3.196 \(\int \frac{\sqrt{c+d x^3}}{a+b x} \, dx\)

Optimal. Leaf size=1482 \[ \text{result too large to display} \]

[Out]

(2*Sqrt[c + d*x^3])/(3*b) - (2*a*d^(1/3)*Sqrt[c + d*x^3])/(b^2*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (c^(1/6)
*Sqrt[b*c^(1/3) - a*d^(1/3)]*Sqrt[b^2*c^(2/3) + a*b*c^(1/3)*d^(1/3) + a^2*d^(2/3)]*(c^(1/3) + d^(1/3)*x)*Sqrt[
(c^(2/3)*(1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*ArcTanh[(Sq
rt[2 - Sqrt[3]]*Sqrt[b^2*c^(2/3) + a*b*c^(1/3)*d^(1/3) + a^2*d^(2/3)]*Sqrt[1 - ((1 - Sqrt[3])*c^(1/3) + d^(1/3
)*x)^2/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2])/(3^(1/4)*Sqrt[b]*c^(1/6)*Sqrt[b*c^(1/3) - a*d^(1/3)]*Sqrt[7 - 4
*Sqrt[3] + ((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)^2/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2])])/(b^(5/2)*Sqrt[(c^(1
/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3]) + (3^(1/4)*Sqrt[2 - Sqrt[3]
]*a*c^(1/3)*d^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1
/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)],
-7 - 4*Sqrt[3]])/(b^2*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*
x^3]) + (2*Sqrt[2 + Sqrt[3]]*a*((1 - Sqrt[3])*b*c^(1/3) + a*d^(1/3))*d^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/
3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*c
^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*b^3*Sqrt[(c^(1/3)*(c^(1/3)
+ d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3]) - (2*Sqrt[2 + Sqrt[3]]*(b^3*c - a^3*d)*
(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*
EllipticF[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(3
^(1/4)*b^3*((1 + Sqrt[3])*b*c^(1/3) - a*d^(1/3))*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) +
d^(1/3)*x)^2]*Sqrt[c + d*x^3]) - (4*3^(1/4)*Sqrt[2 + Sqrt[3]]*c^(1/3)*(b^3*c - a^3*d)*(c^(1/3) + d^(1/3)*x)*S
qrt[(c^(2/3)*(1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Ellipti
cPi[((1 + Sqrt[3])*b*c^(1/3) - a*d^(1/3))^2/((1 - Sqrt[3])*b*c^(1/3) - a*d^(1/3))^2, -ArcSin[((1 - Sqrt[3])*c^
(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(b^2*(2*b^2*c^(2/3) + 2*a*b*c^(1/3)*
d^(1/3) - a^2*d^(2/3))*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*
x^3])

________________________________________________________________________________________

Rubi [A]  time = 2.81458, antiderivative size = 1482, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.632, Rules used = {2147, 261, 1878, 218, 1877, 2136, 2142, 2113, 537, 571, 93, 208} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x^3]/(a + b*x),x]

[Out]

(2*Sqrt[c + d*x^3])/(3*b) - (2*a*d^(1/3)*Sqrt[c + d*x^3])/(b^2*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (c^(1/6)
*Sqrt[b*c^(1/3) - a*d^(1/3)]*Sqrt[b^2*c^(2/3) + a*b*c^(1/3)*d^(1/3) + a^2*d^(2/3)]*(c^(1/3) + d^(1/3)*x)*Sqrt[
(c^(2/3)*(1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*ArcTanh[(Sq
rt[2 - Sqrt[3]]*Sqrt[b^2*c^(2/3) + a*b*c^(1/3)*d^(1/3) + a^2*d^(2/3)]*Sqrt[1 - ((1 - Sqrt[3])*c^(1/3) + d^(1/3
)*x)^2/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2])/(3^(1/4)*Sqrt[b]*c^(1/6)*Sqrt[b*c^(1/3) - a*d^(1/3)]*Sqrt[7 - 4
*Sqrt[3] + ((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)^2/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2])])/(b^(5/2)*Sqrt[(c^(1
/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3]) + (3^(1/4)*Sqrt[2 - Sqrt[3]
]*a*c^(1/3)*d^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1
/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)],
-7 - 4*Sqrt[3]])/(b^2*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*
x^3]) + (2*Sqrt[2 + Sqrt[3]]*a*((1 - Sqrt[3])*b*c^(1/3) + a*d^(1/3))*d^(1/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/
3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*c
^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*b^3*Sqrt[(c^(1/3)*(c^(1/3)
+ d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3]) - (2*Sqrt[2 + Sqrt[3]]*(b^3*c - a^3*d)*
(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*
EllipticF[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(3
^(1/4)*b^3*((1 + Sqrt[3])*b*c^(1/3) - a*d^(1/3))*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) +
d^(1/3)*x)^2]*Sqrt[c + d*x^3]) - (4*3^(1/4)*Sqrt[2 + Sqrt[3]]*c^(1/3)*(b^3*c - a^3*d)*(c^(1/3) + d^(1/3)*x)*S
qrt[(c^(2/3)*(1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Ellipti
cPi[((1 + Sqrt[3])*b*c^(1/3) - a*d^(1/3))^2/((1 - Sqrt[3])*b*c^(1/3) - a*d^(1/3))^2, -ArcSin[((1 - Sqrt[3])*c^
(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(b^2*(2*b^2*c^(2/3) + 2*a*b*c^(1/3)*
d^(1/3) - a^2*d^(2/3))*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*
x^3])

Rule 2147

Int[Sqrt[(a_) + (b_.)*(x_)^3]/((c_) + (d_.)*(x_)), x_Symbol] :> Dist[b/d, Int[x^2/Sqrt[a + b*x^3], x], x] + (D
ist[(b*c)/d^3, Int[(c - d*x)/Sqrt[a + b*x^3], x], x] - Dist[(b*c^3 - a*d^3)/d^3, Int[1/((c + d*x)*Sqrt[a + b*x
^3]), x], x]) /; FreeQ[{a, b, c, d}, x]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 1878

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a,
3]]}, Dist[(c*r - (1 - Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 - Sqrt[3])*s + r*x)
/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((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]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

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]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rule 2136

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[b/a, 3]}, -Dist[q/((1 + Sqrt[
3])*d - c*q), Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/((1 + Sqrt[3])*d - c*q), Int[(1 + Sqrt[3] + q*x)/((c + d*
x)*Sqrt[a + b*x^3]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0]

Rule 2142

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{q = Simplify[((1 +
Sqrt[3])*f)/e]}, Dist[(4*3^(1/4)*Sqrt[2 - Sqrt[3]]*f*(1 + q*x)*Sqrt[(1 - q*x + q^2*x^2)/(1 + Sqrt[3] + q*x)^2
])/(q*Sqrt[a + b*x^3]*Sqrt[(1 + q*x)/(1 + Sqrt[3] + q*x)^2]), Subst[Int[1/(((1 - Sqrt[3])*d - c*q + ((1 + Sqrt
[3])*d - c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]), x], x, (-1 + Sqrt[3] - q*x)/(1 + Sqrt[3] + q*x)], x
]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 + 3*Sqrt[3])*a*f^3, 0] && NeQ[b*c^
3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]

Rule 2113

Int[1/(((a_) + (b_.)*(x_))*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[a, Int[1/((
a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Dist[b, Int[x/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[
e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 571

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n], x] /; FreeQ[{a, b, c, d, e,
f, m, n, p, q, r}, x] && EqQ[m - n + 1, 0]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d x^3}}{a+b x} \, dx &=\frac{(a d) \int \frac{a-b x}{\sqrt{c+d x^3}} \, dx}{b^3}+\frac{d \int \frac{x^2}{\sqrt{c+d x^3}} \, dx}{b}-\left (-c+\frac{a^3 d}{b^3}\right ) \int \frac{1}{(a+b x) \sqrt{c+d x^3}} \, dx\\ &=\frac{2 \sqrt{c+d x^3}}{3 b}-\frac{\left (a d^{2/3}\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\sqrt{c+d x^3}} \, dx}{b^2}+\frac{\left (a \left (a+\frac{\left (1-\sqrt{3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{b^3}+\frac{\left (b \left (c-\frac{a^3 d}{b^3}\right )\right ) \int \frac{1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{(a+b x) \sqrt{c+d x^3}} \, dx}{b+\sqrt{3} b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}}-\frac{\left (\sqrt [3]{d} \left (c-\frac{a^3 d}{b^3}\right )\right ) \int \frac{1}{\sqrt{c+d x^3}} \, dx}{\left (1+\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}}\\ &=\frac{2 \sqrt{c+d x^3}}{3 b}-\frac{2 a \sqrt [3]{d} \sqrt{c+d x^3}}{b^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} a \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{b^2 \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{2+\sqrt{3}} a \left (a+\frac{\left (1-\sqrt{3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^3 \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{2 \sqrt{2+\sqrt{3}} \left (c-\frac{a^3 d}{b^3}\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b \left (c-\frac{a^3 d}{b^3}\right ) \left (1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt{\frac{1-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\left (1-\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}+\left (\left (1+\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right ) x\right ) \sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx,x,\frac{-1+\sqrt{3}-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}\right )}{\left (b+\sqrt{3} b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right ) \sqrt{\frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}} \sqrt{c+d x^3}}\\ &=\frac{2 \sqrt{c+d x^3}}{3 b}-\frac{2 a \sqrt [3]{d} \sqrt{c+d x^3}}{b^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} a \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{b^2 \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{2+\sqrt{3}} a \left (a+\frac{\left (1-\sqrt{3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^3 \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{2 \sqrt{2+\sqrt{3}} \left (c-\frac{a^3 d}{b^3}\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b \left (c-\frac{a^3 d}{b^3}\right ) \left (1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt{\frac{1-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2} \left (\left (\left (1-\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2-\left (\left (1+\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2 x^2\right )} \, dx,x,\frac{-1+\sqrt{3}-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}\right )}{\sqrt{\frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}} \sqrt{c+d x^3}}+\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b \left (b-\sqrt{3} b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right ) \left (c-\frac{a^3 d}{b^3}\right ) \left (1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt{\frac{1-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2} \left (\left (\left (1-\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2-\left (\left (1+\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2 x^2\right )} \, dx,x,\frac{-1+\sqrt{3}-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}\right )}{\left (b+\sqrt{3} b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right ) \sqrt{\frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}} \sqrt{c+d x^3}}\\ &=\frac{2 \sqrt{c+d x^3}}{3 b}-\frac{2 a \sqrt [3]{d} \sqrt{c+d x^3}}{b^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} a \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{b^2 \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{2+\sqrt{3}} a \left (a+\frac{\left (1-\sqrt{3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^3 \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{2 \sqrt{2+\sqrt{3}} \left (c-\frac{a^3 d}{b^3}\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{c} \left (b^3 c-a^3 d\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3} \left (1-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \Pi \left (\frac{\left (\left (1+\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2};-\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{b^2 \left (2 b^2 c^{2/3}+2 a b \sqrt [3]{c} \sqrt [3]{d}-a^2 d^{2/3}\right ) \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\left (2 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b \left (c-\frac{a^3 d}{b^3}\right ) \left (1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt{\frac{1-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{7-4 \sqrt{3}+x} \left (\left (\left (1-\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2-\left (\left (1+\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2 x\right )} \, dx,x,\frac{\left (-1+\sqrt{3}-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}\right )}{\sqrt{\frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}} \sqrt{c+d x^3}}\\ &=\frac{2 \sqrt{c+d x^3}}{3 b}-\frac{2 a \sqrt [3]{d} \sqrt{c+d x^3}}{b^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} a \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{b^2 \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{2+\sqrt{3}} a \left (a+\frac{\left (1-\sqrt{3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^3 \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{2 \sqrt{2+\sqrt{3}} \left (c-\frac{a^3 d}{b^3}\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{c} \left (b^3 c-a^3 d\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3} \left (1-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \Pi \left (\frac{\left (\left (1+\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2};-\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{b^2 \left (2 b^2 c^{2/3}+2 a b \sqrt [3]{c} \sqrt [3]{d}-a^2 d^{2/3}\right ) \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{\left (4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b \left (c-\frac{a^3 d}{b^3}\right ) \left (1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right ) \sqrt{\frac{1-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{-\left (\left (1-\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2+\left (\left (1+\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2-\left (\left (\left (1-\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2+\left (7-4 \sqrt{3}\right ) \left (\left (1+\sqrt{3}\right ) b-\frac{a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2\right ) x^2} \, dx,x,\frac{\sqrt{1-\frac{\left (-1+\sqrt{3}-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}}{\sqrt{7-4 \sqrt{3}+\frac{\left (-1+\sqrt{3}-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}}}\right )}{\sqrt{\frac{1+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}}{\left (1+\sqrt{3}+\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2}} \sqrt{c+d x^3}}\\ &=\frac{2 \sqrt{c+d x^3}}{3 b}-\frac{2 a \sqrt [3]{d} \sqrt{c+d x^3}}{b^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac{\sqrt [6]{c} \sqrt{b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt{b^2 c^{2/3}+a b \sqrt [3]{c} \sqrt [3]{d}+a^2 d^{2/3}} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3} \left (1-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \tanh ^{-1}\left (\frac{\sqrt{2-\sqrt{3}} \sqrt{b^2 c^{2/3}+a b \sqrt [3]{c} \sqrt [3]{d}+a^2 d^{2/3}} \sqrt{1-\frac{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}}}{\sqrt [4]{3} \sqrt{b} \sqrt [6]{c} \sqrt{b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt{7-4 \sqrt{3}+\frac{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}}}\right )}{b^{5/2} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} a \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{b^2 \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{2 \sqrt{2+\sqrt{3}} a \left (a+\frac{\left (1-\sqrt{3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) d^{2/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^3 \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{2 \sqrt{2+\sqrt{3}} \left (c-\frac{a^3 d}{b^3}\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{c} \left (b^3 c-a^3 d\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3} \left (1-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+\frac{d^{2/3} x^2}{c^{2/3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \Pi \left (\frac{\left (\left (1+\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt{3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2};-\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right )|-7-4 \sqrt{3}\right )}{b^2 \left (2 b^2 c^{2/3}+2 a b \sqrt [3]{c} \sqrt [3]{d}-a^2 d^{2/3}\right ) \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}\\ \end{align*}

Mathematica [C]  time = 2.0305, size = 820, normalized size = 0.55 \[ \frac{2 \left (\frac{\sqrt [3]{-1} \sqrt{3} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c} d \sqrt{\frac{\sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}} \sqrt{\frac{d^{2/3} x^2}{c^{2/3}}-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+1} \Pi \left (\frac{i \sqrt{3} b \sqrt [3]{c}}{\sqrt [3]{d} a+\sqrt [3]{-1} b \sqrt [3]{c}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}}\right )|\sqrt [3]{-1}\right ) a^3}{b^2 \left (\sqrt [3]{d} a+\sqrt [3]{-1} b \sqrt [3]{c}\right )}-\frac{3^{3/4} d^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt{\frac{\sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}} \sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{d} x}{\sqrt [3]{c}}} F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}}\right )|\sqrt [3]{-1}\right ) a^2}{b^2 \sqrt{\frac{(-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}}}+\frac{3^{3/4} \sqrt [3]{c} \sqrt [3]{d} \left (\sqrt [3]{-1} \sqrt [3]{c}-\sqrt [3]{d} x\right ) \sqrt{-\frac{2 i \sqrt [3]{d} x}{\sqrt [3]{c}}+\sqrt{3}+i} \sqrt{\frac{i \left (\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right )}{3 i+\sqrt{3}}} \left (\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{d} x}{\sqrt [3]{c}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )+F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt [6]{-1}-\frac{i \sqrt [3]{d} x}{\sqrt [3]{c}}}}{\sqrt [4]{3}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right ) a}{b \sqrt{\frac{(-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}}}+d x^3+c-\frac{3 i b c^{4/3} \sqrt{\frac{\sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}} \sqrt{\frac{d^{2/3} x^2}{c^{2/3}}-\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+1} \Pi \left (\frac{i \sqrt{3} b \sqrt [3]{c}}{\sqrt [3]{d} a+\sqrt [3]{-1} b \sqrt [3]{c}};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{d} a+\sqrt [3]{-1} b \sqrt [3]{c}}\right )}{3 b \sqrt{d x^3+c}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c + d*x^3]/(a + b*x),x]

[Out]

(2*(c + d*x^3 - (3^(3/4)*a^2*d^(2/3)*((-1)^(1/3)*c^(1/3) - d^(1/3)*x)*Sqrt[(c^(1/3) + d^(1/3)*x)/((1 + (-1)^(1
/3))*c^(1/3))]*Sqrt[(-1)^(1/6) - (I*d^(1/3)*x)/c^(1/3)]*EllipticF[ArcSin[Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)
/((1 + (-1)^(1/3))*c^(1/3))]], (-1)^(1/3)])/(b^2*Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/
3))]) + (3^(3/4)*a*c^(1/3)*d^(1/3)*((-1)^(1/3)*c^(1/3) - d^(1/3)*x)*Sqrt[I + Sqrt[3] - ((2*I)*d^(1/3)*x)/c^(1/
3)]*Sqrt[(I*(1 + (d^(1/3)*x)/c^(1/3)))/(3*I + Sqrt[3])]*((-1 + (-1)^(2/3))*EllipticE[ArcSin[Sqrt[(-1)^(1/6) -
(I*d^(1/3)*x)/c^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))] + EllipticF[ArcSin[Sqrt[(-1)^(1/6) - (I*d^(1/3)
*x)/c^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]))/(b*Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/
3))*c^(1/3))]) - ((3*I)*b*c^(4/3)*Sqrt[(c^(1/3) + d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]*Sqrt[1 - (d^(1/3)*x)/
c^(1/3) + (d^(2/3)*x^2)/c^(2/3)]*EllipticPi[(I*Sqrt[3]*b*c^(1/3))/((-1)^(1/3)*b*c^(1/3) + a*d^(1/3)), ArcSin[S
qrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]], (-1)^(1/3)])/((-1)^(1/3)*b*c^(1/3) + a*d^(1
/3)) + ((-1)^(1/3)*Sqrt[3]*(1 + (-1)^(1/3))*a^3*c^(1/3)*d*Sqrt[(c^(1/3) + d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3)
)]*Sqrt[1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)]*EllipticPi[(I*Sqrt[3]*b*c^(1/3))/((-1)^(1/3)*b*c^(1/3
) + a*d^(1/3)), ArcSin[Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]], (-1)^(1/3)])/(b^2*(
(-1)^(1/3)*b*c^(1/3) + a*d^(1/3)))))/(3*b*Sqrt[c + d*x^3])

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Maple [A]  time = 0.151, size = 1126, normalized size = 0.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)^(1/2)/(b*x+a),x)

[Out]

2/3*(d*x^3+c)^(1/2)/b-2/3*I*a^2/b^3*3^(1/2)*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)
^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2
)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d
*x^3+c)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d
^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+2/
3*I*a/b^2*3^(1/2)*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)
^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2
/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c
*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(
-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2
)/d*(-c*d^2)^(1/3)))^(1/2))+1/d*(-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/
d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(
1/2)/d*(-c*d^2)^(1/3)))^(1/2)))+2/3*I*(a^3*d-b^3*c)/b^4*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/
2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1
/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-
c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)/(-1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)+a/b)*EllipticPi(1/3*
3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),I*3^(1/2)/d
*(-c*d^2)^(1/3)/(-1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)+a/b),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d
*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{3} + c}}{b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^(1/2)/(b*x+a),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^3 + c)/(b*x + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{3} + c}}{b x + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^(1/2)/(b*x+a),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^3 + c)/(b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x^{3}}}{a + b x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c)**(1/2)/(b*x+a),x)

[Out]

Integral(sqrt(c + d*x**3)/(a + b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x^{3} + c}}{b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^(1/2)/(b*x+a),x, algorithm="giac")

[Out]

integrate(sqrt(d*x^3 + c)/(b*x + a), x)