3.7.100 \(\int \frac {(2+3 x)^3}{\sqrt [3]{4+27 x^2}} \, dx\) [700]

Optimal. Leaf size=558 \[ \frac {1}{30} (2+3 x)^2 \left (4+27 x^2\right )^{2/3}+\frac {4}{35} (7+4 x) \left (4+27 x^2\right )^{2/3}-\frac {96 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )}+\frac {16 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{21\ 3^{3/4} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}}-\frac {32\ 2^{5/6} \left (2^{2/3}-\sqrt [3]{4+27 x^2}\right ) \sqrt {\frac {2 \sqrt [3]{2}+2^{2/3} \sqrt [3]{4+27 x^2}+\left (4+27 x^2\right )^{2/3}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {2^{2/3} \left (1+\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}{2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{63 \sqrt [4]{3} x \sqrt {-\frac {2^{2/3}-\sqrt [3]{4+27 x^2}}{\left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )^2}}} \]

[Out]

1/30*(2+3*x)^2*(27*x^2+4)^(2/3)+4/35*(7+4*x)*(27*x^2+4)^(2/3)-96/7*x/(-(27*x^2+4)^(1/3)+2^(2/3)*(1-3^(1/2)))-3
2/189*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)+16/63*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)

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Rubi [A]
time = 0.29, antiderivative size = 558, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {757, 794, 241, 310, 225, 1893} \begin {gather*} -\frac {32\ 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}} F\left (\text {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 )}{63 \sqrt [4]{3} x \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 {16 \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 (\text {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\ 3^{3/4} x \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 {1}{30} \left (27 x^2+4\right )^{2/3} (3 x+2)^2+\frac {4}{35} (4 x+7) \left (27 x^2+4\right )^{2/3}-\frac {96 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{27 x^2+4}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^3/(4 + 27*x^2)^(1/3),x]

[Out]

((2 + 3*x)^2*(4 + 27*x^2)^(2/3))/30 + (4*(7 + 4*x)*(4 + 27*x^2)^(2/3))/35 - (96*x)/(7*(2^(2/3)*(1 - Sqrt[3]) -
(4 + 27*x^2)^(1/3))) + (16*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]])/(21*
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)]) - (32*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]])/(63*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 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 794

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

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*} \int \frac {(2+3 x)^3}{\sqrt [3]{4+27 x^2}} \, dx &=\frac {1}{30} (2+3 x)^2 \left (4+27 x^2\right )^{2/3}+\frac {1}{90} \int \frac {(2+3 x) (288+864 x)}{\sqrt [3]{4+27 x^2}} \, dx\\ &=\frac {1}{30} (2+3 x)^2 \left (4+27 x^2\right )^{2/3}+\frac {4}{35} (7+4 x) \left (4+27 x^2\right )^{2/3}+\frac {32}{7} \int \frac {1}{\sqrt [3]{4+27 x^2}} \, dx\\ &=\frac {1}{30} (2+3 x)^2 \left (4+27 x^2\right )^{2/3}+\frac {4}{35} (7+4 x) \left (4+27 x^2\right )^{2/3}+\frac {\left (16 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{7 \sqrt {3} x}\\ &=\frac {1}{30} (2+3 x)^2 \left (4+27 x^2\right )^{2/3}+\frac {4}{35} (7+4 x) \left (4+27 x^2\right )^{2/3}-\frac {\left (16 \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 \sqrt {3} x}+\frac {\left (32 \sqrt [6]{2} \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4+x^3}} \, dx,x,\sqrt [3]{4+27 x^2}\right )}{7 \sqrt {3 \left (2-\sqrt {3}\right )} x}\\ &=\frac {1}{30} (2+3 x)^2 \left (4+27 x^2\right )^{2/3}+\frac {4}{35} (7+4 x) \left (4+27 x^2\right )^{2/3}-\frac {96 x}{7 \left (2^{2/3} \left (1-\sqrt {3}\right )-\sqrt [3]{4+27 x^2}\right )}+\frac {16 \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 )}{21\ 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 {32\ 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 )}{63 \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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 15.08, size = 53, normalized size = 0.09 \begin {gather*} \frac {1}{210} \left (4+27 x^2\right )^{2/3} \left (196+180 x+63 x^2\right )+\frac {16}{7} \sqrt [3]{2} x \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {27 x^2}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4 + 27*x^2)^(2/3)*(196 + 180*x + 63*x^2))/210 + (16*2^(1/3)*x*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*x^2)/4])
/7

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.50, size = 40, normalized size = 0.07

method	result	size
risch	\(\frac {\left (63 x^{2}+180 x +196\right ) \left (27 x^{2}+4\right )^{\frac {2}{3}}}{210}+\frac {16 \,2^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {27 x^{2}}{4}\right )}{7}\)	\(40\)
meijerg	\(4 \,2^{\frac {1}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {27 x^{2}}{4}\right )+9 \,2^{\frac {1}{3}} x^{2} \hypergeom \left (\left [\frac {1}{3}, 1\right ], \left [2\right ], -\frac {27 x^{2}}{4}\right )+9 \,2^{\frac {1}{3}} x^{3} \hypergeom \left (\left [\frac {1}{3}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], -\frac {27 x^{2}}{4}\right )+\frac {27 \,2^{\frac {1}{3}} x^{4} \hypergeom \left (\left [\frac {1}{3}, 2\right ], \left [3\right ], -\frac {27 x^{2}}{4}\right )}{8}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210*(63*x^2+180*x+196)*(27*x^2+4)^(2/3)+16/7*2^(1/3)*x*hypergeom([1/3,1/2],[3/2],-27/4*x^2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)/(27*x^2 + 4)^(1/3), x)

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Sympy [A]
time = 3.00, size = 85, normalized size = 0.15 \begin {gather*} 9 \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 )} + \frac {3 x^{2} \left (27 x^{2} + 4\right )^{\frac {2}{3}}}{10} + 4 \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 {14 \left (27 x^{2} + 4\right )^{\frac {2}{3}}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9*2**(1/3)*x**3*hyper((1/3, 3/2), (5/2,), 27*x**2*exp_polar(I*pi)/4) + 3*x**2*(27*x**2 + 4)**(2/3)/10 + 4*2**(
1/3)*x*hyper((1/3, 1/2), (3/2,), 27*x**2*exp_polar(I*pi)/4) + 14*(27*x**2 + 4)**(2/3)/15

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^3}{{\left (27\,x^2+4\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^3/(27*x^2 + 4)^(1/3),x)

[Out]

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

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