Integrand size = 24, antiderivative size = 276 \[ \int \frac {x^4 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\frac {x^2 \sqrt [3]{a+b x^3}}{3 d}+\frac {(3 b c-a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{2/3} d^2}-\frac {c^{2/3} \sqrt [3]{b c-a d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {c^{2/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^2}+\frac {(3 b c-a d) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{6 b^{2/3} d^2}-\frac {c^{2/3} \sqrt [3]{b c-a d} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d^2} \]
1/3*x^2*(b*x^3+a)^(1/3)/d+1/6*c^(2/3)*(-a*d+b*c)^(1/3)*ln(d*x^3+c)/d^2+1/6 *(-a*d+3*b*c)*ln(b^(1/3)*x-(b*x^3+a)^(1/3))/b^(2/3)/d^2-1/2*c^(2/3)*(-a*d+ b*c)^(1/3)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/d^2+1/9*(-a*d+3* b*c)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(2/3)/d^2*3^(1/ 2)-1/3*c^(2/3)*(-a*d+b*c)^(1/3)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3) /(b*x^3+a)^(1/3))*3^(1/2))/d^2*3^(1/2)
Result contains complex when optimal does not.
Time = 4.47 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.69 \[ \int \frac {x^4 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\frac {12 d x^2 \sqrt [3]{a+b x^3}+\frac {4 \sqrt {3} (3 b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{b^{2/3}}+6 \sqrt {-6-6 i \sqrt {3}} c^{2/3} \sqrt [3]{b c-a d} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )+\frac {4 (3 b c-a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{b^{2/3}}+6 \left (1-i \sqrt {3}\right ) c^{2/3} \sqrt [3]{b c-a d} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )+\frac {2 (-3 b c+a d) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{b^{2/3}}+3 i \left (i+\sqrt {3}\right ) c^{2/3} \sqrt [3]{b c-a d} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{36 d^2} \]
(12*d*x^2*(a + b*x^3)^(1/3) + (4*Sqrt[3]*(3*b*c - a*d)*ArcTan[(Sqrt[3]*b^( 1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))])/b^(2/3) + 6*Sqrt[-6 - (6*I)*Sq rt[3]]*c^(2/3)*(b*c - a*d)^(1/3)*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*( b*c - a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))] + (4*(3*b *c - a*d)*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/b^(2/3) + 6*(1 - I*Sqrt[3 ])*c^(2/3)*(b*c - a*d)^(1/3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c ^(1/3)*(a + b*x^3)^(1/3)] + (2*(-3*b*c + a*d)*Log[b^(2/3)*x^2 + b^(1/3)*x* (a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/b^(2/3) + (3*I)*(I + Sqrt[3])*c^(2 /3)*(b*c - a*d)^(1/3)*Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[3])*c^(1/ 3)*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(1/3) + I*(I + Sqrt[3])*c^(2/3)*(a + b* x^3)^(2/3)])/(36*d^2)
Time = 0.42 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {978, 1054, 2009}
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.
| \(\displaystyle \int \frac {x^4 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx\) |
| \(\Big \downarrow \) 978 |
| \(\displaystyle \frac {x^2 \sqrt [3]{a+b x^3}}{3 d}-\frac {\int \frac {x \left ((3 b c-a d) x^3+2 a c\right )}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{3 d}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {x^2 \sqrt [3]{a+b x^3}}{3 d}-\frac {\int \left (\frac {(3 b c-a d) x}{d \left (b x^3+a\right )^{2/3}}+\frac {3 \left (a c d-b c^2\right ) x}{d \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}\right )dx}{3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^2 \sqrt [3]{a+b x^3}}{3 d}-\frac {-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) (3 b c-a d)}{\sqrt {3} b^{2/3} d}+\frac {\sqrt {3} c^{2/3} \sqrt [3]{b c-a d} \arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{d}-\frac {(3 b c-a d) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3} d}-\frac {c^{2/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{2 d}+\frac {3 c^{2/3} \sqrt [3]{b c-a d} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d}}{3 d}\) |
(x^2*(a + b*x^3)^(1/3))/(3*d) - (-(((3*b*c - a*d)*ArcTan[(1 + (2*b^(1/3)*x )/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/3)*d)) + (Sqrt[3]*c^(2/3)*(b* c - a*d)^(1/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1 /3)))/Sqrt[3]])/d - (c^(2/3)*(b*c - a*d)^(1/3)*Log[c + d*x^3])/(2*d) - ((3 *b*c - a*d)*Log[b^(1/3)*x - (a + b*x^3)^(1/3)])/(2*b^(2/3)*d) + (3*c^(2/3) *(b*c - a*d)^(1/3)*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)]) /(2*d))/(3*d)
3.7.66.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Simp[e^n/(b*(m + n*(p + q) + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c , d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Time = 4.86 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.42
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} \left (a d -3 b c \right ) \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{6}+\left (-a d \,b^{\frac {2}{3}}+b^{\frac {5}{3}} c \right ) \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\sqrt {3}\, \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} \left (a d -3 b c \right ) \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )}{3}+\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} \left (a d -3 b c \right ) \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{3}-\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{2} \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} d \,b^{\frac {2}{3}}+\left (a d \,b^{\frac {2}{3}}-b^{\frac {5}{3}} c \right ) \left (\arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}+\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )}{3 b^{\frac {2}{3}} \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} d^{2}}\) | \(393\) |
-1/3/b^(2/3)/((a*d-b*c)/c)^(2/3)*(-1/6*((a*d-b*c)/c)^(2/3)*(a*d-3*b*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)+(-a*d*b^(2/3) +b^(5/3)*c)*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)-1/3*3^(1/2)*((a* d-b*c)/c)^(2/3)*(a*d-3*b*c)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/3 ))/b^(1/3)/x)+1/3*((a*d-b*c)/c)^(2/3)*(a*d-3*b*c)*ln((-b^(1/3)*x+(b*x^3+a) ^(1/3))/x)-(b*x^3+a)^(1/3)*x^2*((a*d-b*c)/c)^(2/3)*d*b^(2/3)+(a*d*b^(2/3)- b^(5/3)*c)*(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)+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)))/d^2
Leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (222) = 444\).
Time = 0.34 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.64 \[ \int \frac {x^4 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\frac {6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{2} d x^{2} + 6 \, \sqrt {3} {\left (-b c^{3} + a c^{2} d\right )}^{\frac {1}{3}} b^{2} \arctan \left (-\frac {\sqrt {3} {\left (b c^{2} - a c d\right )} x + 2 \, \sqrt {3} {\left (-b c^{3} + a c^{2} d\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{3 \, {\left (b c^{2} - a c d\right )} x}\right ) + 6 \, {\left (-b c^{3} + a c^{2} d\right )}^{\frac {1}{3}} b^{2} \log \left (\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} c + {\left (-b c^{3} + a c^{2} d\right )}^{\frac {1}{3}} x}{x}\right ) - 3 \, {\left (-b c^{3} + a c^{2} d\right )}^{\frac {1}{3}} b^{2} \log \left (\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} c^{2} - {\left (-b c^{3} + a c^{2} d\right )}^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} c x + {\left (-b c^{3} + a c^{2} d\right )}^{\frac {2}{3}} x^{2}}{x^{2}}\right ) - 2 \, \sqrt {3} {\left (3 \, b^{2} c - a b d\right )} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} b x - 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}}}{3 \, b^{2} x}\right ) + 2 \, \left (-b^{2}\right )^{\frac {2}{3}} {\left (3 \, b c - a d\right )} \log \left (-\frac {\left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) - \left (-b^{2}\right )^{\frac {2}{3}} {\left (3 \, b c - a d\right )} \log \left (-\frac {\left (-b^{2}\right )^{\frac {1}{3}} b x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right )}{18 \, b^{2} d^{2}} \]
1/18*(6*(b*x^3 + a)^(1/3)*b^2*d*x^2 + 6*sqrt(3)*(-b*c^3 + a*c^2*d)^(1/3)*b ^2*arctan(-1/3*(sqrt(3)*(b*c^2 - a*c*d)*x + 2*sqrt(3)*(-b*c^3 + a*c^2*d)^( 2/3)*(b*x^3 + a)^(1/3))/((b*c^2 - a*c*d)*x)) + 6*(-b*c^3 + a*c^2*d)^(1/3)* b^2*log(((b*x^3 + a)^(1/3)*c + (-b*c^3 + a*c^2*d)^(1/3)*x)/x) - 3*(-b*c^3 + a*c^2*d)^(1/3)*b^2*log(((b*x^3 + a)^(2/3)*c^2 - (-b*c^3 + a*c^2*d)^(1/3) *(b*x^3 + a)^(1/3)*c*x + (-b*c^3 + a*c^2*d)^(2/3)*x^2)/x^2) - 2*sqrt(3)*(3 *b^2*c - a*b*d)*sqrt(-(-b^2)^(1/3))*arctan(-1/3*(sqrt(3)*(-b^2)^(1/3)*b*x - 2*sqrt(3)*(b*x^3 + a)^(1/3)*(-b^2)^(2/3))*sqrt(-(-b^2)^(1/3))/(b^2*x)) + 2*(-b^2)^(2/3)*(3*b*c - a*d)*log(-((-b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b)/ x) - (-b^2)^(2/3)*(3*b*c - 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))/(b^2*d^2)
\[ \int \frac {x^4 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\int \frac {x^{4} \sqrt [3]{a + b x^{3}}}{c + d x^{3}}\, dx \]
\[ \int \frac {x^4 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{4}}{d x^{3} + c} \,d x } \]
\[ \int \frac {x^4 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{4}}{d x^{3} + c} \,d x } \]
Timed out. \[ \int \frac {x^4 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\int \frac {x^4\,{\left (b\,x^3+a\right )}^{1/3}}{d\,x^3+c} \,d x \]