\(\int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx\) [2499]

Optimal result

Integrand size = 20, antiderivative size = 603 \[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\frac {90 \sqrt [3]{5} (1-x)}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}-\frac {5^{5/6} \sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{108 \sqrt {2} \sqrt [4]{3} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac {5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right ),-7+4 \sqrt {3}\right )}{54\ 3^{3/4} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}} \]

[Out]

1/12*(27*x^2-54*x+52)^(2/3)+90*5^(1/3)*(1-x)/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2))+1/162*5^(5/6)
*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticF((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30+30*3^(1/2))/(-10^(1
/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2)),2*I-I*3^(1/2))*((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2/
3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2))^2)^(1/2)*3^(1/4)/(1-x)/((-30
+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2))^2)^(1/2)-1/324*5^(5/6
)*(30-10^(1/3)*(2700+(-54+54*x)^2)^(1/3))*EllipticE((-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30+30*3^(1/2))/(-10^(
1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2)),2*I-I*3^(1/2))*((900+30*10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+10^(2
/3)*(2700+(-54+54*x)^2)^(2/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2))^2)^(1/2)*(1/2+1/2*3^(1/2))*
3^(3/4)/(1-x)/((-30+10^(1/3)*(2700+(-54+54*x)^2)^(1/3))/(-10^(1/3)*(2700+(-54+54*x)^2)^(1/3)+30-30*3^(1/2))^2)
^(1/2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 603, 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} \[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right ),-7+4 \sqrt {3}\right )}{54\ 3^{3/4} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}-\frac {5^{5/6} \sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right ) \sqrt {\frac {10^{2/3} \left ((54 x-54)^2+2700\right )^{2/3}+30 \sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}+900}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} E\left (\arcsin \left (\frac {30 \left (1+\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}\right )|-7+4 \sqrt {3}\right )}{108 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}}{\left (30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}\right )^2}} (1-x)}+\frac {1}{12} \left (27 x^2-54 x+52\right )^{2/3}+\frac {90 \sqrt [3]{5} (1-x)}{30 \left (1-\sqrt {3}\right )-\sqrt [3]{10} \sqrt [3]{(54 x-54)^2+2700}} \]

[In]

Int[(2 + 3*x)/(52 - 54*x + 27*x^2)^(1/3),x]

[Out]

(52 - 54*x + 27*x^2)^(2/3)/12 + (90*5^(1/3)*(1 - x))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3
)) - (5^(5/6)*Sqrt[2 + Sqrt[3]]*(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)*(2700 +
(-54 + 54*x)^2)^(1/3) + 10^(2/3)*(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54
*x)^2)^(1/3))^2]*EllipticE[ArcSin[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]
) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(108*Sqrt[2]*3^(1/4)*(1 - x)*Sqrt[-((30 - 10^(1
/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2)]) + (5^(5/6)
*(30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))*Sqrt[(900 + 30*10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3) + 10^(2/3
)*(2700 + (-54 + 54*x)^2)^(2/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))^2]*EllipticF[ArcS
in[(30*(1 + Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(1 - Sqrt[3]) - 10^(1/3)*(2700 + (-54 + 54*
x)^2)^(1/3))], -7 + 4*Sqrt[3]])/(54*3^(3/4)*(1 - x)*Sqrt[-((30 - 10^(1/3)*(2700 + (-54 + 54*x)^2)^(1/3))/(30*(
1 - Sqrt[3]) - 10^(1/3)*(2700 + (-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 (52-54 x+27 x^2\right )^{2/3}+5 \int \frac {1}{\sqrt [3]{52-54 x+27 x^2}} \, dx \\ & = \frac {1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\frac {1}{54} \sqrt [3]{5} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{2700}}} \, dx,x,-54+54 x\right ) \\ & = \frac {1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\frac {\left (5 \sqrt [3]{5} \sqrt {(-54+54 x)^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\frac {\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{2 \sqrt {3} (-54+54 x)} \\ & = \frac {1}{12} \left (52-54 x+27 x^2\right )^{2/3}-\frac {\left (5 \sqrt [3]{5} \sqrt {(-54+54 x)^2}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\frac {\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{2 \sqrt {3} (-54+54 x)}+\frac {\left (5 \sqrt [3]{5} \left (1+\sqrt {3}\right ) \sqrt {(-54+54 x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\frac {\sqrt [3]{2700+(-54+54 x)^2}}{3\ 10^{2/3}}\right )}{2 \sqrt {3} (-54+54 x)} \\ & = \frac {1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\frac {90 \sqrt [3]{5} (1-x)}{30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}-\frac {5^{5/6} \sqrt {2+\sqrt {3}} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {30+30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{108 \sqrt {2} \sqrt [4]{3} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}}}+\frac {5^{5/6} \left (30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right ) \sqrt {\frac {900+30 \sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}+10^{2/3} \left (2700+(-54+54 x)^2\right )^{2/3}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {30+30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{54\ 3^{3/4} (1-x) \sqrt {-\frac {30-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 x)^2}}{\left (30-30 \sqrt {3}-\sqrt [3]{10} \sqrt [3]{2700+(-54+54 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 = 47, normalized size of antiderivative = 0.08 \[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\frac {1}{12} \left (52-54 x+27 x^2\right )^{2/3}+\sqrt [3]{5} (-1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {27}{25} (-1+x)^2\right ) \]

[In]

Integrate[(2 + 3*x)/(52 - 54*x + 27*x^2)^(1/3),x]

[Out]

(52 - 54*x + 27*x^2)^(2/3)/12 + 5^(1/3)*(-1 + x)*Hypergeometric2F1[1/3, 1/2, 3/2, (-27*(-1 + x)^2)/25]

Maple [F]

\[\int \frac {2+3 x}{\left (27 x^{2}-54 x +52\right )^{\frac {1}{3}}}d x\]

[In]

int((2+3*x)/(27*x^2-54*x+52)^(1/3),x)

[Out]

int((2+3*x)/(27*x^2-54*x+52)^(1/3),x)

Fricas [F]

\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((2+3*x)/(27*x^2-54*x+52)^(1/3),x, algorithm="fricas")

[Out]

integral((3*x + 2)/(27*x^2 - 54*x + 52)^(1/3), x)

Sympy [F]

\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {3 x + 2}{\sqrt [3]{27 x^{2} - 54 x + 52}}\, dx \]

[In]

integrate((2+3*x)/(27*x**2-54*x+52)**(1/3),x)

[Out]

Integral((3*x + 2)/(27*x**2 - 54*x + 52)**(1/3), x)

Maxima [F]

\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((2+3*x)/(27*x^2-54*x+52)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*x + 2)/(27*x^2 - 54*x + 52)^(1/3), x)

Giac [F]

\[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int { \frac {3 \, x + 2}{{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((2+3*x)/(27*x^2-54*x+52)^(1/3),x, algorithm="giac")

[Out]

integrate((3*x + 2)/(27*x^2 - 54*x + 52)^(1/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {2+3 x}{\sqrt [3]{52-54 x+27 x^2}} \, dx=\int \frac {3\,x+2}{{\left (27\,x^2-54\,x+52\right )}^{1/3}} \,d x \]

[In]

int((3*x + 2)/(27*x^2 - 54*x + 52)^(1/3),x)

[Out]

int((3*x + 2)/(27*x^2 - 54*x + 52)^(1/3), x)