Integrand size = 17, antiderivative size = 61 \[ \int \frac {1}{(1-x)^{10/3} (1+x)^{2/3}} \, dx=\frac {3 \sqrt [3]{1+x}}{14 (1-x)^{7/3}}+\frac {9 \sqrt [3]{1+x}}{56 (1-x)^{4/3}}+\frac {27 \sqrt [3]{1+x}}{112 \sqrt [3]{1-x}} \] Output:
3/14*(1+x)^(1/3)/(1-x)^(7/3)+9/56*(1+x)^(1/3)/(1-x)^(4/3)+27/112*(1+x)^(1/ 3)/(1-x)^(1/3)
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(1-x)^{10/3} (1+x)^{2/3}} \, dx=\frac {3 \sqrt [3]{1+x} \left (23-24 x+9 x^2\right )}{112 (1-x)^{7/3}} \] Input:
Integrate[1/((1 - x)^(10/3)*(1 + x)^(2/3)),x]
Output:
(3*(1 + x)^(1/3)*(23 - 24*x + 9*x^2))/(112*(1 - x)^(7/3))
Time = 0.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {55, 55, 48}
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.
\(\displaystyle \int \frac {1}{(1-x)^{10/3} (x+1)^{2/3}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {3}{7} \int \frac {1}{(1-x)^{7/3} (x+1)^{2/3}}dx+\frac {3 \sqrt [3]{x+1}}{14 (1-x)^{7/3}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {3}{7} \left (\frac {3}{8} \int \frac {1}{(1-x)^{4/3} (x+1)^{2/3}}dx+\frac {3 \sqrt [3]{x+1}}{8 (1-x)^{4/3}}\right )+\frac {3 \sqrt [3]{x+1}}{14 (1-x)^{7/3}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {3}{7} \left (\frac {9 \sqrt [3]{x+1}}{16 \sqrt [3]{1-x}}+\frac {3 \sqrt [3]{x+1}}{8 (1-x)^{4/3}}\right )+\frac {3 \sqrt [3]{x+1}}{14 (1-x)^{7/3}}\) |
Input:
Int[1/((1 - x)^(10/3)*(1 + x)^(2/3)),x]
Output:
(3*(1 + x)^(1/3))/(14*(1 - x)^(7/3)) + (3*((3*(1 + x)^(1/3))/(8*(1 - x)^(4 /3)) + (9*(1 + x)^(1/3))/(16*(1 - x)^(1/3))))/7
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.41
method | result | size |
gosper | \(\frac {3 \left (1+x \right )^{\frac {1}{3}} \left (9 x^{2}-24 x +23\right )}{112 \left (1-x \right )^{\frac {7}{3}}}\) | \(25\) |
orering | \(-\frac {3 \left (1+x \right )^{\frac {1}{3}} \left (-1+x \right ) \left (9 x^{2}-24 x +23\right )}{112 \left (1-x \right )^{\frac {10}{3}}}\) | \(28\) |
risch | \(\frac {3 \left (\left (1-x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}} \left (9 x^{3}-15 x^{2}-x +23\right )}{112 \left (1+x \right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} \left (-1+x \right )^{2} \left (-\left (-1+x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}\) | \(60\) |
Input:
int(1/(1-x)^(10/3)/(1+x)^(2/3),x,method=_RETURNVERBOSE)
Output:
3/112*(1+x)^(1/3)/(1-x)^(7/3)*(9*x^2-24*x+23)
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(1-x)^{10/3} (1+x)^{2/3}} \, dx=-\frac {3 \, {\left (9 \, x^{2} - 24 \, x + 23\right )} {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}}}{112 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \] Input:
integrate(1/(1-x)^(10/3)/(1+x)^(2/3),x, algorithm="fricas")
Output:
-3/112*(9*x^2 - 24*x + 23)*(x + 1)^(1/3)*(-x + 1)^(2/3)/(x^3 - 3*x^2 + 3*x - 1)
Result contains complex when optimal does not.
Time = 21.78 (sec) , antiderivative size = 563, normalized size of antiderivative = 9.23 \[ \int \frac {1}{(1-x)^{10/3} (1+x)^{2/3}} \, dx =\text {Too large to display} \] Input:
integrate(1/(1-x)**(10/3)/(1+x)**(2/3),x)
Output:
Piecewise((9*(x + 1)**2*exp(I*pi/3)*gamma(1/3)/(36*(-1 + 2/(x + 1))**(1/3) *(x + 1)**2*exp(I*pi/3)*gamma(10/3) - 144*(-1 + 2/(x + 1))**(1/3)*(x + 1)* exp(I*pi/3)*gamma(10/3) + 144*(-1 + 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(10 /3)) - 42*(x + 1)*exp(I*pi/3)*gamma(1/3)/(36*(-1 + 2/(x + 1))**(1/3)*(x + 1)**2*exp(I*pi/3)*gamma(10/3) - 144*(-1 + 2/(x + 1))**(1/3)*(x + 1)*exp(I* pi/3)*gamma(10/3) + 144*(-1 + 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(10/3)) + 56*exp(I*pi/3)*gamma(1/3)/(36*(-1 + 2/(x + 1))**(1/3)*(x + 1)**2*exp(I*pi /3)*gamma(10/3) - 144*(-1 + 2/(x + 1))**(1/3)*(x + 1)*exp(I*pi/3)*gamma(10 /3) + 144*(-1 + 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(10/3)), 1/Abs(x + 1) > 1/2), (9*(x + 1)**2*gamma(1/3)/(36*(1 - 2/(x + 1))**(1/3)*(x + 1)**2*exp( I*pi/3)*gamma(10/3) - 144*(1 - 2/(x + 1))**(1/3)*(x + 1)*exp(I*pi/3)*gamma (10/3) + 144*(1 - 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(10/3)) - 42*(x + 1)* gamma(1/3)/(36*(1 - 2/(x + 1))**(1/3)*(x + 1)**2*exp(I*pi/3)*gamma(10/3) - 144*(1 - 2/(x + 1))**(1/3)*(x + 1)*exp(I*pi/3)*gamma(10/3) + 144*(1 - 2/( x + 1))**(1/3)*exp(I*pi/3)*gamma(10/3)) + 56*gamma(1/3)/(36*(1 - 2/(x + 1) )**(1/3)*(x + 1)**2*exp(I*pi/3)*gamma(10/3) - 144*(1 - 2/(x + 1))**(1/3)*( x + 1)*exp(I*pi/3)*gamma(10/3) + 144*(1 - 2/(x + 1))**(1/3)*exp(I*pi/3)*ga mma(10/3)), True))
\[ \int \frac {1}{(1-x)^{10/3} (1+x)^{2/3}} \, dx=\int { \frac {1}{{\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {10}{3}}} \,d x } \] Input:
integrate(1/(1-x)^(10/3)/(1+x)^(2/3),x, algorithm="maxima")
Output:
integrate(1/((x + 1)^(2/3)*(-x + 1)^(10/3)), x)
\[ \int \frac {1}{(1-x)^{10/3} (1+x)^{2/3}} \, dx=\int { \frac {1}{{\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {10}{3}}} \,d x } \] Input:
integrate(1/(1-x)^(10/3)/(1+x)^(2/3),x, algorithm="giac")
Output:
integrate(1/((x + 1)^(2/3)*(-x + 1)^(10/3)), x)
Timed out. \[ \int \frac {1}{(1-x)^{10/3} (1+x)^{2/3}} \, dx=\int \frac {1}{{\left (1-x\right )}^{10/3}\,{\left (x+1\right )}^{2/3}} \,d x \] Input:
int(1/((1 - x)^(10/3)*(x + 1)^(2/3)),x)
Output:
int(1/((1 - x)^(10/3)*(x + 1)^(2/3)), x)
\[ \int \frac {1}{(1-x)^{10/3} (1+x)^{2/3}} \, dx=-\left (\int \frac {1}{\left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} x^{3}-3 \left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} x^{2}+3 \left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} x -\left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}}}d x \right ) \] Input:
int(1/(1-x)^(10/3)/(1+x)^(2/3),x)
Output:
- int(1/((x + 1)**(2/3)*( - x + 1)**(1/3)*x**3 - 3*(x + 1)**(2/3)*( - x + 1)**(1/3)*x**2 + 3*(x + 1)**(2/3)*( - x + 1)**(1/3)*x - (x + 1)**(2/3)*( - x + 1)**(1/3)),x)