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

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {757, 655, 241, 310, 225, 1893} \[ \int \frac {(2+3 x)^2}{\sqrt [3]{4+27 x^2}} \, dx=-\frac {8\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right ),-7+4 \sqrt {3}\right )}{21 \sqrt [4]{3} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}+\frac {4 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{27 x^2+4}\right ) \sqrt {\frac {\left (27 x^2+4\right )^{2/3}+2^{2/3} \sqrt [3]{27 x^2+4}+2 \sqrt [3]{2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} E\left (\arcsin \left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}}\right )|-7+4 \sqrt {3}\right )}{7\ 3^{3/4} \sqrt {-\frac {2^{2/3}-\sqrt [3]{27 x^2+4}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )^2}} x}-\frac {72 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )}+\frac {1}{21} (3 x+2) \left (27 x^2+4\right )^{2/3}+\frac {5}{21} \left (27 x^2+4\right )^{2/3} \]

[In]

Int[(2 + 3*x)^2/(4 + 27*x^2)^(1/3),x]

[Out]

(5*(4 + 27*x^2)^(2/3))/21 + ((2 + 3*x)*(4 + 27*x^2)^(2/3))/21 - (72*x)/(7*(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2
)^(1/3))) + (4*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 + Sq
rt[3]) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(7*3^(3/4)*x*Sqrt
[-((2^(2/3) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[3]) - (4 + 27*x^2)^(1/3))^2)]) - (8*2^(5/6)*(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]])/(21*3^(1/4)*x*Sqrt[-((2^(2/3) - (4 + 27*x^2)^(1/3))/(2^(2/3)*(1 - Sqrt[
3]) - (4 + 27*x^2)^(1/3))^2)])

Rule 225

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]*Sq
rt[(-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 241

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Dist[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 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x]
, x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 655

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

Rule 757

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

Rule 1893

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[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
] + Simp[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 - Sqrt[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*Sqr
t[3])*a*d^3, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{21} (2+3 x) \left (4+27 x^2\right )^{2/3}+\frac {1}{63} \int \frac {216+540 x}{\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 {24}{7} \int \frac {1}{\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 {\left (4 \sqrt {3} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{7 x} \\ & = \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 {\left (4 \sqrt {3} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {2^{2/3} \left (1+\sqrt {3}\right )-x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{7 x}+\frac {\left (4\ 2^{2/3} \sqrt {3} \left (1+\sqrt {3}\right ) \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{7 x} \\ & = \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 (\sin ^{-1}\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}} F\left (\sin ^{-1}\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}}} \\ \end{align*}

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 \[ \int \frac {(2+3 x)^2}{\sqrt [3]{4+27 x^2}} \, dx=\frac {1}{21} \left ((7+3 x) \left (4+27 x^2\right )^{2/3}+36 \sqrt [3]{2} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27 x^2}{4}\right )\right ) \]

[In]

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

[Out]

((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

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	\(2 \,2^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^{2}}{4}\right )+3 \,2^{\frac {1}{3}} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},1;2;-\frac {27 x^{2}}{4}\right )+\frac {3 \,2^{\frac {1}{3}} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {3}{2};\frac {5}{2};-\frac {27 x^{2}}{4}\right )}{2}\)	\(57\)

[In]

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

[Out]

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)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((3*x + 2)^2/(27*x^2 + 4)^(1/3),x)

[Out]

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