\(\int \frac {(a-b x)^{2/3}}{(a+b x)^{2/3}} \, dx\) [156]

Optimal result

Integrand size = 20, antiderivative size = 111 \[ \int \frac {(a-b x)^{2/3}}{(a+b x)^{2/3}} \, dx=\frac {(a-b x)^{2/3} \sqrt [3]{a+b x}}{b}+\frac {4 a \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a-b x}}{\sqrt {3} \sqrt [3]{a+b x}}\right )}{\sqrt {3} b}+\frac {2 a \log (a+b x)}{3 b}+\frac {2 a \log \left (1+\frac {\sqrt [3]{a-b x}}{\sqrt [3]{a+b x}}\right )}{b} \] Output:

(-b*x+a)^(2/3)*(b*x+a)^(1/3)/b-4/3*a*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+2/3*a*ln(b*x+a)/b+2*a*ln(1+(-b*x+a)^(1/ 
3)/(b*x+a)^(1/3))/b
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.33 \[ \int \frac {(a-b x)^{2/3}}{(a+b x)^{2/3}} \, dx=\frac {3 (a-b x)^{2/3} \sqrt [3]{a+b x}-4 \sqrt {3} a \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{-2 \sqrt [3]{a-b x}+\sqrt [3]{a+b x}}\right )+4 a \log \left (b \left (\sqrt [3]{a-b x}+\sqrt [3]{a+b x}\right )\right )-2 a \log \left ((a-b x)^{2/3}-\sqrt [3]{a-b x} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{3 b} \] Input:

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

Output:

(3*(a - b*x)^(2/3)*(a + b*x)^(1/3) - 4*Sqrt[3]*a*ArcTan[(Sqrt[3]*(a + b*x) 
^(1/3))/(-2*(a - b*x)^(1/3) + (a + b*x)^(1/3))] + 4*a*Log[b*((a - b*x)^(1/ 
3) + (a + b*x)^(1/3))] - 2*a*Log[(a - b*x)^(2/3) - (a - b*x)^(1/3)*(a + b* 
x)^(1/3) + (a + b*x)^(2/3)])/(3*b)
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {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.

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 72

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

Input:

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

Output:

((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 + Log[a + b*x]/(2*b) + (3* 
Log[1 + (a - b*x)^(1/3)/(a + b*x)^(1/3)])/(2*b)))/3
 

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]

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

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 [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.50 \[ \int \frac {(a-b x)^{2/3}}{(a+b x)^{2/3}} \, dx=-\frac {4 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b x - a\right )} + 2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}}{3 \, {\left (b x - a\right )}}\right ) + 2 \, a \log \left (\frac {b x - {\left (b x + a\right )}^{\frac {2}{3}} {\left (-b x + a\right )}^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} - a}{b x - a}\right ) - 4 \, a \log \left (-\frac {b x - {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} - a}{b x - a}\right ) - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}}{3 \, b} \] Input:

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

Output:

-1/3*(4*sqrt(3)*a*arctan(1/3*(sqrt(3)*(b*x - a) + 2*sqrt(3)*(b*x + a)^(1/3 
)*(-b*x + a)^(2/3))/(b*x - a)) + 2*a*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)) - 4*a*log(-(b*x 
- (b*x + a)^(1/3)*(-b*x + a)^(2/3) - a)/(b*x - a)) - 3*(b*x + a)^(1/3)*(-b 
*x + a)^(2/3))/b
 

Sympy [F]

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

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

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

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

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

Output:

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