\(\int \frac {(a+b x)^3}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx\) [3029]

Optimal result

Integrand size = 33, antiderivative size = 1388 \[ \int \frac {(a+b x)^3}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\frac {3 (a+b x)^2 (c+d x)^{2/3}}{14 d^2 \sqrt [3]{b c+a d+2 b d x}}+\frac {9 (b c-a d) (c+d x)^{2/3} (b c-7 a d-6 b d x)}{112 d^4 \sqrt [3]{b c+a d+2 b d x}}+\frac {81 (b c-a d)^2 \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}}{112 b^{2/3} d^6 \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 {81 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (b c-a d)^{8/3} \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 (\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 )}{224 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) \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 {27\ 3^{3/4} (b c-a d)^{8/3} \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}} \operatorname {EllipticF}\left (\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 )}{56 \sqrt {2} 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) \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]

3/14*(b*x+a)^2*(d*x+c)^(2/3)/d^2/(2*b*d*x+a*d+b*c)^(1/3)+9/112*(-a*d+b*c)*(d*x+c)^(2/3)*(-6*b*d*x-7*a*d+b*c)/d
^4/(2*b*d*x+a*d+b*c)^(1/3)+81/112*(-a*d+b*c)^2*((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^6/(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)))+27/112*3^(3/4)*(-a*d+b*c)^(8/3)*((
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))*EllipticF((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^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^(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)-81/224*3^(1/4)*(-a*d+b*c)^(8/3)*((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^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)/(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)

Rubi [A] (warning: unable to verify)

Time = 1.41 (sec) , antiderivative size = 1388, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {102, 151, 64, 637, 309, 224, 1891} \[ \int \frac {(a+b x)^3}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=-\frac {81 \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 (\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)^{8/3}}{224 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) \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 {27\ 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}} \operatorname {EllipticF}\left (\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)^{8/3}}{56 \sqrt {2} 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) \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 {81 \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)^2}{112 b^{2/3} d^6 \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 (c+d x)^{2/3} (b c-7 a d-6 b d x) (b c-a d)}{112 d^4 \sqrt [3]{b c+a d+2 b d x}}+\frac {3 (a+b x)^2 (c+d x)^{2/3}}{14 d^2 \sqrt [3]{b c+a d+2 b d x}} \]

[In]

Int[(a + b*x)^3/((c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(4/3)),x]

[Out]

(3*(a + b*x)^2*(c + d*x)^(2/3))/(14*d^2*(b*c + a*d + 2*b*d*x)^(1/3)) + (9*(b*c - a*d)*(c + d*x)^(2/3)*(b*c - 7
*a*d - 6*b*d*x))/(112*d^4*(b*c + a*d + 2*b*d*x)^(1/3)) + (81*(b*c - a*d)^2*((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])/(112*b^(2/3)*d^6*(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))) - (81*3^(1/4)*Sqrt[2 - Sqrt[3]]*(b*c - a*d)^(8/3)*((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]])/(224*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)*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]) + (27*
3^(3/4)*(b*c - a*d)^(8/3)*((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 + S
qrt[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]])/(56*Sqrt[2]*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)*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 102

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Dist[
(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m +
1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 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*} \text {integral}& = \frac {3 (a+b x)^2 (c+d x)^{2/3}}{14 d^2 \sqrt [3]{b c+a d+2 b d x}}+\frac {3 \int \frac {(a+b x) \left (-2 b (b c-a d) (b c+2 a d)-6 b^2 d (b c-a d) x\right )}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx}{14 b d^2} \\ & = \frac {3 (a+b x)^2 (c+d x)^{2/3}}{14 d^2 \sqrt [3]{b c+a d+2 b d x}}+\frac {9 (b c-a d) (c+d x)^{2/3} (b c-7 a d-6 b d x)}{112 d^4 \sqrt [3]{b c+a d+2 b d x}}+\frac {\left (27 (b c-a d)^2\right ) \int \frac {1}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx}{56 d^3} \\ & = \frac {3 (a+b x)^2 (c+d x)^{2/3}}{14 d^2 \sqrt [3]{b c+a d+2 b d x}}+\frac {9 (b c-a d) (c+d x)^{2/3} (b c-7 a d-6 b d x)}{112 d^4 \sqrt [3]{b c+a d+2 b d x}}+\frac {\left (27 (b c-a d)^2 \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}{56 d^3 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \\ & = \frac {3 (a+b x)^2 (c+d x)^{2/3}}{14 d^2 \sqrt [3]{b c+a d+2 b d x}}+\frac {9 (b c-a d) (c+d x)^{2/3} (b c-7 a d-6 b d x)}{112 d^4 \sqrt [3]{b c+a d+2 b d x}}+\frac {\left (81 (b c-a d)^2 \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 )}{56 d^3 \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 (a+b x)^2 (c+d x)^{2/3}}{14 d^2 \sqrt [3]{b c+a d+2 b d x}}+\frac {9 (b c-a d) (c+d x)^{2/3} (b c-7 a d-6 b d x)}{112 d^4 \sqrt [3]{b c+a d+2 b d x}}+\frac {\left (81 (b c-a d)^2 \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 )}{112 \sqrt [3]{b} d^3 \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 (81 \left (1-\sqrt {3}\right ) (b c-a d)^{8/3} \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 )}{112 \sqrt [3]{b} d^3 \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 (a+b x)^2 (c+d x)^{2/3}}{14 d^2 \sqrt [3]{b c+a d+2 b d x}}+\frac {9 (b c-a d) (c+d x)^{2/3} (b c-7 a d-6 b d x)}{112 d^4 \sqrt [3]{b c+a d+2 b d x}}+\frac {81 (b c-a d)^2 \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}}{112 b^{2/3} d^6 \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 {81 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (b c-a d)^{8/3} \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 )}{224 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) \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 {27\ 3^{3/4} (b c-a d)^{8/3} \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 )}{56 \sqrt {2} 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) \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*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.20 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.10 \[ \int \frac {(a+b x)^3}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\frac {3 (c+d x)^{2/3} \left (29 a^2 d^2+2 a b d (-12 c+17 d x)+b^2 \left (3 c^2-18 c d x+8 d^2 x^2\right )+27 (b c-a d)^2 \sqrt [3]{\frac {a d+b (c+2 d x)}{-b c+a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {2 b (c+d x)}{b c-a d}\right )\right )}{112 d^4 \sqrt [3]{a d+b (c+2 d x)}} \]

[In]

Integrate[(a + b*x)^3/((c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(4/3)),x]

[Out]

(3*(c + d*x)^(2/3)*(29*a^2*d^2 + 2*a*b*d*(-12*c + 17*d*x) + b^2*(3*c^2 - 18*c*d*x + 8*d^2*x^2) + 27*(b*c - a*d
)^2*((a*d + b*(c + 2*d*x))/(-(b*c) + a*d))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (2*b*(c + d*x))/(b*c - a*d)]
))/(112*d^4*(a*d + b*(c + 2*d*x))^(1/3))

Maple [F]

\[\int \frac {\left (b x +a \right )^{3}}{\left (d x +c \right )^{\frac {1}{3}} \left (2 b d x +a d +b c \right )^{\frac {4}{3}}}d x\]

[In]

int((b*x+a)^3/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(4/3),x)

[Out]

int((b*x+a)^3/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(4/3),x)

Fricas [F]

\[ \int \frac {(a+b x)^3}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{3}}{{\left (2 \, b d x + b c + a d\right )}^{\frac {4}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((b*x+a)^3/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(4/3),x, algorithm="fricas")

[Out]

integral((b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*(2*b*d*x + b*c + a*d)^(2/3)*(d*x + c)^(2/3)/(4*b^2*d^3*x^3
+ b^2*c^3 + 2*a*b*c^2*d + a^2*c*d^2 + 4*(2*b^2*c*d^2 + a*b*d^3)*x^2 + (5*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x)
, x)

Sympy [F]

\[ \int \frac {(a+b x)^3}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\int \frac {\left (a + b x\right )^{3}}{\sqrt [3]{c + d x} \left (a d + b c + 2 b d x\right )^{\frac {4}{3}}}\, dx \]

[In]

integrate((b*x+a)**3/(d*x+c)**(1/3)/(2*b*d*x+a*d+b*c)**(4/3),x)

[Out]

Integral((a + b*x)**3/((c + d*x)**(1/3)*(a*d + b*c + 2*b*d*x)**(4/3)), x)

Maxima [F]

\[ \int \frac {(a+b x)^3}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{3}}{{\left (2 \, b d x + b c + a d\right )}^{\frac {4}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((b*x+a)^3/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(4/3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^3/((2*b*d*x + b*c + a*d)^(4/3)*(d*x + c)^(1/3)), x)

Giac [F]

\[ \int \frac {(a+b x)^3}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{3}}{{\left (2 \, b d x + b c + a d\right )}^{\frac {4}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((b*x+a)^3/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)^(4/3),x, algorithm="giac")

[Out]

integrate((b*x + a)^3/((2*b*d*x + b*c + a*d)^(4/3)*(d*x + c)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^3}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx=\int \frac {{\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{1/3}\,{\left (a\,d+b\,c+2\,b\,d\,x\right )}^{4/3}} \,d x \]

[In]

int((a + b*x)^3/((c + d*x)^(1/3)*(a*d + b*c + 2*b*d*x)^(4/3)),x)

[Out]

int((a + b*x)^3/((c + d*x)^(1/3)*(a*d + b*c + 2*b*d*x)^(4/3)), x)