3]{27 x^2+4}-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}}}\right )}{2 \sqrt {3} x}+2 \left (-\frac {\arctan \left (\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{27 x^2+4}}+\frac {1}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{12 \sqrt [3]{2}}-\frac {\log (3 x+2)}{12 \sqrt [3]{2}}\right )\right )-\frac {\left (27 x^2+4\right )^{2/3}}{18 (3 x+2)}\right )-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3}{16} \left (\frac {1}{6} \left (\frac {\sqrt {x^2} \left (-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right ),-7+4 \sqrt {3}\right )}{3\ 3^{3/4} \sqrt {x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}}}+\frac {\sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {x^2} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}}}-\frac {6 \sqrt {3} \sqrt {x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )}{2 \sqrt {3} x}+2 \left (-\frac {\arctan \left (\frac {\sqrt [3]{2} (2-3 x)}{\sqrt {3} \sqrt [3]{27 x^2+4}}+\frac {1}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{12 \sqrt [3]{2}}-\frac {\log (3 x+2)}{12 \sqrt [3]{2}}\right )\right )-\frac {\left (27 x^2+4\right )^{2/3}}{18 (3 x+2)}\right )-\frac {\left (27 x^2+4\right )^{2/3}}{96 (3 x+2)^2}\)

input

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

output

-1/96*(4 + 27*x^2)^(2/3)/(2 + 3*x)^2 + (3*(-1/18*(4 + 27*x^2)^(2/3)/(2 + 3 
*x) + ((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))*S 
qrt[(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 + Sqr 
t[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)*Sqrt[x^2]*Sqrt[-((2^(2/3) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - S 
qrt[3]) - (4 + 27*x^2)^(1/3))^2)])))/(2*Sqrt[3]*x) + 2*(-1/6*ArcTan[1/Sqrt 
[3] + (2^(1/3)*(2 - 3*x))/(Sqrt[3]*(4 + 27*x^2)^(1/3))]/(2^(1/3)*Sqrt[3]) 
- Log[2 + 3*x]/(12*2^(1/3)) + Log[54 - 81*x - 27*2^(2/3)*(4 + 27*x^2)^(1/3 
)]/(12*2^(1/3))))/6))/16
 

3.8.5.3.1 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 498

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)/((n + 1)*(b*c^2 + a*d^2))), x] + S 
imp[b/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n 
 + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n 
, -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp 
lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
 

rule 501

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(1/3)), x_Symbol] :> With[ 
{q = Rt[6*b^2*(d^2/c^2), 3]}, Simp[(-Sqrt[3])*b*d*(ArcTan[1/Sqrt[3] + 2*b*( 
(c - d*x)/(Sqrt[3]*c*q*(a + b*x^2)^(1/3)))]/(c^2*q^2)), x] + (-Simp[3*b*d*( 
Log[c + d*x]/(2*c^2*q^2)), x] + Simp[3*b*d*(Log[b*c - b*d*x - c*q*(a + b*x^ 
2)^(1/3)]/(2*c^2*q^2)), x])] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 - 3*a*d 
^2, 0]
 

rule 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

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]
 

3.8.5.4 Maple [F]

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

input

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

output

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

3.8.5.5 Fricas [F]

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

input

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

output

integral((27*x^2 + 4)^(2/3)/(729*x^5 + 1458*x^4 + 1080*x^3 + 432*x^2 + 144 
*x + 32), x)
 

3.8.5.6 Sympy [F]

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

input

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

output

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

3.8.5.7 Maxima [F]

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

input

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

output

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

3.8.5.8 Giac [F]

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

input

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

output

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

3.8.5.9 Mupad [F(-1)]

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

input

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

output

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