3.26.8 \(\int \frac {1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx\) [2508]

Optimal. Leaf size=696 \[ -\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac {9 (1+x)}{2 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{12\ 2^{2/3} \sqrt {3}}-\frac {\sqrt {2+\sqrt {3}} \left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{72 \sqrt {2} \sqrt [4]{3} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}+\frac {\left (6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right ) \sqrt {\frac {1+\sqrt [3]{28+54 x+27 x^2}+\left (28+54 x+27 x^2\right )^{2/3}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {6 \left (1+\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}\right )|-7+4 \sqrt {3}\right )}{36\ 3^{3/4} (1+x) \sqrt {-\frac {6-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}}{\left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{108+(54+54 x)^2}\right )^2}}}-\frac {\log (2+3 x)}{24\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}} \]

[Out]

-1/24*(27*x^2+54*x+28)^(2/3)/(2+3*x)^2+1/12*(27*x^2+54*x+28)^(2/3)/(2+3*x)-1/48*ln(2+3*x)*2^(1/3)+1/48*ln(-108
-81*x+27*2^(1/3)*(27*x^2+54*x+28)^(1/3))*2^(1/3)+9/2*(1+x)/(-2^(1/3)*(108+(54+54*x)^2)^(1/3)+6-6*3^(1/2))-1/72
*arctan(1/3*3^(1/2)+1/3*2^(2/3)*(4+3*x)/(27*x^2+54*x+28)^(1/3)*3^(1/2))*2^(1/3)*3^(1/2)+1/108*(6-2^(1/3)*(108+
(54+54*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)-1/432*(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)

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Rubi [A]
time = 0.42, antiderivative size = 696, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {758, 848, 857, 633, 241, 310, 225, 1893, 766} \begin {gather*} \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}} F\left (\text {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 )}{36\ 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 (\text {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 )}{72 \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 {\text {ArcTan}\left (\frac {2^{2/3} (3 x+4)}{\sqrt {3} \sqrt [3]{27 x^2+54 x+28}}+\frac {1}{\sqrt {3}}\right )}{12\ 2^{2/3} \sqrt {3}}+\frac {\left (27 x^2+54 x+28\right )^{2/3}}{12 (3 x+2)}-\frac {\left (27 x^2+54 x+28\right )^{2/3}}{24 (3 x+2)^2}+\frac {\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{24\ 2^{2/3}}+\frac {9 (x+1)}{2 \left (6 \left (1-\sqrt {3}\right )-\sqrt [3]{2} \sqrt [3]{(54 x+54)^2+108}\right )}-\frac {\log (3 x+2)}{24\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(28 + 54*x + 27*x^2)^(2/3)/(2 + 3*x)^2 + (28 + 54*x + 27*x^2)^(2/3)/(12*(2 + 3*x)) + (9*(1 + x))/(2*(6*(
1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))) - ArcTan[1/Sqrt[3] + (2^(2/3)*(4 + 3*x))/(Sqrt[3]*(28 + 5
4*x + 27*x^2)^(1/3))]/(12*2^(2/3)*Sqrt[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]])/(72*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)*(10
8 + (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]])/(36*3^(3/4)*(1 + x)*Sqrt[-((6 - 2^(1/3)*(10
8 + (54 + 54*x)^2)^(1/3))/(6*(1 - Sqrt[3]) - 2^(1/3)*(108 + (54 + 54*x)^2)^(1/3))^2)]) - Log[2 + 3*x]/(24*2^(2
/3)) + Log[-108 - 81*x + 27*2^(1/3)*(28 + 54*x + 27*x^2)^(1/3)]/(24*2^(2/3))

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 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 766

Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[-3*c*e^2*(2*c
*d - b*e), 3]}, Simp[(-Sqrt[3])*c*e*(ArcTan[1/Sqrt[3] - 2*((c*d - b*e - c*e*x)/(Sqrt[3]*q*(a + b*x + c*x^2)^(1
/3)))]/q^2), x] + (-Simp[3*c*e*(Log[d + e*x]/(2*q^2)), x] + Simp[3*c*e*(Log[c*d - b*e - c*e*x + q*(a + b*x + c
*x^2)^(1/3)]/(2*q^2)), x])] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && EqQ[c^2*d^2 - b*c*d*e + b^2
*e^2 - 3*a*c*e^2, 0] && NegQ[c*e^2*(2*c*d - b*e)]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

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 {1}{(2+3 x)^3 \sqrt [3]{28+54 x+27 x^2}} \, dx &=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}-\frac {1}{72} \int \frac {108+54 x}{(2+3 x)^2 \sqrt [3]{28+54 x+27 x^2}} \, dx\\ &=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac {\int \frac {-648-1944 x}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx}{2592}\\ &=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {1}{4} \int \frac {1}{\sqrt [3]{28+54 x+27 x^2}} \, dx+\frac {1}{4} \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx\\ &=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{12\ 2^{2/3} \sqrt {3}}-\frac {\log (2+3 x)}{24\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}}-\frac {1}{216} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\frac {x^2}{108}}} \, dx,x,54+54 x\right )\\ &=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{12\ 2^{2/3} \sqrt {3}}-\frac {\log (2+3 x)}{24\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{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 )}{8 \sqrt {3} (54+54 x)}\\ &=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{12\ 2^{2/3} \sqrt {3}}-\frac {\log (2+3 x)}{24\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{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 )}{8 \sqrt {3} (54+54 x)}-\frac {\left (\sqrt {\frac {1}{6} \left (2+\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 )}{4 (54+54 x)}\\ &=-\frac {\left (28+54 x+27 x^2\right )^{2/3}}{24 (2+3 x)^2}+\frac {\left (28+54 x+27 x^2\right )^{2/3}}{12 (2+3 x)}+\frac {3 (1+x)}{4 \left (1-\sqrt {3}-\sqrt [3]{28+54 x+27 x^2}\right )}-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{12\ 2^{2/3} \sqrt {3}}-\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 )}{24\ 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 {\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 )}{18 \sqrt {2} \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}}}-\frac {\log (2+3 x)}{24\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{24\ 2^{2/3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 17.59, size = 233, normalized size = 0.33 \begin {gather*} \frac {\frac {162 (1+2 x) \left (28+54 x+27 x^2\right )}{(2+3 x)^2}-54 \sqrt [3]{3} \sqrt [3]{\frac {9-i \sqrt {3}+9 x}{2+3 x}} \sqrt [3]{\frac {9+i \sqrt {3}+9 x}{2+3 x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};-\frac {3+i \sqrt {3}}{6+9 x},\frac {-3+i \sqrt {3}}{6+9 x}\right )+9 i 2^{2/3} 3^{5/6} \sqrt [3]{-9 i+\sqrt {3}-9 i x} \left (9 i+\sqrt {3}+9 i x\right ) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {9 i+\sqrt {3}+9 i x}{2 \sqrt {3}}\right )}{1296 \sqrt [3]{28+54 x+27 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((162*(1 + 2*x)*(28 + 54*x + 27*x^2))/(2 + 3*x)^2 - 54*3^(1/3)*((9 - I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((9 + I
*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -((3 + I*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(
6 + 9*x)] + (9*I)*2^(2/3)*3^(5/6)*(-9*I + Sqrt[3] - (9*I)*x)^(1/3)*(9*I + Sqrt[3] + (9*I)*x)*Hypergeometric2F1
[1/3, 2/3, 5/3, (9*I + Sqrt[3] + (9*I)*x)/(2*Sqrt[3])])/(1296*(28 + 54*x + 27*x^2)^(1/3))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (2+3 x \right )^{3} \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

integral((27*x^2 + 54*x + 28)^(2/3)/(729*x^5 + 2916*x^4 + 4644*x^3 + 3672*x^2 + 1440*x + 224), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x + 2\right )^{3} \sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (3\,x+2\right )}^3\,{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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