\(\int (a-b x)^{2/3} (a+b x)^{2/3} \, dx\) [163]

Optimal result

Integrand size = 20, antiderivative size = 657 \[ \int (a-b x)^{2/3} (a+b x)^{2/3} \, dx=\frac {3}{7} x (a-b x)^{2/3} (a+b x)^{2/3}-\frac {12 a^2 x \sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{7 \sqrt [3]{a-b x} \sqrt [3]{a+b x} \left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )}-\frac {6 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^4 \sqrt [3]{1-\frac {b^2 x^2}{a^2}} \left (1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right ) \sqrt {\frac {1+\sqrt [3]{1-\frac {b^2 x^2}{a^2}}+\left (1-\frac {b^2 x^2}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}\right )|-7+4 \sqrt {3}\right )}{7 b^2 x \sqrt [3]{a-b x} \sqrt [3]{a+b x} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}}}+\frac {4 \sqrt {2} 3^{3/4} a^4 \sqrt [3]{1-\frac {b^2 x^2}{a^2}} \left (1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right ) \sqrt {\frac {1+\sqrt [3]{1-\frac {b^2 x^2}{a^2}}+\left (1-\frac {b^2 x^2}{a^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}\right ),-7+4 \sqrt {3}\right )}{7 b^2 x \sqrt [3]{a-b x} \sqrt [3]{a+b x} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}}{\left (1-\sqrt {3}-\sqrt [3]{1-\frac {b^2 x^2}{a^2}}\right )^2}}} \] Output:

3/7*x*(-b*x+a)^(2/3)*(b*x+a)^(2/3)-12/7*a^2*x*(1-b^2*x^2/a^2)^(1/3)/(-b*x+ 
a)^(1/3)/(b*x+a)^(1/3)/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3))-6/7*3^(1/4)*(1/2* 
6^(1/2)+1/2*2^(1/2))*a^4*(1-b^2*x^2/a^2)^(1/3)*(1-(1-b^2*x^2/a^2)^(1/3))*( 
(1+(1-b^2*x^2/a^2)^(1/3)+(1-b^2*x^2/a^2)^(2/3))/(1-3^(1/2)-(1-b^2*x^2/a^2) 
^(1/3))^2)^(1/2)*EllipticE((1+3^(1/2)-(1-b^2*x^2/a^2)^(1/3))/(1-3^(1/2)-(1 
-b^2*x^2/a^2)^(1/3)),2*I-I*3^(1/2))/b^2/x/(-b*x+a)^(1/3)/(b*x+a)^(1/3)/(-( 
1-(1-b^2*x^2/a^2)^(1/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3))^2)^(1/2)+4/7*2^ 
(1/2)*3^(3/4)*a^4*(1-b^2*x^2/a^2)^(1/3)*(1-(1-b^2*x^2/a^2)^(1/3))*((1+(1-b 
^2*x^2/a^2)^(1/3)+(1-b^2*x^2/a^2)^(2/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3)) 
^2)^(1/2)*EllipticF((1+3^(1/2)-(1-b^2*x^2/a^2)^(1/3))/(1-3^(1/2)-(1-b^2*x^ 
2/a^2)^(1/3)),2*I-I*3^(1/2))/b^2/x/(-b*x+a)^(1/3)/(b*x+a)^(1/3)/(-(1-(1-b^ 
2*x^2/a^2)^(1/3))/(1-3^(1/2)-(1-b^2*x^2/a^2)^(1/3))^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.10 \[ \int (a-b x)^{2/3} (a+b x)^{2/3} \, dx=-\frac {3\ 2^{2/3} (a-b x)^{5/3} (a+b x)^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {5}{3},\frac {8}{3},\frac {a-b x}{2 a}\right )}{5 b \left (\frac {a+b x}{a}\right )^{2/3}} \] Input:

Integrate[(a - b*x)^(2/3)*(a + b*x)^(2/3),x]
 

Output:

(-3*2^(2/3)*(a - b*x)^(5/3)*(a + b*x)^(2/3)*Hypergeometric2F1[-2/3, 5/3, 8 
/3, (a - b*x)/(2*a)])/(5*b*((a + b*x)/a)^(2/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.51 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {46, 211, 233, 833, 760, 2418}

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.

\(\displaystyle \int (a-b x)^{2/3} (a+b x)^{2/3} \, dx\)

\(\Big \downarrow \) 46

\(\displaystyle \frac {(a-b x)^{2/3} (a+b x)^{2/3} \int \left (a^2-b^2 x^2\right )^{2/3}dx}{\left (a^2-b^2 x^2\right )^{2/3}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(a-b x)^{2/3} (a+b x)^{2/3} \left (\frac {4}{7} a^2 \int \frac {1}{\sqrt [3]{a^2-b^2 x^2}}dx+\frac {3}{7} x \left (a^2-b^2 x^2\right )^{2/3}\right )}{\left (a^2-b^2 x^2\right )^{2/3}}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {(a-b x)^{2/3} (a+b x)^{2/3} \left (\frac {3}{7} x \left (a^2-b^2 x^2\right )^{2/3}-\frac {6 a^2 \sqrt {-b^2 x^2} \int \frac {\sqrt [3]{a^2-b^2 x^2}}{\sqrt {-b^2 x^2}}d\sqrt [3]{a^2-b^2 x^2}}{7 b^2 x}\right )}{\left (a^2-b^2 x^2\right )^{2/3}}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {(a-b x)^{2/3} (a+b x)^{2/3} \left (\frac {3}{7} x \left (a^2-b^2 x^2\right )^{2/3}-\frac {6 a^2 \sqrt {-b^2 x^2} \left (\left (1+\sqrt {3}\right ) a^{2/3} \int \frac {1}{\sqrt {-b^2 x^2}}d\sqrt [3]{a^2-b^2 x^2}-\int \frac {\left (1+\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}{\sqrt {-b^2 x^2}}d\sqrt [3]{a^2-b^2 x^2}\right )}{7 b^2 x}\right )}{\left (a^2-b^2 x^2\right )^{2/3}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {(a-b x)^{2/3} (a+b x)^{2/3} \left (\frac {3}{7} x \left (a^2-b^2 x^2\right )^{2/3}-\frac {6 a^2 \sqrt {-b^2 x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}{\sqrt {-b^2 x^2}}d\sqrt [3]{a^2-b^2 x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) a^{2/3} \left (a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right ) \sqrt {\frac {a^{4/3}+\left (a^2-b^2 x^2\right )^{2/3}+a^{2/3} \sqrt [3]{a^2-b^2 x^2}}{\left (\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}{\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b^2 x^2} \sqrt {-\frac {a^{2/3} \left (a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )^2}}}\right )}{7 b^2 x}\right )}{\left (a^2-b^2 x^2\right )^{2/3}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {(a-b x)^{2/3} (a+b x)^{2/3} \left (\frac {3}{7} x \left (a^2-b^2 x^2\right )^{2/3}-\frac {6 a^2 \sqrt {-b^2 x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) a^{2/3} \left (a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right ) \sqrt {\frac {a^{4/3}+\left (a^2-b^2 x^2\right )^{2/3}+a^{2/3} \sqrt [3]{a^2-b^2 x^2}}{\left (\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}{\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b^2 x^2} \sqrt {-\frac {a^{2/3} \left (a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{2/3} \left (a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right ) \sqrt {\frac {a^{4/3}+\left (a^2-b^2 x^2\right )^{2/3}+a^{2/3} \sqrt [3]{a^2-b^2 x^2}}{\left (\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}{\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b^2 x^2} \sqrt {-\frac {a^{2/3} \left (a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}\right )^2}}}-\frac {2 \sqrt {-b^2 x^2}}{\left (1-\sqrt {3}\right ) a^{2/3}-\sqrt [3]{a^2-b^2 x^2}}\right )}{7 b^2 x}\right )}{\left (a^2-b^2 x^2\right )^{2/3}}\)

Input:

Int[(a - b*x)^(2/3)*(a + b*x)^(2/3),x]
 

Output:

((a - b*x)^(2/3)*(a + b*x)^(2/3)*((3*x*(a^2 - b^2*x^2)^(2/3))/7 - (6*a^2*S 
qrt[-(b^2*x^2)]*((-2*Sqrt[-(b^2*x^2)])/((1 - Sqrt[3])*a^(2/3) - (a^2 - b^2 
*x^2)^(1/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(2/3)*(a^(2/3) - (a^2 - b^2*x^ 
2)^(1/3))*Sqrt[(a^(4/3) + a^(2/3)*(a^2 - b^2*x^2)^(1/3) + (a^2 - b^2*x^2)^ 
(2/3))/((1 - Sqrt[3])*a^(2/3) - (a^2 - b^2*x^2)^(1/3))^2]*EllipticE[ArcSin 
[((1 + Sqrt[3])*a^(2/3) - (a^2 - b^2*x^2)^(1/3))/((1 - Sqrt[3])*a^(2/3) - 
(a^2 - b^2*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(Sqrt[-(b^2*x^2)]*Sqrt[-((a^(2/3 
)*(a^(2/3) - (a^2 - b^2*x^2)^(1/3)))/((1 - Sqrt[3])*a^(2/3) - (a^2 - b^2*x 
^2)^(1/3))^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*a^(2/3)*(a^(2/3) - (a 
^2 - b^2*x^2)^(1/3))*Sqrt[(a^(4/3) + a^(2/3)*(a^2 - b^2*x^2)^(1/3) + (a^2 
- b^2*x^2)^(2/3))/((1 - Sqrt[3])*a^(2/3) - (a^2 - b^2*x^2)^(1/3))^2]*Ellip 
ticF[ArcSin[((1 + Sqrt[3])*a^(2/3) - (a^2 - b^2*x^2)^(1/3))/((1 - Sqrt[3]) 
*a^(2/3) - (a^2 - b^2*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-(b^2*x 
^2)]*Sqrt[-((a^(2/3)*(a^(2/3) - (a^2 - b^2*x^2)^(1/3)))/((1 - Sqrt[3])*a^( 
2/3) - (a^2 - b^2*x^2)^(1/3))^2)])))/(7*b^2*x)))/(a^2 - b^2*x^2)^(2/3)
 

Defintions of rubi rules used

rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(a 
 + b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m])   I 
nt[(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 
 0] &&  !IntegerQ[2*m]
 

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \left (-b x +a \right )^{\frac {2}{3}} \left (b x +a \right )^{\frac {2}{3}}d x\]

Input:

int((-b*x+a)^(2/3)*(b*x+a)^(2/3),x)
 

Output:

int((-b*x+a)^(2/3)*(b*x+a)^(2/3),x)
 

Fricas [F]

\[ \int (a-b x)^{2/3} (a+b x)^{2/3} \, dx=\int { {\left (b x + a\right )}^{\frac {2}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} \,d x } \] Input:

integrate((-b*x+a)^(2/3)*(b*x+a)^(2/3),x, algorithm="fricas")
 

Output:

integral((b*x + a)^(2/3)*(-b*x + a)^(2/3), x)
 

Sympy [F]

\[ \int (a-b x)^{2/3} (a+b x)^{2/3} \, dx=\int \left (a - b x\right )^{\frac {2}{3}} \left (a + b x\right )^{\frac {2}{3}}\, dx \] Input:

integrate((-b*x+a)**(2/3)*(b*x+a)**(2/3),x)
 

Output:

Integral((a - b*x)**(2/3)*(a + b*x)**(2/3), x)
 

Maxima [F]

\[ \int (a-b x)^{2/3} (a+b x)^{2/3} \, dx=\int { {\left (b x + a\right )}^{\frac {2}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} \,d x } \] Input:

integrate((-b*x+a)^(2/3)*(b*x+a)^(2/3),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(2/3)*(-b*x + a)^(2/3), x)
 

Giac [F]

\[ \int (a-b x)^{2/3} (a+b x)^{2/3} \, dx=\int { {\left (b x + a\right )}^{\frac {2}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} \,d x } \] Input:

integrate((-b*x+a)^(2/3)*(b*x+a)^(2/3),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(2/3)*(-b*x + a)^(2/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int (a-b x)^{2/3} (a+b x)^{2/3} \, dx=\int {\left (a+b\,x\right )}^{2/3}\,{\left (a-b\,x\right )}^{2/3} \,d x \] Input:

int((a + b*x)^(2/3)*(a - b*x)^(2/3),x)
 

Output:

int((a + b*x)^(2/3)*(a - b*x)^(2/3), x)
 

Reduce [F]

\[ \int (a-b x)^{2/3} (a+b x)^{2/3} \, dx=\frac {3 \left (b x +a \right )^{\frac {2}{3}} \left (-b x +a \right )^{\frac {2}{3}} x}{7}+\frac {4 \left (\int \frac {\left (b x +a \right )^{\frac {2}{3}} \left (-b x +a \right )^{\frac {2}{3}}}{-b^{2} x^{2}+a^{2}}d x \right ) a^{2}}{7} \] Input:

int((-b*x+a)^(2/3)*(b*x+a)^(2/3),x)
                                                                                    
                                                                                    
 

Output:

(3*(a + b*x)**(2/3)*(a - b*x)**(2/3)*x + 4*int(((a + b*x)**(2/3)*(a - b*x) 
**(2/3))/(a**2 - b**2*x**2),x)*a**2)/7