Integrand size = 20, antiderivative size = 112 \[ \int \left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \, dx=\frac {7}{18} a x \left (a+b x^3\right )^{2/3}-\frac {1}{6} x \left (a+b x^3\right )^{5/3}+\frac {7 a^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{b}}-\frac {7 a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 \sqrt [3]{b}} \] Output:
7/18*a*x*(b*x^3+a)^(2/3)-1/6*x*(b*x^3+a)^(5/3)+7/27*a^2*arctan(1/3*(1+2*b^ (1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/b^(1/3)-7/18*a^2*ln(-b^(1/3)*x+( b*x^3+a)^(1/3))/b^(1/3)
Time = 0.40 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.39 \[ \int \left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \, dx=\frac {3 \sqrt [3]{b} \left (a+b x^3\right )^{2/3} \left (4 a x-3 b x^4\right )+14 \sqrt {3} a^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-14 a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+7 a^2 \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{54 \sqrt [3]{b}} \] Input:
Integrate[(a - b*x^3)*(a + b*x^3)^(2/3),x]
Output:
(3*b^(1/3)*(a + b*x^3)^(2/3)*(4*a*x - 3*b*x^4) + 14*Sqrt[3]*a^2*ArcTan[(Sq rt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))] - 14*a^2*Log[-(b^(1/3) *x) + (a + b*x^3)^(1/3)] + 7*a^2*Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^( 1/3) + (a + b*x^3)^(2/3)])/(54*b^(1/3))
Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {913, 748, 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.
| \(\displaystyle \int \left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \, dx\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {7}{6} a \int \left (b x^3+a\right )^{2/3}dx-\frac {1}{6} x \left (a+b x^3\right )^{5/3}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {7}{6} a \left (\frac {2}{3} a \int \frac {1}{\sqrt [3]{b x^3+a}}dx+\frac {1}{3} x \left (a+b x^3\right )^{2/3}\right )-\frac {1}{6} x \left (a+b x^3\right )^{5/3}\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle \frac {7}{6} a \left (\frac {2}{3} a \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )+\frac {1}{3} x \left (a+b x^3\right )^{2/3}\right )-\frac {1}{6} x \left (a+b x^3\right )^{5/3}\) |
Input:
Int[(a - b*x^3)*(a + b*x^3)^(2/3),x]
Output:
-1/6*(x*(a + b*x^3)^(5/3)) + (7*a*((x*(a + b*x^3)^(2/3))/3 + (2*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))/6
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Simp[a*n*(p/(n*p + 1)) Int[(a + b*x^n)^(p - 1), x], x] /; Fre eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat or[p + 1/n], Denominator[p]])
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 = 1.20 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.29
| method | result | size |
| pseudoelliptic | \(\frac {-9 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{\frac {4}{3}} x^{4}+12 a x \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{\frac {1}{3}}-14 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) a^{2}-14 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a^{2}+7 \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a^{2}}{54 b^{\frac {1}{3}}}\) | \(144\) |
Input:
int((-b*x^3+a)*(b*x^3+a)^(2/3),x,method=_RETURNVERBOSE)
Output:
1/54*(-9*(b*x^3+a)^(2/3)*b^(4/3)*x^4+12*a*x*(b*x^3+a)^(2/3)*b^(1/3)-14*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^2-14*ln ((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)*a^2+7*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^2)/b^(1/3)
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (85) = 170\).
Time = 0.11 (sec) , antiderivative size = 399, normalized size of antiderivative = 3.56 \[ \int \left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \, dx=\left [\frac {21 \, \sqrt {\frac {1}{3}} a^{2} b \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} b x^{3} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 2 \, a\right ) - 14 \, a^{2} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 7 \, a^{2} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left (3 \, b^{2} x^{4} - 4 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b}, -\frac {42 \, \sqrt {\frac {1}{3}} a^{2} b \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{x}\right ) + 14 \, a^{2} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 7 \, a^{2} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (3 \, b^{2} x^{4} - 4 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b}\right ] \] Input:
integrate((-b*x^3+a)*(b*x^3+a)^(2/3),x, algorithm="fricas")
Output:
[1/54*(21*sqrt(1/3)*a^2*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) - 14*a^2 *(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + 7*a^2*(-b)^(2/3)*l og(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)*x + (b*x^3 + a)^(2/3))/x ^2) - 3*(3*b^2*x^4 - 4*a*b*x)*(b*x^3 + a)^(2/3))/b, -1/54*(42*sqrt(1/3)*a^ 2*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) + 14*a^2*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - 7*a^2*(-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) + 3*(3*b^2*x^4 - 4*a*b*x)*(b*x^3 + a)^(2/3))/b]
Result contains complex when optimal does not.
Time = 2.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.71 \[ \int \left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \, dx=\frac {a^{\frac {5}{3}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {a^{\frac {2}{3}} b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} \] Input:
integrate((-b*x**3+a)*(b*x**3+a)**(2/3),x)
Output:
a**(5/3)*x*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a) /(3*gamma(4/3)) - a**(2/3)*b*x**4*gamma(4/3)*hyper((-2/3, 4/3), (7/3,), b* x**3*exp_polar(I*pi)/a)/(3*gamma(7/3))
Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (85) = 170\).
Time = 0.12 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.88 \[ \int \left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \, dx=-\frac {1}{9} \, {\left (\frac {2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {1}{3}}} + \frac {2 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {1}{3}}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{{\left (b - \frac {b x^{3} + a}{x^{3}}\right )} x^{2}}\right )} a - \frac {1}{54} \, {\left (\frac {2 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} - \frac {a^{2} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {2 \, a^{2} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}} + \frac {3 \, {\left (\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{2} b}{x^{2}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{b^{3} - \frac {2 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2} b}{x^{6}}}\right )} b \] Input:
integrate((-b*x^3+a)*(b*x^3+a)^(2/3),x, algorithm="maxima")
Output:
-1/9*(2*sqrt(3)*a*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^( 1/3))/b^(1/3) - a*log(b^(2/3) + (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a)^ (2/3)/x^2)/b^(1/3) + 2*a*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(1/3) + 3*( b*x^3 + a)^(2/3)*a/((b - (b*x^3 + a)/x^3)*x^2))*a - 1/54*(2*sqrt(3)*a^2*ar ctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3))/b^(4/3) - a^2* 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^2*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(4/3) + 3*((b*x^3 + a)^(2/3 )*a^2*b/x^2 + 2*(b*x^3 + a)^(5/3)*a^2/x^5)/(b^3 - 2*(b*x^3 + a)*b^2/x^3 + (b*x^3 + a)^2*b/x^6))*b
\[ \int \left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \, dx=\int { -{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b x^{3} - a\right )} \,d x } \] Input:
integrate((-b*x^3+a)*(b*x^3+a)^(2/3),x, algorithm="giac")
Output:
integrate(-(b*x^3 + a)^(2/3)*(b*x^3 - a), x)
Timed out. \[ \int \left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \, dx=\int {\left (b\,x^3+a\right )}^{2/3}\,\left (a-b\,x^3\right ) \,d x \] Input:
int((a + b*x^3)^(2/3)*(a - b*x^3),x)
Output:
int((a + b*x^3)^(2/3)*(a - b*x^3), x)
\[ \int \left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \, dx=\frac {2 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a x}{9}-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} b \,x^{4}}{6}+\frac {7 \left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}d x \right ) a^{2}}{9} \] Input:
int((-b*x^3+a)*(b*x^3+a)^(2/3),x)
Output:
(4*(a + b*x**3)**(2/3)*a*x - 3*(a + b*x**3)**(2/3)*b*x**4 + 14*int((a + b* x**3)**(2/3)/(a + b*x**3),x)*a**2)/18