\(\int (3+4 x+2 x^2)^{2/3} \, dx\) [140]

Optimal result

Integrand size = 14, antiderivative size = 21 \[ \int \left (3+4 x+2 x^2\right )^{2/3} \, dx=(1+x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {3}{2},-2 (1+x)^2\right ) \] Output:

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

Mathematica [A] (verified)

Time = 6.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (3+4 x+2 x^2\right )^{2/3} \, dx=(1+x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {3}{2},-2 (1+x)^2\right ) \] Input:

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

Output:

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

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(367\) vs. \(2(21)=42\).

Time = 0.71 (sec) , antiderivative size = 367, normalized size of antiderivative = 17.48, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1087, 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.

\(\displaystyle \int \left (2 x^2+4 x+3\right )^{2/3} \, dx\)

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{7} \int \frac {1}{\sqrt [3]{\frac {1}{8} (4 x+4)^2+1}}d(4 x+4)+\frac {3}{7} \left (2 x^2+4 x+3\right )^{2/3} (x+1)\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3 \sqrt {2} \sqrt {(4 x+4)^2} \int \frac {2 \sqrt {2} \sqrt [3]{\frac {1}{8} (4 x+4)^2+1}}{\sqrt {(4 x+4)^2}}d\sqrt [3]{\frac {1}{8} (4 x+4)^2+1}}{7 (4 x+4)}+\frac {3}{7} \left (2 x^2+4 x+3\right )^{2/3} (x+1)\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3 \sqrt {2} \sqrt {(4 x+4)^2} \left (\left (1+\sqrt {3}\right ) \int \frac {2 \sqrt {2}}{\sqrt {(4 x+4)^2}}d\sqrt [3]{\frac {1}{8} (4 x+4)^2+1}-\int \frac {2 \sqrt {2} \left (-4 x+\sqrt {3}-3\right )}{\sqrt {(4 x+4)^2}}d\sqrt [3]{\frac {1}{8} (4 x+4)^2+1}\right )}{7 (4 x+4)}+\frac {3}{7} \left (2 x^2+4 x+3\right )^{2/3} (x+1)\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3 \sqrt {2} \sqrt {(4 x+4)^2} \left (-\int \frac {2 \sqrt {2} \left (-4 x+\sqrt {3}-3\right )}{\sqrt {(4 x+4)^2}}d\sqrt [3]{\frac {1}{8} (4 x+4)^2+1}-\frac {4 \sqrt {2 \left (2-\sqrt {3}\right )} \left (1+\sqrt {3}\right ) (-4 x-3) \sqrt {\frac {4 x+\left (\frac {1}{8} (4 x+4)^2+1\right )^{2/3}+5}{\left (-4 x-\sqrt {3}-3\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-4 x+\sqrt {3}-3}{-4 x-\sqrt {3}-3}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {-4 x-3}{\left (-4 x-\sqrt {3}-3\right )^2}} \sqrt {(4 x+4)^2}}\right )}{7 (4 x+4)}+\frac {3}{7} \left (2 x^2+4 x+3\right )^{2/3} (x+1)\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3 \sqrt {2} \sqrt {(4 x+4)^2} \left (-\frac {4 \sqrt {2 \left (2-\sqrt {3}\right )} \left (1+\sqrt {3}\right ) (-4 x-3) \sqrt {\frac {4 x+\left (\frac {1}{8} (4 x+4)^2+1\right )^{2/3}+5}{\left (-4 x-\sqrt {3}-3\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-4 x+\sqrt {3}-3}{-4 x-\sqrt {3}-3}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {-4 x-3}{\left (-4 x-\sqrt {3}-3\right )^2}} \sqrt {(4 x+4)^2}}+\frac {2 \sqrt [4]{3} \sqrt {2 \left (2+\sqrt {3}\right )} (-4 x-3) \sqrt {\frac {4 x+\left (\frac {1}{8} (4 x+4)^2+1\right )^{2/3}+5}{\left (-4 x-\sqrt {3}-3\right )^2}} E\left (\arcsin \left (\frac {-4 x+\sqrt {3}-3}{-4 x-\sqrt {3}-3}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {-4 x-3}{\left (-4 x-\sqrt {3}-3\right )^2}} \sqrt {(4 x+4)^2}}-\frac {\sqrt {(4 x+4)^2}}{\sqrt {2} \left (-4 x-\sqrt {3}-3\right )}\right )}{7 (4 x+4)}+\frac {3}{7} \left (2 x^2+4 x+3\right )^{2/3} (x+1)\)

Input:

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

Output:

(3*(1 + x)*(3 + 4*x + 2*x^2)^(2/3))/7 + (3*Sqrt[2]*Sqrt[(4 + 4*x)^2]*(-(Sq 
rt[(4 + 4*x)^2]/(Sqrt[2]*(-3 - Sqrt[3] - 4*x))) + (2*3^(1/4)*Sqrt[2*(2 + S 
qrt[3])]*(-3 - 4*x)*Sqrt[(5 + 4*x + (1 + (4 + 4*x)^2/8)^(2/3))/(-3 - Sqrt[ 
3] - 4*x)^2]*EllipticE[ArcSin[(-3 + Sqrt[3] - 4*x)/(-3 - Sqrt[3] - 4*x)], 
-7 + 4*Sqrt[3]])/(Sqrt[-((-3 - 4*x)/(-3 - Sqrt[3] - 4*x)^2)]*Sqrt[(4 + 4*x 
)^2]) - (4*Sqrt[2*(2 - Sqrt[3])]*(1 + Sqrt[3])*(-3 - 4*x)*Sqrt[(5 + 4*x + 
(1 + (4 + 4*x)^2/8)^(2/3))/(-3 - Sqrt[3] - 4*x)^2]*EllipticF[ArcSin[(-3 + 
Sqrt[3] - 4*x)/(-3 - Sqrt[3] - 4*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-((-3 
 - 4*x)/(-3 - Sqrt[3] - 4*x)^2)]*Sqrt[(4 + 4*x)^2])))/(7*(4 + 4*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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 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]

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

Input:

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

Output:

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

Fricas [F]

\[ \int \left (3+4 x+2 x^2\right )^{2/3} \, dx=\int { {\left (2 \, x^{2} + 4 \, x + 3\right )}^{\frac {2}{3}} \,d x } \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

\[ \int \left (3+4 x+2 x^2\right )^{2/3} \, dx=\int { {\left (2 \, x^{2} + 4 \, x + 3\right )}^{\frac {2}{3}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \left (3+4 x+2 x^2\right )^{2/3} \, dx=\int { {\left (2 \, x^{2} + 4 \, x + 3\right )}^{\frac {2}{3}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (3+4 x+2 x^2\right )^{2/3} \, dx=\frac {3 \left (2 x^{2}+4 x +3\right )^{\frac {2}{3}} x}{7}+\frac {9 \left (2 x^{2}+4 x +3\right )^{\frac {2}{3}}}{14}-\frac {4 \left (\int \frac {x}{\left (2 x^{2}+4 x +3\right )^{\frac {1}{3}}}d x \right )}{7} \] Input:

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

Output:

(6*(2*x**2 + 4*x + 3)**(2/3)*x + 9*(2*x**2 + 4*x + 3)**(2/3) - 8*int(((2*x 
**2 + 4*x + 3)**(2/3)*x)/(2*x**2 + 4*x + 3),x))/14