\(\int \frac {(2+3 x)^4}{\sqrt [3]{28+54 x+27 x^2}} \, dx\) [709]

Optimal result

Integrand size = 22, antiderivative size = 98 \[ \int \frac {(2+3 x)^4}{\sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {11}{195} (2+3 x)^2 \left (28+54 x+27 x^2\right )^{2/3}+\frac {1}{39} (2+3 x)^3 \left (28+54 x+27 x^2\right )^{2/3}+\frac {4}{455} (8+29 x) \left (28+54 x+27 x^2\right )^{2/3}+\frac {16}{91} (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-27 (1+x)^2\right ) \] Output:

-11/195*(2+3*x)^2*(27*x^2+54*x+28)^(2/3)+1/39*(2+3*x)^3*(27*x^2+54*x+28)^( 
2/3)+4/455*(8+29*x)*(27*x^2+54*x+28)^(2/3)+16/91*(1+x)*hypergeom([1/3, 1/2 
],[3/2],-27*(1+x)^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 10.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.58 \[ \int \frac {(2+3 x)^4}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {\left (28+54 x+27 x^2\right )^{2/3} \left (68+684 x+1197 x^2+945 x^3\right )+240 (1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-27 (1+x)^2\right )}{1365} \] Input:

Integrate[(2 + 3*x)^4/(28 + 54*x + 27*x^2)^(1/3),x]
 

Output:

((28 + 54*x + 27*x^2)^(2/3)*(68 + 684*x + 1197*x^2 + 945*x^3) + 240*(1 + x 
)*Hypergeometric2F1[1/3, 1/2, 3/2, -27*(1 + x)^2])/1365
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(427\) vs. \(2(98)=196\).

Time = 0.52 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.36, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1166, 27, 1236, 27, 1225, 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.

\(\displaystyle \int \frac {(3 x+2)^4}{\sqrt [3]{27 x^2+54 x+28}} \, dx\)

\(\Big \downarrow \) 1166

\(\displaystyle \frac {1}{117} \int -\frac {18 (3 x+2)^2 (33 x+28)}{\sqrt [3]{27 x^2+54 x+28}}dx+\frac {1}{39} \left (27 x^2+54 x+28\right )^{2/3} (3 x+2)^3\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{13} \int \frac {(3 x+2)^2 (33 x+28)}{\sqrt [3]{27 x^2+54 x+28}}dx\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{13} \left (\frac {1}{90} \int -\frac {36 (3 x+2) (87 x+80)}{\sqrt [3]{27 x^2+54 x+28}}dx+\frac {11}{30} \left (27 x^2+54 x+28\right )^{2/3} (3 x+2)^2\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{13} \left (\frac {11}{30} (3 x+2)^2 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{5} \int \frac {(3 x+2) (87 x+80)}{\sqrt [3]{27 x^2+54 x+28}}dx\right )\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{13} \left (\frac {11}{30} (3 x+2)^2 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{5} \left (\frac {20}{7} \int \frac {1}{\sqrt [3]{27 x^2+54 x+28}}dx+\frac {1}{7} \left (27 x^2+54 x+28\right )^{2/3} (29 x+8)\right )\right )\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{13} \left (\frac {11}{30} (3 x+2)^2 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{5} \left (\frac {10}{189} \int \frac {1}{\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}d(54 x+54)+\frac {1}{7} \left (27 x^2+54 x+28\right )^{2/3} (29 x+8)\right )\right )\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{13} \left (\frac {11}{30} (3 x+2)^2 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{5} \left (\frac {10 \sqrt {(54 x+54)^2} \int \frac {6 \sqrt {3} \sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}}{7 \sqrt {3} (54 x+54)}+\frac {1}{7} \left (27 x^2+54 x+28\right )^{2/3} (29 x+8)\right )\right )\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{13} \left (\frac {11}{30} (3 x+2)^2 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{5} \left (\frac {10 \sqrt {(54 x+54)^2} \left (\left (1+\sqrt {3}\right ) \int \frac {6 \sqrt {3}}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}-\int \frac {6 \sqrt {3} \left (-54 x+\sqrt {3}-53\right )}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}\right )}{7 \sqrt {3} (54 x+54)}+\frac {1}{7} \left (27 x^2+54 x+28\right )^{2/3} (29 x+8)\right )\right )\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{13} \left (\frac {11}{30} (3 x+2)^2 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{5} \left (\frac {10 \sqrt {(54 x+54)^2} \left (-\int \frac {6 \sqrt {3} \left (-54 x+\sqrt {3}-53\right )}{\sqrt {(54 x+54)^2}}d\sqrt [3]{\frac {1}{108} (54 x+54)^2+1}-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}\right )}{7 \sqrt {3} (54 x+54)}+\frac {1}{7} \left (27 x^2+54 x+28\right )^{2/3} (29 x+8)\right )\right )\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {1}{39} (3 x+2)^3 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{13} \left (\frac {11}{30} (3 x+2)^2 \left (27 x^2+54 x+28\right )^{2/3}-\frac {2}{5} \left (\frac {10 \sqrt {(54 x+54)^2} \left (-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}+\frac {6\ 3^{3/4} \sqrt {2+\sqrt {3}} (-54 x-53) \sqrt {\frac {54 x+\left (\frac {1}{108} (54 x+54)^2+1\right )^{2/3}+55}{\left (-54 x-\sqrt {3}-53\right )^2}} E\left (\arcsin \left (\frac {-54 x+\sqrt {3}-53}{-54 x-\sqrt {3}-53}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-54 x-53}{\left (-54 x-\sqrt {3}-53\right )^2}} \sqrt {(54 x+54)^2}}-\frac {\sqrt {(54 x+54)^2}}{3 \sqrt {3} \left (-54 x-\sqrt {3}-53\right )}\right )}{7 \sqrt {3} (54 x+54)}+\frac {1}{7} \left (27 x^2+54 x+28\right )^{2/3} (29 x+8)\right )\right )\)

Input:

Int[(2 + 3*x)^4/(28 + 54*x + 27*x^2)^(1/3),x]
 

Output:

((2 + 3*x)^3*(28 + 54*x + 27*x^2)^(2/3))/39 - (2*((11*(2 + 3*x)^2*(28 + 54 
*x + 27*x^2)^(2/3))/30 - (2*(((8 + 29*x)*(28 + 54*x + 27*x^2)^(2/3))/7 + ( 
10*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)*Sqrt[(55 + 54*x + (1 + (54 + 54*x 
)^2/108)^(2/3))/(-53 - Sqrt[3] - 54*x)^2]*EllipticF[ArcSin[(-53 + Sqrt[3] 
- 54*x)/(-53 - Sqrt[3] - 54*x)], -7 + 4*Sqrt[3]])/(Sqrt[-((-53 - 54*x)/(-5 
3 - Sqrt[3] - 54*x)^2)]*Sqrt[(54 + 54*x)^2])))/(7*Sqrt[3]*(54 + 54*x))))/5 
))/13
 

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 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 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 
1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])
 

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]

\[\int \frac {\left (3 x +2\right )^{4}}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x\]

Input:

int((3*x+2)^4/(27*x^2+54*x+28)^(1/3),x)
 

Output:

int((3*x+2)^4/(27*x^2+54*x+28)^(1/3),x)
 

Fricas [F]

\[ \int \frac {(2+3 x)^4}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{4}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)^4/(27*x^2+54*x+28)^(1/3),x, algorithm="fricas")
 

Output:

integral((81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)/(27*x^2 + 54*x + 28)^(1/ 
3), x)
 

Sympy [F]

\[ \int \frac {(2+3 x)^4}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {\left (3 x + 2\right )^{4}}{\sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \] Input:

integrate((2+3*x)**4/(27*x**2+54*x+28)**(1/3),x)
 

Output:

Integral((3*x + 2)**4/(27*x**2 + 54*x + 28)**(1/3), x)
 

Maxima [F]

\[ \int \frac {(2+3 x)^4}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{4}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)^4/(27*x^2+54*x+28)^(1/3),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((3*x + 2)^4/(27*x^2 + 54*x + 28)^(1/3), x)
 

Giac [F]

\[ \int \frac {(2+3 x)^4}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{4}}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate((2+3*x)^4/(27*x^2+54*x+28)^(1/3),x, algorithm="giac")
 

Output:

integrate((3*x + 2)^4/(27*x^2 + 54*x + 28)^(1/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+3 x)^4}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {{\left (3\,x+2\right )}^4}{{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \] Input:

int((3*x + 2)^4/(54*x + 27*x^2 + 28)^(1/3),x)
 

Output:

int((3*x + 2)^4/(54*x + 27*x^2 + 28)^(1/3), x)
 

Reduce [F]

\[ \int \frac {(2+3 x)^4}{\sqrt [3]{28+54 x+27 x^2}} \, dx=81 \left (\int \frac {x^{4}}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x \right )+216 \left (\int \frac {x^{3}}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x \right )+216 \left (\int \frac {x^{2}}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x \right )+96 \left (\int \frac {x}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x \right )+16 \left (\int \frac {1}{\left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x \right ) \] Input:

int((2+3*x)^4/(27*x^2+54*x+28)^(1/3),x)
 

Output:

81*int(x**4/(27*x**2 + 54*x + 28)**(1/3),x) + 216*int(x**3/(27*x**2 + 54*x 
 + 28)**(1/3),x) + 216*int(x**2/(27*x**2 + 54*x + 28)**(1/3),x) + 96*int(x 
/(27*x**2 + 54*x + 28)**(1/3),x) + 16*int(1/(27*x**2 + 54*x + 28)**(1/3),x 
)