\(\int \frac {1}{(a+b x)^{10/3} \sqrt [3]{c+d x}} \, dx\) [1600]

Optimal result

Integrand size = 19, antiderivative size = 1372 \[ \int \frac {1}{(a+b x)^{10/3} \sqrt [3]{c+d x}} \, dx=-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {15 d^{7/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{14 \sqrt [3]{2} b^{2/3} (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}-\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} d^{7/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{28 \sqrt [3]{2} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}+\frac {5\ 3^{3/4} d^{7/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{7\ 2^{5/6} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}} \]

[Out]

-3/7*(d*x+c)^(2/3)/(-a*d+b*c)/(b*x+a)^(7/3)+15/28*d*(d*x+c)^(2/3)/(-a*d+b*c)^2/(b*x+a)^(4/3)-15/14*d^2*(d*x+c)
^(2/3)/(-a*d+b*c)^3/(b*x+a)^(1/3)+15/28*d^(7/3)*((b*x+a)*(d*x+c))^(1/3)*((2*b*d*x+a*d+b*c)^2)^(1/2)*((a*d+b*(2
*d*x+c))^2)^(1/2)*2^(2/3)/b^(2/3)/(-a*d+b*c)^3/(b*x+a)^(1/3)/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)/(2^(2/3)*b^(1/3)*
d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))+5/14*3^(3/4)*d^(7/3)*((b*x+a)*(d*x+c))^(1/3)*((-
a*d+b*c)^(2/3)+2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3))*EllipticF((2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d
*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1-3^(1/2)))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(
1+3^(1/2))),I*3^(1/2)+2*I)*((2*b*d*x+a*d+b*c)^2)^(1/2)*(((-a*d+b*c)^(4/3)-2^(2/3)*b^(1/3)*d^(1/3)*(-a*d+b*c)^(
2/3)*((b*x+a)*(d*x+c))^(1/3)+2*2^(1/3)*b^(2/3)*d^(2/3)*((b*x+a)*(d*x+c))^(2/3))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x
+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))^2)^(1/2)*2^(1/6)/b^(2/3)/(-a*d+b*c)^(7/3)/(b*x+a)^(1/3)/(d*x+
c)^(1/3)/(2*b*d*x+a*d+b*c)/((a*d+b*(2*d*x+c))^2)^(1/2)/((-a*d+b*c)^(2/3)*((-a*d+b*c)^(2/3)+2^(2/3)*b^(1/3)*d^(
1/3)*((b*x+a)*(d*x+c))^(1/3))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))^2
)^(1/2)-15/56*3^(1/4)*d^(7/3)*((b*x+a)*(d*x+c))^(1/3)*((-a*d+b*c)^(2/3)+2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+
c))^(1/3))*EllipticE((2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1-3^(1/2)))/(2^(2/3)*b
^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*((2*b*d*x+a*d+b*c)^2)^(1/2
)*(1/2*6^(1/2)-1/2*2^(1/2))*(((-a*d+b*c)^(4/3)-2^(2/3)*b^(1/3)*d^(1/3)*(-a*d+b*c)^(2/3)*((b*x+a)*(d*x+c))^(1/3
)+2*2^(1/3)*b^(2/3)*d^(2/3)*((b*x+a)*(d*x+c))^(2/3))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*
c)^(2/3)*(1+3^(1/2)))^2)^(1/2)*2^(2/3)/b^(2/3)/(-a*d+b*c)^(7/3)/(b*x+a)^(1/3)/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)/
((a*d+b*(2*d*x+c))^2)^(1/2)/((-a*d+b*c)^(2/3)*((-a*d+b*c)^(2/3)+2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3
))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(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 = 1.70 (sec) , antiderivative size = 1372, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {53, 64, 637, 309, 224, 1891} \[ \int \frac {1}{(a+b x)^{10/3} \sqrt [3]{c+d x}} \, dx=-\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right ) d^{7/3}}{28 \sqrt [3]{2} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}+\frac {5\ 3^{3/4} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)} (b c-a d)^{2/3}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right ),-7-4 \sqrt {3}\right ) d^{7/3}}{7\ 2^{5/6} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}+\frac {15 \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2} d^{7/3}}{14 \sqrt [3]{2} b^{2/3} (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}-\frac {15 (c+d x)^{2/3} d^2}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {15 (c+d x)^{2/3} d}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}} \]

[In]

Int[1/((a + b*x)^(10/3)*(c + d*x)^(1/3)),x]

[Out]

(-3*(c + d*x)^(2/3))/(7*(b*c - a*d)*(a + b*x)^(7/3)) + (15*d*(c + d*x)^(2/3))/(28*(b*c - a*d)^2*(a + b*x)^(4/3
)) - (15*d^2*(c + d*x)^(2/3))/(14*(b*c - a*d)^3*(a + b*x)^(1/3)) + (15*d^(7/3)*((a + b*x)*(c + d*x))^(1/3)*Sqr
t[(b*c + a*d + 2*b*d*x)^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])/(14*2^(1/3)*b^(2/3)*(b*c - a*d)^3*(a + b*x)^(1/3)*(c
+ d*x)^(1/3)*(b*c + a*d + 2*b*d*x)*((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c +
d*x))^(1/3))) - (15*3^(1/4)*Sqrt[2 - Sqrt[3]]*d^(7/3)*((a + b*x)*(c + d*x))^(1/3)*Sqrt[(b*c + a*d + 2*b*d*x)^
2]*((b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2^(2/3)
*b^(1/3)*d^(1/3)*(b*c - a*d)^(2/3)*((a + b*x)*(c + d*x))^(1/3) + 2*2^(1/3)*b^(2/3)*d^(2/3)*((a + b*x)*(c + d*x
))^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*EllipticE
[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))/((1 + Sqrt[3])
*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))], -7 - 4*Sqrt[3]])/(28*2^(1/3)*b^(2/
3)*(b*c - a*d)^(7/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d
)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(
1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*Sqrt[(a*d + b*(c + 2*d*x))^2]) + (5*3^(3/4)*d^(7/3)*((a + b*x)*(c
+ d*x))^(1/3)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*((b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x)
)^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2^(2/3)*b^(1/3)*d^(1/3)*(b*c - a*d)^(2/3)*((a + b*x)*(c + d*x))^(1/3) + 2*2
^(1/3)*b^(2/3)*d^(2/3)*((a + b*x)*(c + d*x))^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)
*((a + b*x)*(c + d*x))^(1/3))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*(
(a + b*x)*(c + d*x))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(
1/3))], -7 - 4*Sqrt[3]])/(7*2^(5/6)*b^(2/3)*(b*c - a*d)^(7/3)*(a + b*x)^(1/3)*(c + d*x)^(1/3)*(b*c + a*d + 2*b
*d*x)*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3)))/((1 +
Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*Sqrt[(a*d + b*(c + 2*d*x
))^2])

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 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 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}-\frac {(5 d) \int \frac {1}{(a+b x)^{7/3} \sqrt [3]{c+d x}} \, dx}{7 (b c-a d)} \\ & = -\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}+\frac {\left (5 d^2\right ) \int \frac {1}{(a+b x)^{4/3} \sqrt [3]{c+d x}} \, dx}{14 (b c-a d)^2} \\ & = -\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {\left (5 d^3\right ) \int \frac {1}{\sqrt [3]{a+b x} \sqrt [3]{c+d x}} \, dx}{14 (b c-a d)^3} \\ & = -\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {\left (5 d^3 \sqrt [3]{(a+b x) (c+d x)}\right ) \int \frac {1}{\sqrt [3]{a c+(b c+a d) x+b d x^2}} \, dx}{14 (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x}} \\ & = -\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {\left (15 d^3 \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{14 (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)} \\ & = -\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {\left (15 d^{8/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \text {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} x}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{14\ 2^{2/3} \sqrt [3]{b} (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)}-\frac {\left (15 \left (1-\sqrt {3}\right ) d^{8/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{14\ 2^{2/3} \sqrt [3]{b} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x)} \\ & = -\frac {3 (c+d x)^{2/3}}{7 (b c-a d) (a+b x)^{7/3}}+\frac {15 d (c+d x)^{2/3}}{28 (b c-a d)^2 (a+b x)^{4/3}}-\frac {15 d^2 (c+d x)^{2/3}}{14 (b c-a d)^3 \sqrt [3]{a+b x}}+\frac {15 d^{7/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{14 \sqrt [3]{2} b^{2/3} (b c-a d)^3 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}-\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} d^{7/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{28 \sqrt [3]{2} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}+\frac {5\ 3^{3/4} d^{7/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{7\ 2^{5/6} b^{2/3} (b c-a d)^{7/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.05 \[ \int \frac {1}{(a+b x)^{10/3} \sqrt [3]{c+d x}} \, dx=-\frac {3 \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},\frac {1}{3},-\frac {4}{3},\frac {d (a+b x)}{-b c+a d}\right )}{7 b (a+b x)^{7/3} \sqrt [3]{c+d x}} \]

[In]

Integrate[1/((a + b*x)^(10/3)*(c + d*x)^(1/3)),x]

[Out]

(-3*((b*(c + d*x))/(b*c - a*d))^(1/3)*Hypergeometric2F1[-7/3, 1/3, -4/3, (d*(a + b*x))/(-(b*c) + a*d)])/(7*b*(
a + b*x)^(7/3)*(c + d*x)^(1/3))

Maple [F]

\[\int \frac {1}{\left (b x +a \right )^{\frac {10}{3}} \left (d x +c \right )^{\frac {1}{3}}}d x\]

[In]

int(1/(b*x+a)^(10/3)/(d*x+c)^(1/3),x)

[Out]

int(1/(b*x+a)^(10/3)/(d*x+c)^(1/3),x)

Fricas [F]

\[ \int \frac {1}{(a+b x)^{10/3} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {10}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(10/3)/(d*x+c)^(1/3),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{(a+b x)^{10/3} \sqrt [3]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {10}{3}} \sqrt [3]{c + d x}}\, dx \]

[In]

integrate(1/(b*x+a)**(10/3)/(d*x+c)**(1/3),x)

[Out]

Integral(1/((a + b*x)**(10/3)*(c + d*x)**(1/3)), x)

Maxima [F]

\[ \int \frac {1}{(a+b x)^{10/3} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {10}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(10/3)/(d*x+c)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(10/3)*(d*x + c)^(1/3)), x)

Giac [F]

\[ \int \frac {1}{(a+b x)^{10/3} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {10}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(b*x+a)^(10/3)/(d*x+c)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(10/3)*(d*x + c)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{10/3} \sqrt [3]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{10/3}\,{\left (c+d\,x\right )}^{1/3}} \,d x \]

[In]

int(1/((a + b*x)^(10/3)*(c + d*x)^(1/3)),x)

[Out]

int(1/((a + b*x)^(10/3)*(c + d*x)^(1/3)), x)