\(\int \frac {(a+b x^3)^{5/3}}{c+d x^3} \, dx\) [87]

Optimal result

Integrand size = 21, antiderivative size = 273 \[ \int \frac {\left (a+b x^3\right )^{5/3}}{c+d x^3} \, dx=\frac {b x \left (a+b x^3\right )^{2/3}}{3 d}-\frac {b^{2/3} (3 b c-5 a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} d^2}+\frac {(b c-a d)^{5/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} d^2}+\frac {(b c-a d)^{5/3} \log \left (c+d x^3\right )}{6 c^{2/3} d^2}-\frac {(b c-a d)^{5/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} d^2}+\frac {b^{2/3} (3 b c-5 a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{6 d^2} \]

[Out]

1/3*b*x*(b*x^3+a)^(2/3)/d+1/6*(-a*d+b*c)^(5/3)*ln(d*x^3+c)/c^(2/3)/d^2-1/2*(-a*d+b*c)^(5/3)*ln((-a*d+b*c)^(1/3
)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(2/3)/d^2+1/6*b^(2/3)*(-5*a*d+3*b*c)*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/d^2-1/9*b^(
2/3)*(-5*a*d+3*b*c)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/d^2*3^(1/2)+1/3*(-a*d+b*c)^(5/3)*arcta
n(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))/c^(2/3)/d^2*3^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {427, 544, 245, 384} \[ \int \frac {\left (a+b x^3\right )^{5/3}}{c+d x^3} \, dx=-\frac {b^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) (3 b c-5 a d)}{3 \sqrt {3} d^2}+\frac {(b c-a d)^{5/3} \arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} d^2}+\frac {b^{2/3} (3 b c-5 a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{6 d^2}+\frac {(b c-a d)^{5/3} \log \left (c+d x^3\right )}{6 c^{2/3} d^2}-\frac {(b c-a d)^{5/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} d^2}+\frac {b x \left (a+b x^3\right )^{2/3}}{3 d} \]

[In]

Int[(a + b*x^3)^(5/3)/(c + d*x^3),x]

[Out]

(b*x*(a + b*x^3)^(2/3))/(3*d) - (b^(2/3)*(3*b*c - 5*a*d)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]]
)/(3*Sqrt[3]*d^2) + ((b*c - a*d)^(5/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3
]])/(Sqrt[3]*c^(2/3)*d^2) + ((b*c - a*d)^(5/3)*Log[c + d*x^3])/(6*c^(2/3)*d^2) - ((b*c - a*d)^(5/3)*Log[((b*c
- a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(2*c^(2/3)*d^2) + (b^(2/3)*(3*b*c - 5*a*d)*Log[-(b^(1/3)*x) + (a
+ b*x^3)^(1/3)])/(6*d^2)

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

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

Rule 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 544

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

Rubi steps \begin{align*} \text {integral}& = \frac {b x \left (a+b x^3\right )^{2/3}}{3 d}+\frac {\int \frac {-a (b c-3 a d)-b (3 b c-5 a d) x^3}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{3 d} \\ & = \frac {b x \left (a+b x^3\right )^{2/3}}{3 d}-\frac {(b (3 b c-5 a d)) \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{3 d^2}+\frac {(b c-a d)^2 \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{d^2} \\ & = \frac {b x \left (a+b x^3\right )^{2/3}}{3 d}-\frac {b^{2/3} (3 b c-5 a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} d^2}+\frac {(b c-a d)^{5/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} d^2}+\frac {(b c-a d)^{5/3} \log \left (c+d x^3\right )}{6 c^{2/3} d^2}-\frac {(b c-a d)^{5/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} d^2}+\frac {b^{2/3} (3 b c-5 a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{6 d^2} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 10.51 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+b x^3\right )^{5/3}}{c+d x^3} \, dx=\frac {3 b \sqrt [3]{b c-a d} (-3 b c+5 a d) x^4 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 \sqrt [3]{c} \left (6 a b c^{2/3} \sqrt [3]{b c-a d} x+6 b^2 c^{2/3} \sqrt [3]{b c-a d} x^4+2 \sqrt {3} a (-b c+3 a d) \sqrt [3]{a+b x^3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )+2 a (b c-3 a d) \sqrt [3]{a+b x^3} \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )-a b c \sqrt [3]{a+b x^3} \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+3 a^2 d \sqrt [3]{a+b x^3} \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{36 c d \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}} \]

[In]

Integrate[(a + b*x^3)^(5/3)/(c + d*x^3),x]

[Out]

(3*b*(b*c - a*d)^(1/3)*(-3*b*c + 5*a*d)*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((
d*x^3)/c)] + 2*c^(1/3)*(6*a*b*c^(2/3)*(b*c - a*d)^(1/3)*x + 6*b^2*c^(2/3)*(b*c - a*d)^(1/3)*x^4 + 2*Sqrt[3]*a*
(-(b*c) + 3*a*d)*(a + b*x^3)^(1/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] +
2*a*(b*c - 3*a*d)*(a + b*x^3)^(1/3)*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] - a*b*c*(a + b*x^3
)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1
/3)] + 3*a^2*d*(a + b*x^3)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d
)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(36*c*d*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3))

Maple [A] (verified)

Time = 4.57 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.47

method	result	size
pseudoelliptic	\(-\frac {-\frac {5 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c \left (a d \,b^{\frac {2}{3}}-\frac {3 b^{\frac {5}{3}} c}{5}\right ) \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{3}-2 \left (a d -b c \right )^{2} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )+\frac {10 \sqrt {3}\, \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c \left (a d \,b^{\frac {2}{3}}-\frac {3 b^{\frac {5}{3}} c}{5}\right ) \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )}{3}+\frac {10 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c \left (a d \,b^{\frac {2}{3}}-\frac {3 b^{\frac {5}{3}} c}{5}\right ) \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{3}-2 \left (b \,x^{3}+a \right )^{\frac {2}{3}} x b c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} d +\left (-2 \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )\right ) \left (a d -b c \right )^{2}}{6 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c \,d^{2}}\)	\(402\)

[In]

int((b*x^3+a)^(5/3)/(d*x^3+c),x,method=_RETURNVERBOSE)

[Out]

-1/6*(-5/3*((a*d-b*c)/c)^(1/3)*c*(a*d*b^(2/3)-3/5*b^(5/3)*c)*ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+
a)^(2/3))/x^2)-2*(a*d-b*c)^2*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)+10/3*3^(1/2)*((a*d-b*c)/c)^(1/3)*c*
(a*d*b^(2/3)-3/5*b^(5/3)*c)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/3))/b^(1/3)/x)+10/3*((a*d-b*c)/c)^(1/
3)*c*(a*d*b^(2/3)-3/5*b^(5/3)*c)*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)-2*(b*x^3+a)^(2/3)*x*b*c*((a*d-b*c)/c)^(1/3
)*d+(-2*arctan(1/3*3^(1/2)*(((a*d-b*c)/c)^(1/3)*x-2*(b*x^3+a)^(1/3))/((a*d-b*c)/c)^(1/3)/x)*3^(1/2)+ln((((a*d-
b*c)/c)^(2/3)*x^2-((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2))*(a*d-b*c)^2)/((a*d-b*c)/c)^(1/3
)/c/d^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (220) = 440\).

Time = 0.76 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b x^3\right )^{5/3}}{c+d x^3} \, dx=\frac {6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b d x + 6 \, \sqrt {3} {\left (b c - a d\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (b c - a d\right )} x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} c \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}}}{3 \, {\left (b c - a d\right )} x}\right ) + 2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} {\left (3 \, b c - 5 \, a d\right )} \arctan \left (-\frac {\sqrt {3} b x - 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {1}{3}}}{3 \, b x}\right ) - 6 \, {\left (b c - a d\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (\frac {c x \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )}}{x}\right ) - 2 \, \left (-b^{2}\right )^{\frac {1}{3}} {\left (3 \, b c - 5 \, a d\right )} \log \left (-\frac {\left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + \left (-b^{2}\right )^{\frac {1}{3}} {\left (3 \, b c - 5 \, a d\right )} \log \left (-\frac {\left (-b^{2}\right )^{\frac {1}{3}} b x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right ) + 3 \, {\left (b c - a d\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (-\frac {{\left (b c - a d\right )} x^{2} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} c x \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b c - a d\right )}}{x^{2}}\right )}{18 \, d^{2}} \]

[In]

integrate((b*x^3+a)^(5/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

1/18*(6*(b*x^3 + a)^(2/3)*b*d*x + 6*sqrt(3)*(b*c - a*d)*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3)*arctan(-1/
3*(sqrt(3)*(b*c - a*d)*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*c*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3))/((b*c -
a*d)*x)) + 2*sqrt(3)*(-b^2)^(1/3)*(3*b*c - 5*a*d)*arctan(-1/3*(sqrt(3)*b*x - 2*sqrt(3)*(b*x^3 + a)^(1/3)*(-b^2
)^(1/3))/(b*x)) - 6*(b*c - a*d)*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3)*log((c*x*((b^2*c^2 - 2*a*b*c*d + a
^2*d^2)/c^2)^(2/3) - (b*x^3 + a)^(1/3)*(b*c - a*d))/x) - 2*(-b^2)^(1/3)*(3*b*c - 5*a*d)*log(-((-b^2)^(2/3)*x -
(b*x^3 + a)^(1/3)*b)/x) + (-b^2)^(1/3)*(3*b*c - 5*a*d)*log(-((-b^2)^(1/3)*b*x^2 - (b*x^3 + a)^(1/3)*(-b^2)^(2
/3)*x - (b*x^3 + a)^(2/3)*b)/x^2) + 3*(b*c - a*d)*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3)*log(-((b*c - a*d
)*x^2*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3) + (b*x^3 + a)^(1/3)*c*x*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2
)^(2/3) + (b*x^3 + a)^(2/3)*(b*c - a*d))/x^2))/d^2

Sympy [F]

\[ \int \frac {\left (a+b x^3\right )^{5/3}}{c+d x^3} \, dx=\int \frac {\left (a + b x^{3}\right )^{\frac {5}{3}}}{c + d x^{3}}\, dx \]

[In]

integrate((b*x**3+a)**(5/3)/(d*x**3+c),x)

[Out]

Integral((a + b*x**3)**(5/3)/(c + d*x**3), x)

Maxima [F]

\[ \int \frac {\left (a+b x^3\right )^{5/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{d x^{3} + c} \,d x } \]

[In]

integrate((b*x^3+a)^(5/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(5/3)/(d*x^3 + c), x)

Giac [F]

\[ \int \frac {\left (a+b x^3\right )^{5/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{d x^{3} + c} \,d x } \]

[In]

integrate((b*x^3+a)^(5/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(5/3)/(d*x^3 + c), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{5/3}}{c+d x^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{5/3}}{d\,x^3+c} \,d x \]

[In]

int((a + b*x^3)^(5/3)/(c + d*x^3),x)

[Out]

int((a + b*x^3)^(5/3)/(c + d*x^3), x)