\(\int \frac {(b+x^3)^3}{\sqrt [3]{a+x^3}} \, dx\) [2529]

Optimal result

Integrand size = 17, antiderivative size = 212 \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{162} \left (a+x^3\right )^{2/3} \left (28 a^2 x-108 a b x+162 b^2 x-21 a x^4+81 b x^4+18 x^7\right )+\frac {1}{243} \left (-14 \sqrt {3} a^3+54 \sqrt {3} a^2 b-81 \sqrt {3} a b^2+81 \sqrt {3} b^3\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )+\frac {1}{243} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\frac {1}{486} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \log \left (x^2+x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}\right ) \]

[Out]

1/162*(x^3+a)^(2/3)*(18*x^7-21*a*x^4+81*b*x^4+28*a^2*x-108*a*b*x+162*b^2*x)+1/243*(-14*3^(1/2)*a^3+54*3^(1/2)*
a^2*b-81*3^(1/2)*a*b^2+81*3^(1/2)*b^3)*arctan(3^(1/2)*x/(x+2*(x^3+a)^(1/3)))+1/243*(14*a^3-54*a^2*b+81*a*b^2-8
1*b^3)*ln(-x+(x^3+a)^(1/3))+1/486*(-14*a^3+54*a^2*b-81*a*b^2+81*b^3)*ln(x^2+x*(x^3+a)^(1/3)+(x^3+a)^(2/3))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {427, 542, 396, 245} \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{162} x \left (28 a^2-87 a b+99 b^2\right ) \left (a+x^3\right )^{2/3}-\frac {\left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {1}{162} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (\sqrt [3]{a+x^3}-x\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2-\frac {1}{54} x (7 a-15 b) \left (a+x^3\right )^{2/3} \left (b+x^3\right ) \]

[In]

Int[(b + x^3)^3/(a + x^3)^(1/3),x]

[Out]

((28*a^2 - 87*a*b + 99*b^2)*x*(a + x^3)^(2/3))/162 - ((7*a - 15*b)*x*(a + x^3)^(2/3)*(b + x^3))/54 + (x*(a + x
^3)^(2/3)*(b + x^3)^2)/9 - ((14*a^3 - 54*a^2*b + 81*a*b^2 - 81*b^3)*ArcTan[(1 + (2*x)/(a + x^3)^(1/3))/Sqrt[3]
])/(81*Sqrt[3]) + ((14*a^3 - 54*a^2*b + 81*a*b^2 - 81*b^3)*Log[-x + (a + x^3)^(1/3)])/162

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 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 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 542

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{9} \int \frac {\left (b+x^3\right ) \left (-((a-9 b) b)+(-7 a+15 b) x^3\right )}{\sqrt [3]{a+x^3}} \, dx \\ & = -\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{54} \int \frac {b \left (7 a^2-21 a b+54 b^2\right )+\left (28 a^2-87 a b+99 b^2\right ) x^3}{\sqrt [3]{a+x^3}} \, dx \\ & = \frac {1}{162} \left (28 a^2-87 a b+99 b^2\right ) x \left (a+x^3\right )^{2/3}-\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{81} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \int \frac {1}{\sqrt [3]{a+x^3}} \, dx \\ & = \frac {1}{162} \left (28 a^2-87 a b+99 b^2\right ) x \left (a+x^3\right )^{2/3}-\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2-\frac {\left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{a+x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {1}{162} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (-x+\sqrt [3]{a+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.92 \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{162} x \left (a+x^3\right )^{2/3} \left (28 a^2-108 a b+162 b^2-21 a x^3+81 b x^3+18 x^6\right )-\frac {\left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )}{81 \sqrt {3}}+\frac {1}{243} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\frac {1}{486} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \log \left (x^2+x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}\right ) \]

[In]

Integrate[(b + x^3)^3/(a + x^3)^(1/3),x]

[Out]

(x*(a + x^3)^(2/3)*(28*a^2 - 108*a*b + 162*b^2 - 21*a*x^3 + 81*b*x^3 + 18*x^6))/162 - ((14*a^3 - 54*a^2*b + 81
*a*b^2 - 81*b^3)*ArcTan[(Sqrt[3]*x)/(x + 2*(a + x^3)^(1/3))])/(81*Sqrt[3]) + ((14*a^3 - 54*a^2*b + 81*a*b^2 -
81*b^3)*Log[-x + (a + x^3)^(1/3)])/243 + ((-14*a^3 + 54*a^2*b - 81*a*b^2 + 81*b^3)*Log[x^2 + x*(a + x^3)^(1/3)
+ (a + x^3)^(2/3)])/486

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(-\frac {14 \left (\sqrt {3}\, \left (a^{3}-\frac {27}{7} a^{2} b +\frac {81}{14} a \,b^{2}-\frac {81}{14} b^{3}\right ) \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+a \right )^{\frac {1}{3}}\right )}{3 x}\right )+\frac {27 x^{7} \left (x^{3}+a \right )^{\frac {2}{3}}}{14}-\frac {9 \left (a -\frac {27 b}{7}\right ) \left (x^{3}+a \right )^{\frac {2}{3}} x^{4}}{4}+3 \left (a^{2}-\frac {27}{7} a b +\frac {81}{14} b^{2}\right ) \left (x^{3}+a \right )^{\frac {2}{3}} x +\left (\ln \left (\frac {-x +\left (x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {x^{2}+x \left (x^{3}+a \right )^{\frac {1}{3}}+\left (x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (a^{3}-\frac {27}{7} a^{2} b +\frac {81}{14} a \,b^{2}-\frac {81}{14} b^{3}\right )\right ) a^{3}}{243 \left (x^{2}+x \left (x^{3}+a \right )^{\frac {1}{3}}+\left (x^{3}+a \right )^{\frac {2}{3}}\right )^{3} {\left (x -\left (x^{3}+a \right )^{\frac {1}{3}}\right )}^{3}}\)	\(205\)

[In]

int((x^3+b)^3/(x^3+a)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-14/243*(3^(1/2)*(a^3-27/7*a^2*b+81/14*a*b^2-81/14*b^3)*arctan(1/3*3^(1/2)/x*(x+2*(x^3+a)^(1/3)))+27/14*x^7*(x
^3+a)^(2/3)-9/4*(a-27/7*b)*(x^3+a)^(2/3)*x^4+3*(a^2-27/7*a*b+81/14*b^2)*(x^3+a)^(2/3)*x+(ln((-x+(x^3+a)^(1/3))
/x)-1/2*ln((x^2+x*(x^3+a)^(1/3)+(x^3+a)^(2/3))/x^2))*(a^3-27/7*a^2*b+81/14*a*b^2-81/14*b^3))*a^3/(x^2+x*(x^3+a
)^(1/3)+(x^3+a)^(2/3))^3/(x-(x^3+a)^(1/3))^3

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.90 \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{243} \, \sqrt {3} {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + a\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{243} \, {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \log \left (-\frac {x - {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{486} \, {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \log \left (\frac {x^{2} + {\left (x^{3} + a\right )}^{\frac {1}{3}} x + {\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {1}{162} \, {\left (18 \, x^{7} - 3 \, {\left (7 \, a - 27 \, b\right )} x^{4} + 2 \, {\left (14 \, a^{2} - 54 \, a b + 81 \, b^{2}\right )} x\right )} {\left (x^{3} + a\right )}^{\frac {2}{3}} \]

[In]

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

[Out]

1/243*sqrt(3)*(14*a^3 - 54*a^2*b + 81*a*b^2 - 81*b^3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + a)^(1/3))/x) +
1/243*(14*a^3 - 54*a^2*b + 81*a*b^2 - 81*b^3)*log(-(x - (x^3 + a)^(1/3))/x) - 1/486*(14*a^3 - 54*a^2*b + 81*a*
b^2 - 81*b^3)*log((x^2 + (x^3 + a)^(1/3)*x + (x^3 + a)^(2/3))/x^2) + 1/162*(18*x^7 - 3*(7*a - 27*b)*x^4 + 2*(1
4*a^2 - 54*a*b + 81*b^2)*x)*(x^3 + a)^(2/3)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.94 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.71 \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {b^{3} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {4}{3}\right )} + \frac {b^{2} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {b x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} \Gamma \left (\frac {10}{3}\right )} + \frac {x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {13}{3}\right )} \]

[In]

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

[Out]

b**3*x*gamma(1/3)*hyper((1/3, 1/3), (4/3,), x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gamma(4/3)) + b**2*x**4*gamma(
4/3)*hyper((1/3, 4/3), (7/3,), x**3*exp_polar(I*pi)/a)/(a**(1/3)*gamma(7/3)) + b*x**7*gamma(7/3)*hyper((1/3, 7
/3), (10/3,), x**3*exp_polar(I*pi)/a)/(a**(1/3)*gamma(10/3)) + x**10*gamma(10/3)*hyper((1/3, 10/3), (13/3,), x
**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gamma(13/3))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (184) = 368\).

Time = 0.30 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.26 \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {14}{243} \, \sqrt {3} a^{3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {1}{6} \, {\left (2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 2 \, \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right )\right )} b^{3} - \frac {7}{243} \, a^{3} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {14}{243} \, a^{3} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {1}{6} \, {\left (2 \, \sqrt {3} a \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 2 \, a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {6 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a}{x^{2} {\left (\frac {x^{3} + a}{x^{3}} - 1\right )}}\right )} b^{2} - \frac {1}{18} \, {\left (4 \, \sqrt {3} a^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - 2 \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 4 \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {3 \, {\left (\frac {7 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a^{2}}{x^{2}} - \frac {4 \, {\left (x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{\frac {2 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {{\left (x^{3} + a\right )}^{2}}{x^{6}} - 1}\right )} b + \frac {\frac {67 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a^{3}}{x^{2}} - \frac {77 \, {\left (x^{3} + a\right )}^{\frac {5}{3}} a^{3}}{x^{5}} + \frac {28 \, {\left (x^{3} + a\right )}^{\frac {8}{3}} a^{3}}{x^{8}}}{162 \, {\left (\frac {3 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {3 \, {\left (x^{3} + a\right )}^{2}}{x^{6}} + \frac {{\left (x^{3} + a\right )}^{3}}{x^{9}} - 1\right )}} \]

[In]

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

[Out]

14/243*sqrt(3)*a^3*arctan(1/3*sqrt(3)*(2*(x^3 + a)^(1/3)/x + 1)) - 1/6*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 +
a)^(1/3)/x + 1)) - log((x^3 + a)^(1/3)/x + (x^3 + a)^(2/3)/x^2 + 1) + 2*log((x^3 + a)^(1/3)/x - 1))*b^3 - 7/2
43*a^3*log((x^3 + a)^(1/3)/x + (x^3 + a)^(2/3)/x^2 + 1) + 14/243*a^3*log((x^3 + a)^(1/3)/x - 1) + 1/6*(2*sqrt(
3)*a*arctan(1/3*sqrt(3)*(2*(x^3 + a)^(1/3)/x + 1)) - a*log((x^3 + a)^(1/3)/x + (x^3 + a)^(2/3)/x^2 + 1) + 2*a*
log((x^3 + a)^(1/3)/x - 1) + 6*(x^3 + a)^(2/3)*a/(x^2*((x^3 + a)/x^3 - 1)))*b^2 - 1/18*(4*sqrt(3)*a^2*arctan(1
/3*sqrt(3)*(2*(x^3 + a)^(1/3)/x + 1)) - 2*a^2*log((x^3 + a)^(1/3)/x + (x^3 + a)^(2/3)/x^2 + 1) + 4*a^2*log((x^
3 + a)^(1/3)/x - 1) + 3*(7*(x^3 + a)^(2/3)*a^2/x^2 - 4*(x^3 + a)^(5/3)*a^2/x^5)/(2*(x^3 + a)/x^3 - (x^3 + a)^2
/x^6 - 1))*b + 1/162*(67*(x^3 + a)^(2/3)*a^3/x^2 - 77*(x^3 + a)^(5/3)*a^3/x^5 + 28*(x^3 + a)^(8/3)*a^3/x^8)/(3
*(x^3 + a)/x^3 - 3*(x^3 + a)^2/x^6 + (x^3 + a)^3/x^9 - 1)

Giac [F]

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

[In]

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

[Out]

integrate((x^3 + b)^3/(x^3 + a)^(1/3), x)

Mupad [F(-1)]

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

[In]

int((b + x^3)^3/(a + x^3)^(1/3),x)

[Out]

int((b + x^3)^3/(a + x^3)^(1/3), x)