3.31.24 \(\int \frac {a+b x}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx\) [3024]

Optimal. Leaf size=1326 \[ \frac {3 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{8 d^2}-\frac {9 (b c-a d) \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2}}{8 b^{2/3} d^4 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}+\frac {9 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (b c-a d)^{5/3} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{16 b^{2/3} d^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}-\frac {3\ 3^{3/4} (b c-a d)^{5/3} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt {2} b^{2/3} d^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}} \]

[Out]

3/8*(d*x+c)^(2/3)*(2*b*d*x+a*d+b*c)^(2/3)/d^2-9/8*(-a*d+b*c)*((d*x+c)*(2*b*d*x+a*d+b*c))^(1/3)*(d^2*(4*b*d*x+a
*d+3*b*c)^2)^(1/2)*((d*(a*d+3*b*c)+4*b*d^2*x)^2)^(1/2)/b^(2/3)/d^4/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(1/3)/(4*b*
d*x+a*d+3*b*c)/(2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))-3/8*3^(3/4)*(-a*d+b*
c)^(5/3)*((d*x+c)*(2*b*d*x+a*d+b*c))^(1/3)*((-a*d+b*c)^(2/3)+2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3))*Elli
pticF((2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1-3^(1/2)))/(2*b^(1/3)*((d*x+c)*(a*d+b*(2
*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*((d*(a*d+3*b*c)+4*b*d^2*x)^2)^(1/2)*(((-a*d+b*c)^
(4/3)-2*b^(1/3)*(-a*d+b*c)^(2/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+4*b^(2/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(2/3)
)/(2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))^2)^(1/2)/b^(2/3)/d^2/(d*x+c)^(1/3
)/(2*b*d*x+a*d+b*c)^(1/3)/(4*b*d*x+a*d+3*b*c)*2^(1/2)/(d^2*(4*b*d*x+a*d+3*b*c)^2)^(1/2)/((-a*d+b*c)^(2/3)*((-a
*d+b*c)^(2/3)+2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3))/(2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+
b*c)^(2/3)*(1+3^(1/2)))^2)^(1/2)+9/16*3^(1/4)*(-a*d+b*c)^(5/3)*((d*x+c)*(2*b*d*x+a*d+b*c))^(1/3)*((-a*d+b*c)^(
2/3)+2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3))*EllipticE((2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d
+b*c)^(2/3)*(1-3^(1/2)))/(2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2))),I*3^(1/2)+
2*I)*((d*(a*d+3*b*c)+4*b*d^2*x)^2)^(1/2)*(1/2*6^(1/2)-1/2*2^(1/2))*(((-a*d+b*c)^(4/3)-2*b^(1/3)*(-a*d+b*c)^(2/
3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+4*b^(2/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(2/3))/(2*b^(1/3)*((d*x+c)*(a*d+b*(
2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))^2)^(1/2)/b^(2/3)/d^2/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(1/3)/(4*b
*d*x+a*d+3*b*c)/(d^2*(4*b*d*x+a*d+3*b*c)^2)^(1/2)/((-a*d+b*c)^(2/3)*((-a*d+b*c)^(2/3)+2*b^(1/3)*((d*x+c)*(a*d+
b*(2*d*x+c)))^(1/3))/(2*b^(1/3)*((d*x+c)*(a*d+b*(2*d*x+c)))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 1.05, antiderivative size = 1326, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {81, 64, 637, 309, 224, 1891} \begin {gather*} \frac {9 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (4 b x d^2+(3 b c+a d) d\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))} (b c-a d)^{2/3}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} E\left (\text {ArcSin}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right ) (b c-a d)^{5/3}}{16 b^{2/3} d^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}-\frac {3\ 3^{3/4} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (4 b x d^2+(3 b c+a d) d\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))} (b c-a d)^{2/3}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} F\left (\text {ArcSin}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right ) (b c-a d)^{5/3}}{4 \sqrt {2} b^{2/3} d^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}-\frac {9 \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\left (4 b x d^2+(3 b c+a d) d\right )^2} (b c-a d)}{8 b^{2/3} d^4 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}+\frac {3 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{8 d^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x)/((c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(1/3)),x]

[Out]

(3*(c + d*x)^(2/3)*(b*c + a*d + 2*b*d*x)^(2/3))/(8*d^2) - (9*(b*c - a*d)*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/
3)*Sqrt[d^2*(3*b*c + a*d + 4*b*d*x)^2]*Sqrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2])/(8*b^(2/3)*d^4*(c + d*x)^(1/3)*(
b*c + a*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*
d + b*(c + 2*d*x)))^(1/3))) + (9*3^(1/4)*Sqrt[2 - Sqrt[3]]*(b*c - a*d)^(5/3)*((c + d*x)*(b*c + a*d + 2*b*d*x))
^(1/3)*Sqrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2]*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^
(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2*b^(1/3)*(b*c - a*d)^(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3) + 4*b^(2/
3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(
c + 2*d*x)))^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c +
2*d*x)))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))], -7 - 4
*Sqrt[3]])/(16*b^(2/3)*d^2*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*Sqrt[d^2*(3*b*c
+ a*d + 4*b*d*x)^2]*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^
(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2]) - (3*3^(3/4)
*(b*c - a*d)^(5/3)*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2]*((b*c - a*d)^
(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2*b^(1/3)*(b*c - a*d)^(2/
3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3) + 4*b^(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(2/3))/((1 + Sqrt[3])
*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*(b*
c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/
3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))], -7 - 4*Sqrt[3]])/(4*Sqrt[2]*b^(2/3)*d^2*(c + d*x)^(1/3)*(b*c + a
*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*Sqrt[d^2*(3*b*c + a*d + 4*b*d*x)^2]*Sqrt[((b*c - a*d)^(2/3)*((b*c
- a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3
)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2])

Rule 64

Int[((a_.) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[(a + b*x)^m*((c + d*x)^m/((a + b*x)*
(c + d*x))^m), Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] &&
LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 224

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] && PosQ[a]

Rule 309

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] && PosQ[a]

Rule 637

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[d*(Sqrt[(b + 2*c*x)
^2]/(b + 2*c*x)), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 1891

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] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {a+b x}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx &=\frac {3 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{8 d^2}-\frac {(3 (b c-a d)) \int \frac {1}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx}{4 d}\\ &=\frac {3 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{8 d^2}-\frac {\left (3 (b c-a d) \sqrt [3]{(c+d x) (b c+a d+2 b d x)}\right ) \int \frac {1}{\sqrt [3]{c (b c+a d)+(2 b c d+d (b c+a d)) x+2 b d^2 x^2}} \, dx}{4 d \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}}\\ &=\frac {3 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{8 d^2}-\frac {\left (9 (b c-a d) \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (2 b c d+d (b c+a d)+4 b d^2 x\right )^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-8 b c d^2 (b c+a d)+(2 b c d+d (b c+a d))^2+8 b d^2 x^3}} \, dx,x,\sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{4 d \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} \left (2 b c d+d (b c+a d)+4 b d^2 x\right )}\\ &=\frac {3 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{8 d^2}-\frac {\left (9 (b c-a d) \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (2 b c d+d (b c+a d)+4 b d^2 x\right )^2}\right ) \text {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} x}{\sqrt {-8 b c d^2 (b c+a d)+(2 b c d+d (b c+a d))^2+8 b d^2 x^3}} \, dx,x,\sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{8 \sqrt [3]{b} d \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} \left (2 b c d+d (b c+a d)+4 b d^2 x\right )}-\frac {\left (9 (b c-a d)^{5/3} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (2 b c d+d (b c+a d)+4 b d^2 x\right )^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-8 b c d^2 (b c+a d)+(2 b c d+d (b c+a d))^2+8 b d^2 x^3}} \, dx,x,\sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{4 \sqrt {2 \left (2+\sqrt {3}\right )} \sqrt [3]{b} d \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} \left (2 b c d+d (b c+a d)+4 b d^2 x\right )}\\ &=\frac {3 (c+d x)^{2/3} (b c+a d+2 b d x)^{2/3}}{8 d^2}-\frac {9 (b c-a d) \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2}}{8 b^{2/3} d^4 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}+\frac {9 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (b c-a d)^{5/3} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{16 b^{2/3} d^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}-\frac {3\ 3^{3/4} (b c-a d)^{5/3} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt {2} b^{2/3} d^2 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\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 = 10.10, size = 109, normalized size = 0.08 \begin {gather*} \frac {3 (c+d x)^{2/3} \left (a d+b (c+2 d x)+3 (-b c+a d) \sqrt [3]{\frac {a d+b (c+2 d x)}{-b c+a d}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {2 b (c+d x)}{b c-a d}\right )\right )}{8 d^2 \sqrt [3]{a d+b (c+2 d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/((c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(1/3)),x]

[Out]

(3*(c + d*x)^(2/3)*(a*d + b*(c + 2*d*x) + 3*(-(b*c) + a*d)*((a*d + b*(c + 2*d*x))/(-(b*c) + a*d))^(1/3)*Hyperg
eometric2F1[1/3, 2/3, 5/3, (2*b*(c + d*x))/(b*c - a*d)]))/(8*d^2*(a*d + b*(c + 2*d*x))^(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x}{\sqrt [3]{c + d x} \sqrt [3]{a d + b c + 2 b d x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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