Integrand size = 17, antiderivative size = 41 \[ \int \frac {1}{(1-x)^{7/3} (1+x)^{2/3}} \, dx=\frac {3 \sqrt [3]{1+x}}{8 (1-x)^{4/3}}+\frac {9 \sqrt [3]{1+x}}{16 \sqrt [3]{1-x}} \] Output:
3/8*(1+x)^(1/3)/(1-x)^(4/3)+9/16*(1+x)^(1/3)/(1-x)^(1/3)
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1-x)^{7/3} (1+x)^{2/3}} \, dx=-\frac {3 \sqrt [3]{1+x} (-5+3 x)}{16 (1-x)^{4/3}} \] Input:
Integrate[1/((1 - x)^(7/3)*(1 + x)^(2/3)),x]
Output:
(-3*(1 + x)^(1/3)*(-5 + 3*x))/(16*(1 - x)^(4/3))
Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {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.
\(\displaystyle \int \frac {1}{(1-x)^{7/3} (x+1)^{2/3}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {3}{8} \int \frac {1}{(1-x)^{4/3} (x+1)^{2/3}}dx+\frac {3 \sqrt [3]{x+1}}{8 (1-x)^{4/3}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {9 \sqrt [3]{x+1}}{16 \sqrt [3]{1-x}}+\frac {3 \sqrt [3]{x+1}}{8 (1-x)^{4/3}}\) |
Input:
Int[1/((1 - x)^(7/3)*(1 + x)^(2/3)),x]
Output:
(3*(1 + x)^(1/3))/(8*(1 - x)^(4/3)) + (9*(1 + x)^(1/3))/(16*(1 - x)^(1/3))
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 = 20, normalized size of antiderivative = 0.49
method | result | size |
gosper | \(-\frac {3 \left (-5+3 x \right ) \left (1+x \right )^{\frac {1}{3}}}{16 \left (1-x \right )^{\frac {4}{3}}}\) | \(20\) |
orering | \(\frac {3 \left (1+x \right )^{\frac {1}{3}} \left (-1+x \right ) \left (-5+3 x \right )}{16 \left (1-x \right )^{\frac {7}{3}}}\) | \(23\) |
risch | \(\frac {3 \left (\left (1-x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}} \left (3 x^{2}-2 x -5\right )}{16 \left (1+x \right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} \left (-1+x \right ) \left (-\left (-1+x \right ) \left (1+x \right )^{2}\right )^{\frac {1}{3}}}\) | \(55\) |
Input:
int(1/(1-x)^(7/3)/(1+x)^(2/3),x,method=_RETURNVERBOSE)
Output:
-3/16*(-5+3*x)/(1-x)^(4/3)*(1+x)^(1/3)
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(1-x)^{7/3} (1+x)^{2/3}} \, dx=-\frac {3 \, {\left (3 \, x - 5\right )} {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {2}{3}}}{16 \, {\left (x^{2} - 2 \, x + 1\right )}} \] Input:
integrate(1/(1-x)^(7/3)/(1+x)^(2/3),x, algorithm="fricas")
Output:
-3/16*(3*x - 5)*(x + 1)^(1/3)*(-x + 1)^(2/3)/(x^2 - 2*x + 1)
Result contains complex when optimal does not.
Time = 3.62 (sec) , antiderivative size = 257, normalized size of antiderivative = 6.27 \[ \int \frac {1}{(1-x)^{7/3} (1+x)^{2/3}} \, dx=\begin {cases} \frac {3 \left (x + 1\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right )}{12 \sqrt [3]{-1 + \frac {2}{x + 1}} \left (x + 1\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 24 \sqrt [3]{-1 + \frac {2}{x + 1}} e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} - \frac {8 e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right )}{12 \sqrt [3]{-1 + \frac {2}{x + 1}} \left (x + 1\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 24 \sqrt [3]{-1 + \frac {2}{x + 1}} e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\\frac {3 \left (x + 1\right ) \Gamma \left (\frac {1}{3}\right )}{12 \sqrt [3]{1 - \frac {2}{x + 1}} \left (x + 1\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 24 \sqrt [3]{1 - \frac {2}{x + 1}} e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} - \frac {8 \Gamma \left (\frac {1}{3}\right )}{12 \sqrt [3]{1 - \frac {2}{x + 1}} \left (x + 1\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) - 24 \sqrt [3]{1 - \frac {2}{x + 1}} e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right )} & \text {otherwise} \end {cases} \] Input:
integrate(1/(1-x)**(7/3)/(1+x)**(2/3),x)
Output:
Piecewise((3*(x + 1)*exp(I*pi/3)*gamma(1/3)/(12*(-1 + 2/(x + 1))**(1/3)*(x + 1)*exp(I*pi/3)*gamma(7/3) - 24*(-1 + 2/(x + 1))**(1/3)*exp(I*pi/3)*gamm a(7/3)) - 8*exp(I*pi/3)*gamma(1/3)/(12*(-1 + 2/(x + 1))**(1/3)*(x + 1)*exp (I*pi/3)*gamma(7/3) - 24*(-1 + 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(7/3)), 1/Abs(x + 1) > 1/2), (3*(x + 1)*gamma(1/3)/(12*(1 - 2/(x + 1))**(1/3)*(x + 1)*exp(I*pi/3)*gamma(7/3) - 24*(1 - 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(7 /3)) - 8*gamma(1/3)/(12*(1 - 2/(x + 1))**(1/3)*(x + 1)*exp(I*pi/3)*gamma(7 /3) - 24*(1 - 2/(x + 1))**(1/3)*exp(I*pi/3)*gamma(7/3)), True))
\[ \int \frac {1}{(1-x)^{7/3} (1+x)^{2/3}} \, dx=\int { \frac {1}{{\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {7}{3}}} \,d x } \] Input:
integrate(1/(1-x)^(7/3)/(1+x)^(2/3),x, algorithm="maxima")
Output:
integrate(1/((x + 1)^(2/3)*(-x + 1)^(7/3)), x)
\[ \int \frac {1}{(1-x)^{7/3} (1+x)^{2/3}} \, dx=\int { \frac {1}{{\left (x + 1\right )}^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {7}{3}}} \,d x } \] Input:
integrate(1/(1-x)^(7/3)/(1+x)^(2/3),x, algorithm="giac")
Output:
integrate(1/((x + 1)^(2/3)*(-x + 1)^(7/3)), x)
Timed out. \[ \int \frac {1}{(1-x)^{7/3} (1+x)^{2/3}} \, dx=\int \frac {1}{{\left (1-x\right )}^{7/3}\,{\left (x+1\right )}^{2/3}} \,d x \] Input:
int(1/((1 - x)^(7/3)*(x + 1)^(2/3)),x)
Output:
int(1/((1 - x)^(7/3)*(x + 1)^(2/3)), x)
\[ \int \frac {1}{(1-x)^{7/3} (1+x)^{2/3}} \, dx=\int \frac {1}{\left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} x^{2}-2 \left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}} x +\left (x +1\right )^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}}}d x \] Input:
int(1/(1-x)^(7/3)/(1+x)^(2/3),x)
Output:
int(1/((x + 1)**(2/3)*( - x + 1)**(1/3)*x**2 - 2*(x + 1)**(2/3)*( - x + 1) **(1/3)*x + (x + 1)**(2/3)*( - x + 1)**(1/3)),x)