Integrand size = 20, antiderivative size = 180 \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{18} \left (a+x^3\right )^{2/3} \left (-4 a x+6 b x+6 c x+3 x^4\right )+\frac {1}{27} \left (2 \sqrt {3} a^2-3 \sqrt {3} a b-3 \sqrt {3} a c+9 \sqrt {3} b c\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )+\frac {1}{27} \left (-2 a^2+3 a b+3 a c-9 b c\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\frac {1}{54} \left (2 a^2-3 a b-3 a c+9 b c\right ) \log \left (x^2+x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}\right ) \]
1/18*(x^3+a)^(2/3)*(3*x^4-4*a*x+6*b*x+6*c*x)+1/27*(2*3^(1/2)*a^2-3*3^(1/2) *a*b-3*3^(1/2)*a*c+9*3^(1/2)*b*c)*arctan(3^(1/2)*x/(x+2*(x^3+a)^(1/3)))+1/ 27*(-2*a^2+3*a*b+3*a*c-9*b*c)*ln(-x+(x^3+a)^(1/3))+1/54*(2*a^2-3*a*b-3*a*c +9*b*c)*ln(x^2+x*(x^3+a)^(1/3)+(x^3+a)^(2/3))
Time = 0.61 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.84 \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{54} \left (3 x \left (a+x^3\right )^{2/3} \left (-4 a+6 b+6 c+3 x^3\right )+2 \sqrt {3} \left (2 a^2+9 b c-3 a (b+c)\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )+2 \left (-2 a^2-9 b c+3 a (b+c)\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\left (2 a^2+9 b c-3 a (b+c)\right ) \log \left (x^2+x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}\right )\right ) \]
(3*x*(a + x^3)^(2/3)*(-4*a + 6*b + 6*c + 3*x^3) + 2*Sqrt[3]*(2*a^2 + 9*b*c - 3*a*(b + c))*ArcTan[(Sqrt[3]*x)/(x + 2*(a + x^3)^(1/3))] + 2*(-2*a^2 - 9*b*c + 3*a*(b + c))*Log[-x + (a + x^3)^(1/3)] + (2*a^2 + 9*b*c - 3*a*(b + c))*Log[x^2 + x*(a + x^3)^(1/3) + (a + x^3)^(2/3)])/54
Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.64, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1025, 25, 913, 769}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx\) |
| \(\Big \downarrow \) 1025 |
| \(\displaystyle \frac {1}{6} \int -\frac {(4 a-3 b-6 c) x^3+b (a-6 c)}{\sqrt [3]{x^3+a}}dx+\frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )-\frac {1}{6} \int \frac {(4 a-3 b-6 c) x^3+b (a-6 c)}{\sqrt [3]{x^3+a}}dx\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {1}{6} \left (\frac {2}{3} \left (2 a^2-3 a (b+c)+9 b c\right ) \int \frac {1}{\sqrt [3]{x^3+a}}dx-\frac {1}{3} x \left (a+x^3\right )^{2/3} (4 a-3 b-6 c)\right )+\frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle \frac {1}{6} \left (\frac {2}{3} \left (2 a^2-3 a (b+c)+9 b c\right ) \left (\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{a+x^3}-x\right )\right )-\frac {1}{3} x \left (a+x^3\right )^{2/3} (4 a-3 b-6 c)\right )+\frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )\) |
(x*(a + x^3)^(2/3)*(b + x^3))/6 + (-1/3*((4*a - 3*b - 6*c)*x*(a + x^3)^(2/ 3)) + (2*(2*a^2 + 9*b*c - 3*a*(b + c))*(ArcTan[(1 + (2*x)/(a + x^3)^(1/3)) /Sqrt[3]]/Sqrt[3] - Log[-x + (a + x^3)^(1/3)]/2))/3)/6
3.24.20.3.1 Defintions of rubi rules used
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]]/(Sqrt[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]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(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, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
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] + Simp[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]
Time = 0.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\frac {\sqrt {3}\, \left (a^{2}+\frac {3 \left (-b -c \right ) a}{2}+\frac {9 b c}{2}\right ) \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+a \right )^{\frac {1}{3}}\right )}{3 x}\right )}{3}-\frac {3 x^{4} \left (x^{3}+a \right )^{\frac {2}{3}}}{4}+\left (a -\frac {3 b}{2}-\frac {3 c}{2}\right ) \left (x^{3}+a \right )^{\frac {2}{3}} x -\frac {\left (\ln \left (\frac {x^{2}+x \left (x^{3}+a \right )^{\frac {1}{3}}+\left (x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-x +\left (x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right ) \left (a^{2}+\frac {3 \left (-b -c \right ) a}{2}+\frac {9 b c}{2}\right )}{6}\right ) a^{2}}{9 \left (x^{2}+x \left (x^{3}+a \right )^{\frac {1}{3}}+\left (x^{3}+a \right )^{\frac {2}{3}}\right )^{2} {\left (x -\left (x^{3}+a \right )^{\frac {1}{3}}\right )}^{2}}\) | \(178\) |
-2/9*(1/3*3^(1/2)*(a^2+3/2*(-b-c)*a+9/2*b*c)*arctan(1/3*3^(1/2)/x*(x+2*(x^ 3+a)^(1/3)))-3/4*x^4*(x^3+a)^(2/3)+(a-3/2*b-3/2*c)*(x^3+a)^(2/3)*x-1/6*(ln ((x^2+x*(x^3+a)^(1/3)+(x^3+a)^(2/3))/x^2)-2*ln((-x+(x^3+a)^(1/3))/x))*(a^2 +3/2*(-b-c)*a+9/2*b*c))*a^2/(x^2+x*(x^3+a)^(1/3)+(x^3+a)^(2/3))^2/(x-(x^3+ a)^(1/3))^2
Time = 0.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.88 \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=-\frac {1}{27} \, \sqrt {3} {\left (2 \, a^{2} - 3 \, a b - 3 \, {\left (a - 3 \, b\right )} c\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + a\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{27} \, {\left (2 \, a^{2} - 3 \, a b - 3 \, {\left (a - 3 \, b\right )} c\right )} \log \left (-\frac {x - {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{54} \, {\left (2 \, a^{2} - 3 \, a b - 3 \, {\left (a - 3 \, b\right )} c\right )} \log \left (\frac {x^{2} + {\left (x^{3} + a\right )}^{\frac {1}{3}} x + {\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {1}{18} \, {\left (3 \, x^{4} - 2 \, {\left (2 \, a - 3 \, b - 3 \, c\right )} x\right )} {\left (x^{3} + a\right )}^{\frac {2}{3}} \]
-1/27*sqrt(3)*(2*a^2 - 3*a*b - 3*(a - 3*b)*c)*arctan(1/3*(sqrt(3)*x + 2*sq rt(3)*(x^3 + a)^(1/3))/x) - 1/27*(2*a^2 - 3*a*b - 3*(a - 3*b)*c)*log(-(x - (x^3 + a)^(1/3))/x) + 1/54*(2*a^2 - 3*a*b - 3*(a - 3*b)*c)*log((x^2 + (x^ 3 + a)^(1/3)*x + (x^3 + a)^(2/3))/x^2) + 1/18*(3*x^4 - 2*(2*a - 3*b - 3*c) *x)*(x^3 + a)^(2/3)
Result contains complex when optimal does not.
Time = 3.47 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\frac {b c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {4}{3}\right )} + \frac {b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {10}{3}\right )} \]
b*c*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*x**4*gamma(4/3)*hyper((1/3, 4/3), (7/3,), x**3*exp_po lar(I*pi)/a)/(3*a**(1/3)*gamma(7/3)) + c*x**4*gamma(4/3)*hyper((1/3, 4/3), (7/3,), x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gamma(7/3)) + x**7*gamma(7/3) *hyper((1/3, 7/3), (10/3,), x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*gamma(10/3 ))
Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (152) = 304\).
Time = 0.29 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.29 \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=-\frac {2}{27} \, \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 ) - \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 c + \frac {1}{27} \, 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 ) - \frac {2}{27} \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {1}{18} \, {\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 + \frac {1}{18} \, {\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 )} c - \frac {\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}}}{18 \, {\left (\frac {2 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {{\left (x^{3} + a\right )}^{2}}{x^{6}} - 1\right )}} \]
-2/27*sqrt(3)*a^2*arctan(1/3*sqrt(3)*(2*(x^3 + a)^(1/3)/x + 1)) - 1/6*(2*s qrt(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*c + 1/27*a ^2*log((x^3 + a)^(1/3)/x + (x^3 + a)^(2/3)/x^2 + 1) - 2/27*a^2*log((x^3 + a)^(1/3)/x - 1) + 1/18*(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 + 1 /18*(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)))*c - 1/18*(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)
\[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\int { \frac {{\left (x^{3} + b\right )} {\left (x^{3} + c\right )}}{{\left (x^{3} + a\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\int \frac {\left (x^3+b\right )\,\left (x^3+c\right )}{{\left (x^3+a\right )}^{1/3}} \,d x \]