Integrand size = 20, antiderivative size = 91 \[ \int \frac {a-b x^3}{\sqrt [3]{a+b x^3}} \, dx=-\frac {1}{3} x \left (a+b x^3\right )^{2/3}+\frac {4 a \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b}}-\frac {2 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{b}} \] Output:
-1/3*x*(b*x^3+a)^(2/3)+4/9*a*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^ (1/2))*3^(1/2)/b^(1/3)-2/3*a*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(1/3)
Time = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.54 \[ \int \frac {a-b x^3}{\sqrt [3]{a+b x^3}} \, dx=\frac {-3 \sqrt [3]{b} x \left (a+b x^3\right )^{2/3}+4 \sqrt {3} a \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-4 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+2 a \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{9 \sqrt [3]{b}} \] Input:
Integrate[(a - b*x^3)/(a + b*x^3)^(1/3),x]
Output:
(-3*b^(1/3)*x*(a + b*x^3)^(2/3) + 4*Sqrt[3]*a*ArcTan[(Sqrt[3]*b^(1/3)*x)/( b^(1/3)*x + 2*(a + b*x^3)^(1/3))] - 4*a*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/ 3)] + 2*a*Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3 )])/(9*b^(1/3))
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {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.
| \(\displaystyle \int \frac {a-b x^3}{\sqrt [3]{a+b x^3}} \, dx\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {4}{3} a \int \frac {1}{\sqrt [3]{b x^3+a}}dx-\frac {1}{3} x \left (a+b x^3\right )^{2/3}\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle \frac {4}{3} a \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )-\frac {1}{3} x \left (a+b x^3\right )^{2/3}\) |
Input:
Int[(a - b*x^3)/(a + b*x^3)^(1/3),x]
Output:
-1/3*(x*(a + b*x^3)^(2/3)) + (4*a*(ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^( 1/3))/Sqrt[3]]/(Sqrt[3]*b^(1/3)) - Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)]/( 2*b^(1/3))))/3
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]
Time = 1.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.30
| method | result | size |
| pseudoelliptic | \(-\frac {4 \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) a +\frac {3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} x \,b^{\frac {1}{3}}}{4}+\ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a -\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a}{2}\right )}{9 b^{\frac {1}{3}}}\) | \(118\) |
Input:
int((-b*x^3+a)/(b*x^3+a)^(1/3),x,method=_RETURNVERBOSE)
Output:
-4/9*(3^(1/2)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/3))/b^(1/3)/x)* a+3/4*(b*x^3+a)^(2/3)*x*b^(1/3)+ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)*a-1/2*l n((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)*a)/b^(1/3)
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (68) = 136\).
Time = 0.12 (sec) , antiderivative size = 363, normalized size of antiderivative = 3.99 \[ \int \frac {a-b x^3}{\sqrt [3]{a+b x^3}} \, dx=\left [\frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} b x^{3} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 2 \, a\right ) - 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b x - 4 \, a \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 2 \, a \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{9 \, b}, -\frac {12 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{x}\right ) + 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b x + 4 \, a \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 2 \, a \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{9 \, b}\right ] \] Input:
integrate((-b*x^3+a)/(b*x^3+a)^(1/3),x, algorithm="fricas")
Output:
[1/9*(6*sqrt(1/3)*a*b*sqrt((-b)^(1/3)/b)*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3) *(-b)^(2/3)*x^2 - 3*sqrt(1/3)*((-b)^(1/3)*b*x^3 - (b*x^3 + a)^(1/3)*b*x^2 + 2*(b*x^3 + a)^(2/3)*(-b)^(2/3)*x)*sqrt((-b)^(1/3)/b) + 2*a) - 3*(b*x^3 + a)^(2/3)*b*x - 4*a*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + 2*a*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)*x + (b* x^3 + a)^(2/3))/x^2))/b, -1/9*(12*sqrt(1/3)*a*b*sqrt(-(-b)^(1/3)/b)*arctan (-sqrt(1/3)*((-b)^(1/3)*x - 2*(b*x^3 + a)^(1/3))*sqrt(-(-b)^(1/3)/b)/x) + 3*(b*x^3 + a)^(2/3)*b*x + 4*a*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^( 1/3))/x) - 2*a*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/ 3)*x + (b*x^3 + a)^(2/3))/x^2))/b]
Result contains complex when optimal does not.
Time = 1.47 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int \frac {a-b x^3}{\sqrt [3]{a+b x^3}} \, dx=\frac {a^{\frac {2}{3}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} \] Input:
integrate((-b*x**3+a)/(b*x**3+a)**(1/3),x)
Output:
a**(2/3)*x*gamma(1/3)*hyper((1/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/ (3*gamma(4/3)) - b*x**4*gamma(4/3)*hyper((1/3, 4/3), (7/3,), b*x**3*exp_po lar(I*pi)/a)/(3*a**(1/3)*gamma(7/3))
Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (68) = 136\).
Time = 0.11 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.68 \[ \int \frac {a-b x^3}{\sqrt [3]{a+b x^3}} \, dx=-\frac {1}{6} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {\log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {1}{3}}} + \frac {2 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {1}{3}}}\right )} a - \frac {1}{18} \, {\left (\frac {2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} - \frac {a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {2 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}} - \frac {6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{{\left (b^{2} - \frac {{\left (b x^{3} + a\right )} b}{x^{3}}\right )} x^{2}}\right )} b \] Input:
integrate((-b*x^3+a)/(b*x^3+a)^(1/3),x, algorithm="maxima")
Output:
-1/6*(2*sqrt(3)*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/ 3))/b^(1/3) - log(b^(2/3) + (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a)^(2/3 )/x^2)/b^(1/3) + 2*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(1/3))*a - 1/18*( 2*sqrt(3)*a*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3))/ b^(4/3) - a*log(b^(2/3) + (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a)^(2/3)/ x^2)/b^(4/3) + 2*a*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(4/3) - 6*(b*x^3 + a)^(2/3)*a/((b^2 - (b*x^3 + a)*b/x^3)*x^2))*b
\[ \int \frac {a-b x^3}{\sqrt [3]{a+b x^3}} \, dx=\int { -\frac {b x^{3} - a}{{\left (b x^{3} + a\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((-b*x^3+a)/(b*x^3+a)^(1/3),x, algorithm="giac")
Output:
integrate(-(b*x^3 - a)/(b*x^3 + a)^(1/3), x)
Timed out. \[ \int \frac {a-b x^3}{\sqrt [3]{a+b x^3}} \, dx=\int \frac {a-b\,x^3}{{\left (b\,x^3+a\right )}^{1/3}} \,d x \] Input:
int((a - b*x^3)/(a + b*x^3)^(1/3),x)
Output:
int((a - b*x^3)/(a + b*x^3)^(1/3), x)
\[ \int \frac {a-b x^3}{\sqrt [3]{a+b x^3}} \, dx=-\left (\int \frac {x^{3}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}d x \right ) b +\left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}d x \right ) a \] Input:
int((-b*x^3+a)/(b*x^3+a)^(1/3),x)
Output:
- int(x**3/(a + b*x**3)**(1/3),x)*b + int(1/(a + b*x**3)**(1/3),x)*a