\(\int (a-b x)^{2/3} (a+b x)^{4/3} \, dx\) [154]

Optimal result

Integrand size = 20, antiderivative size = 180 \[ \int (a-b x)^{2/3} (a+b x)^{4/3} \, dx=\frac {8 a^2 (a-b x)^{2/3} \sqrt [3]{a+b x}}{27 b}-\frac {4 a (a-b x)^{5/3} \sqrt [3]{a+b x}}{9 b}-\frac {(a-b x)^{5/3} (a+b x)^{4/3}}{3 b}+\frac {32 a^3 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a-b x}}{\sqrt {3} \sqrt [3]{a+b x}}\right )}{27 \sqrt {3} b}+\frac {16 a^3 \log (a+b x)}{81 b}+\frac {16 a^3 \log \left (1+\frac {\sqrt [3]{a-b x}}{\sqrt [3]{a+b x}}\right )}{27 b} \] Output:

8/27*a^2*(-b*x+a)^(2/3)*(b*x+a)^(1/3)/b-4/9*a*(-b*x+a)^(5/3)*(b*x+a)^(1/3) 
/b-1/3*(-b*x+a)^(5/3)*(b*x+a)^(4/3)/b-32/81*a^3*arctan(-1/3*3^(1/2)+2/3*(- 
b*x+a)^(1/3)*3^(1/2)/(b*x+a)^(1/3))*3^(1/2)/b+16/81*a^3*ln(b*x+a)/b+16/27* 
a^3*ln(1+(-b*x+a)^(1/3)/(b*x+a)^(1/3))/b
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.95 \[ \int (a-b x)^{2/3} (a+b x)^{4/3} \, dx=\frac {3 (a-b x)^{2/3} \sqrt [3]{a+b x} \left (-13 a^2+12 a b x+9 b^2 x^2\right )-32 \sqrt {3} a^3 \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{-2 \sqrt [3]{a-b x}+\sqrt [3]{a+b x}}\right )+32 a^3 \log \left (\sqrt [3]{a-b x}+\sqrt [3]{a+b x}\right )-16 a^3 \log \left ((a-b x)^{2/3}-\sqrt [3]{a-b x} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{81 b} \] Input:

Integrate[(a - b*x)^(2/3)*(a + b*x)^(4/3),x]
 

Output:

(3*(a - b*x)^(2/3)*(a + b*x)^(1/3)*(-13*a^2 + 12*a*b*x + 9*b^2*x^2) - 32*S 
qrt[3]*a^3*ArcTan[(Sqrt[3]*(a + b*x)^(1/3))/(-2*(a - b*x)^(1/3) + (a + b*x 
)^(1/3))] + 32*a^3*Log[(a - b*x)^(1/3) + (a + b*x)^(1/3)] - 16*a^3*Log[(a 
- b*x)^(2/3) - (a - b*x)^(1/3)*(a + b*x)^(1/3) + (a + b*x)^(2/3)])/(81*b)
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {60, 60, 60, 72}

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.

\(\displaystyle \int (a-b x)^{2/3} (a+b x)^{4/3} \, dx\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {8}{9} a \int (a-b x)^{2/3} \sqrt [3]{a+b x}dx-\frac {(a-b x)^{5/3} (a+b x)^{4/3}}{3 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {8}{9} a \left (\frac {1}{3} a \int \frac {(a-b x)^{2/3}}{(a+b x)^{2/3}}dx-\frac {(a-b x)^{5/3} \sqrt [3]{a+b x}}{2 b}\right )-\frac {(a-b x)^{5/3} (a+b x)^{4/3}}{3 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {8}{9} a \left (\frac {1}{3} a \left (\frac {4}{3} a \int \frac {1}{\sqrt [3]{a-b x} (a+b x)^{2/3}}dx+\frac {(a-b x)^{2/3} \sqrt [3]{a+b x}}{b}\right )-\frac {(a-b x)^{5/3} \sqrt [3]{a+b x}}{2 b}\right )-\frac {(a-b x)^{5/3} (a+b x)^{4/3}}{3 b}\)

\(\Big \downarrow \) 72

\(\displaystyle \frac {8}{9} a \left (\frac {1}{3} a \left (\frac {4}{3} a \left (\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a-b x}}{\sqrt {3} \sqrt [3]{a+b x}}\right )}{b}+\frac {\log (a+b x)}{2 b}+\frac {3 \log \left (\frac {\sqrt [3]{a-b x}}{\sqrt [3]{a+b x}}+1\right )}{2 b}\right )+\frac {(a-b x)^{2/3} \sqrt [3]{a+b x}}{b}\right )-\frac {(a-b x)^{5/3} \sqrt [3]{a+b x}}{2 b}\right )-\frac {(a-b x)^{5/3} (a+b x)^{4/3}}{3 b}\)

Input:

Int[(a - b*x)^(2/3)*(a + b*x)^(4/3),x]
 

Output:

-1/3*((a - b*x)^(5/3)*(a + b*x)^(4/3))/b + (8*a*(-1/2*((a - b*x)^(5/3)*(a 
+ b*x)^(1/3))/b + (a*(((a - b*x)^(2/3)*(a + b*x)^(1/3))/b + (4*a*((Sqrt[3] 
*ArcTan[1/Sqrt[3] - (2*(a - b*x)^(1/3))/(Sqrt[3]*(a + b*x)^(1/3))])/b + Lo 
g[a + b*x]/(2*b) + (3*Log[1 + (a - b*x)^(1/3)/(a + b*x)^(1/3)])/(2*b)))/3) 
)/3))/9
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 72

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> 
 With[{q = Rt[-d/b, 3]}, Simp[Sqrt[3]*(q/d)*ArcTan[1/Sqrt[3] - 2*q*((a + b* 
x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))], x] + (Simp[3*(q/(2*d))*Log[q*((a + b* 
x)^(1/3)/(c + d*x)^(1/3)) + 1], x] + Simp[(q/(2*d))*Log[c + d*x], x])] /; F 
reeQ[{a, b, c, d}, x] && NegQ[d/b]
 

Maple [F]

\[\int \left (-b x +a \right )^{\frac {2}{3}} \left (b x +a \right )^{\frac {4}{3}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.06 \[ \int (a-b x)^{2/3} (a+b x)^{4/3} \, dx=-\frac {32 \, \sqrt {3} a^{3} \arctan \left (\frac {\sqrt {3} {\left (b x - a\right )} + 2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}}{3 \, {\left (b x - a\right )}}\right ) + 16 \, a^{3} \log \left (\frac {b x - {\left (b x + a\right )}^{\frac {2}{3}} {\left (-b x + a\right )}^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} - a}{b x - a}\right ) - 32 \, a^{3} \log \left (-\frac {b x - {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} - a}{b x - a}\right ) - 3 \, {\left (9 \, b^{2} x^{2} + 12 \, a b x - 13 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}}{81 \, b} \] Input:

integrate((-b*x+a)^(2/3)*(b*x+a)^(4/3),x, algorithm="fricas")
 

Output:

-1/81*(32*sqrt(3)*a^3*arctan(1/3*(sqrt(3)*(b*x - a) + 2*sqrt(3)*(b*x + a)^ 
(1/3)*(-b*x + a)^(2/3))/(b*x - a)) + 16*a^3*log((b*x - (b*x + a)^(2/3)*(-b 
*x + a)^(1/3) + (b*x + a)^(1/3)*(-b*x + a)^(2/3) - a)/(b*x - a)) - 32*a^3* 
log(-(b*x - (b*x + a)^(1/3)*(-b*x + a)^(2/3) - a)/(b*x - a)) - 3*(9*b^2*x^ 
2 + 12*a*b*x - 13*a^2)*(b*x + a)^(1/3)*(-b*x + a)^(2/3))/b
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((-b*x+a)^(2/3)*(b*x+a)^(4/3),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(4/3)*(-b*x + a)^(2/3), x)
 

Giac [F]

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

integrate((-b*x+a)^(2/3)*(b*x+a)^(4/3),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(4/3)*(-b*x + a)^(2/3), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (a-b x)^{2/3} (a+b x)^{4/3} \, dx=\frac {-45 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +a \right )^{\frac {2}{3}} a^{2}+12 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +a \right )^{\frac {2}{3}} a b x +9 \left (b x +a \right )^{\frac {1}{3}} \left (-b x +a \right )^{\frac {2}{3}} b^{2} x^{2}-32 \left (\int \frac {\left (b x +a \right )^{\frac {1}{3}} \left (-b x +a \right )^{\frac {2}{3}} x}{-b^{2} x^{2}+a^{2}}d x \right ) a^{2} b^{2}}{27 b} \] Input:

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

Output:

( - 45*(a + b*x)**(1/3)*(a - b*x)**(2/3)*a**2 + 12*(a + b*x)**(1/3)*(a - b 
*x)**(2/3)*a*b*x + 9*(a + b*x)**(1/3)*(a - b*x)**(2/3)*b**2*x**2 - 32*int( 
((a + b*x)**(1/3)*(a - b*x)**(2/3)*x)/(a**2 - b**2*x**2),x)*a**2*b**2)/(27 
*b)