\(\int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx\) [2505]

Optimal result

Integrand size = 20, antiderivative size = 560 \[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {1}{12} \left (28+54 x+27 x^2\right )^{2/3}+\frac {18 (1+x)}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}-\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{18 \sqrt {2} \sqrt [4]{3} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}+\frac {\left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{9\ 3^{3/4} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}} \]

[Out]

1/12*(27*x^2+54*x+28)^(2/3)+18*(1+x)/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2))+1/27*(6-2^(1/3)*(108+(54+5
4*x)^2)^(1/3))*EllipticF((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6+6*3^(1/2))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*
3^(1/2)),2*I-I*3^(1/2))*((1+(27*x^2+54*x+28)^(1/3)+(27*x^2+54*x+28)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6
-6*3^(1/2))^2)^(1/2)*3^(1/4)/(1+x)/((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6
*3^(1/2))^2)^(1/2)-1/108*(6-2^(1/3)*(108+(54+54*x)^2)^(1/3))*EllipticE((-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6+6*3
^(1/2))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2)),2*I-I*3^(1/2))*((1+(27*x^2+54*x+28)^(1/3)+(27*x^2+54*x+
28)^(2/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*3^(3/4)/(1+x)*2^(
1/2)/((-6+2^(1/3)*(108+(54+54*x)^2)^(1/3))/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2))^2)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {654, 633, 241, 310, 225, 1893} \[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {\left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right ),-7+4 \sqrt {3}\right )}{9\ 3^{3/4} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}-\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right ) \sqrt {\frac {\left (27 x^2+54 x+28\right )^{2/3}+\sqrt [3]{27 x^2+54 x+28}+1}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} E\left (\arcsin \left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}\right )|-7+4 \sqrt {3}\right )}{18 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )^2}} (x+1)}+\frac {1}{12} \left (27 x^2+54 x+28\right )^{2/3}+\frac {18 (x+1)}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}} \]

[In]

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

[Out]

(28 + 54*x + 27*x^2)^(2/3)/12 + (18*(1 + x))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3)) - (Sqrt[2
+ Sqrt[3]]*(6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))*Sqrt[(1 + (28 + 54*x + 27*x^2)^(1/3) + (28 + 54*x + 27*x
^2)^(2/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2]*EllipticE[ArcSin[(6*(1 + Sqrt[3]) - 2^(1
/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(1
8*Sqrt[2]*3^(1/4)*(1 + x)*Sqrt[-((6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (
54 + 54*x)^2)^(1/3))^2)]) + ((6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))*Sqrt[(1 + (28 + 54*x + 27*x^2)^(1/3) +
(28 + 54*x + 27*x^2)^(2/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcSin[(6*(1
+ Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))], -7
+ 4*Sqrt[3]])/(9*3^(3/4)*(1 + x)*Sqrt[-((6 - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*
(108 + (54 + 54*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 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[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 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[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*} \text {integral}& = \frac {1}{12} \left (28+54 x+27 x^2\right )^{2/3}-\int \frac {1}{\sqrt [3]{28+54 x+27 x^2}} \, dx \\ & = \frac {1}{12} \left (28+54 x+27 x^2\right )^{2/3}-\frac {1}{54} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{108}}} \, dx,x,54+54 x\right ) \\ & = \frac {1}{12} \left (28+54 x+27 x^2\right )^{2/3}-\frac {\sqrt {(54+54 x)^2} \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{2 \sqrt {3} (54+54 x)} \\ & = \frac {1}{12} \left (28+54 x+27 x^2\right )^{2/3}+\frac {\sqrt {(54+54 x)^2} \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{2 \sqrt {3} (54+54 x)}-\frac {\left (\left (1+\sqrt {3}\right ) \sqrt {(54+54 x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{28+54 x+27 x^2}\right )}{2 \sqrt {3} (54+54 x)} \\ & = \frac {1}{12} \left (28+54 x+27 x^2\right )^{2/3}+\frac {3 (1+x)}{1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}-\frac {\sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{28+54 x+27 x^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}{1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{6\ 3^{3/4} (1+x) \sqrt {-\frac {1-\sqrt [3]{28+54 x+27 x^2}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}}}+\frac {\sqrt {2} \left (1-\sqrt [3]{28+54 x+27 x^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}{1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} (1+x) \sqrt {-\frac {1-\sqrt [3]{28+54 x+27 x^2}}{\left (1-\sqrt {3}-\sqrt [3]{28+54 x+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 = 10.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.07 \[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\frac {1}{12} \left (28+54 x+27 x^2\right )^{2/3}-(1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-27 (1+x)^2\right ) \]

[In]

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

[Out]

(28 + 54*x + 27*x^2)^(2/3)/12 - (1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, -27*(1 + x)^2]

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {2+3 x}{\sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {3 x + 2}{\sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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