Integrand size = 26, antiderivative size = 41 \[ \int \frac {x^{7/2}}{(1-x)^2 \left (2 x-x^2\right )^{3/2}} \, dx=\frac {8}{\sqrt {2-x}}+\frac {\sqrt {2-x}}{1-x}-7 \text {arctanh}\left (\sqrt {2-x}\right ) \] Output:
8/(2-x)^(1/2)+(2-x)^(1/2)/(1-x)-7*arctanh((2-x)^(1/2))
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int \frac {x^{7/2}}{(1-x)^2 \left (2 x-x^2\right )^{3/2}} \, dx=\frac {\sqrt {x} \left (-10+9 x+7 \sqrt {-2+x} (-1+x) \arctan \left (\sqrt {-2+x}\right )\right )}{(-1+x) \sqrt {-((-2+x) x)}} \] Input:
Integrate[x^(7/2)/((1 - x)^2*(2*x - x^2)^(3/2)),x]
Output:
(Sqrt[x]*(-10 + 9*x + 7*Sqrt[-2 + x]*(-1 + x)*ArcTan[Sqrt[-2 + x]]))/((-1 + x)*Sqrt[-((-2 + x)*x)])
Time = 0.34 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.80, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1261, 100, 27, 87, 73, 220}
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.
\(\displaystyle \int \frac {x^{7/2}}{(1-x)^2 \left (2 x-x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1261 |
\(\displaystyle \frac {(2-x)^{3/2} x^{3/2} \int \frac {x^2}{(1-x)^2 (2-x)^{3/2}}dx}{\left (2 x-x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {(2-x)^{3/2} x^{3/2} \left (\frac {1}{(1-x) \sqrt {2-x}}-\int \frac {2 x+5}{2 (1-x) (2-x)^{3/2}}dx\right )}{\left (2 x-x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(2-x)^{3/2} x^{3/2} \left (\frac {1}{(1-x) \sqrt {2-x}}-\frac {1}{2} \int \frac {2 x+5}{(1-x) (2-x)^{3/2}}dx\right )}{\left (2 x-x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(2-x)^{3/2} x^{3/2} \left (\frac {1}{2} \left (\frac {18}{\sqrt {2-x}}-7 \int \frac {1}{(1-x) \sqrt {2-x}}dx\right )+\frac {1}{(1-x) \sqrt {2-x}}\right )}{\left (2 x-x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(2-x)^{3/2} x^{3/2} \left (\frac {1}{2} \left (14 \int \frac {1}{1-x}d\sqrt {2-x}+\frac {18}{\sqrt {2-x}}\right )+\frac {1}{(1-x) \sqrt {2-x}}\right )}{\left (2 x-x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {(2-x)^{3/2} x^{3/2} \left (\frac {1}{2} \left (\frac {18}{\sqrt {2-x}}-14 \text {arctanh}\left (\sqrt {2-x}\right )\right )+\frac {1}{(1-x) \sqrt {2-x}}\right )}{\left (2 x-x^2\right )^{3/2}}\) |
Input:
Int[x^(7/2)/((1 - x)^2*(2*x - x^2)^(3/2)),x]
Output:
((2 - x)^(3/2)*x^(3/2)*(1/((1 - x)*Sqrt[2 - x]) + (18/Sqrt[2 - x] - 14*Arc Tanh[Sqrt[2 - x]])/2))/(2*x - x^2)^(3/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) ^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, n}, x] && !IGtQ[n, 0]
Time = 0.54 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.27
method | result | size |
risch | \(\frac {\left (9 x -10\right ) \sqrt {x}}{\left (x -1\right ) \sqrt {-x \left (x -2\right )}}-\frac {7 \,\operatorname {arctanh}\left (\sqrt {2-x}\right ) \sqrt {2-x}\, \sqrt {x}}{\sqrt {-x \left (x -2\right )}}\) | \(52\) |
default | \(\frac {\sqrt {-x \left (x -2\right )}\, \left (7 \ln \left (\sqrt {2-x}-1\right ) \sqrt {2-x}\, x -7 \ln \left (\sqrt {2-x}+1\right ) \sqrt {2-x}\, x -7 \ln \left (\sqrt {2-x}-1\right ) \sqrt {2-x}+7 \ln \left (\sqrt {2-x}+1\right ) \sqrt {2-x}+18 x -20\right )}{2 \sqrt {x}\, \left (x -2\right ) \left (\sqrt {2-x}-1\right ) \left (\sqrt {2-x}+1\right )}\) | \(124\) |
Input:
int(x^(7/2)/(1-x)^2/(-x^2+2*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
(9*x-10)/(x-1)*x^(1/2)/(-x*(x-2))^(1/2)-7*arctanh((2-x)^(1/2))*(2-x)^(1/2) *x^(1/2)/(-x*(x-2))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (34) = 68\).
Time = 0.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 3.27 \[ \int \frac {x^{7/2}}{(1-x)^2 \left (2 x-x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {-x^{2} + 2 \, x} {\left (9 \, x - 10\right )} \sqrt {x} - 7 \, {\left (x^{3} - 3 \, x^{2} + 2 \, x\right )} \log \left (\frac {x^{2} + \sqrt {-x^{2} + 2 \, x} \sqrt {x} - 2 \, x}{x^{2} - 2 \, x}\right ) + 7 \, {\left (x^{3} - 3 \, x^{2} + 2 \, x\right )} \log \left (-\frac {x^{2} - \sqrt {-x^{2} + 2 \, x} \sqrt {x} - 2 \, x}{x^{2} - 2 \, x}\right )}{2 \, {\left (x^{3} - 3 \, x^{2} + 2 \, x\right )}} \] Input:
integrate(x^(7/2)/(1-x)^2/(-x^2+2*x)^(3/2),x, algorithm="fricas")
Output:
-1/2*(2*sqrt(-x^2 + 2*x)*(9*x - 10)*sqrt(x) - 7*(x^3 - 3*x^2 + 2*x)*log((x ^2 + sqrt(-x^2 + 2*x)*sqrt(x) - 2*x)/(x^2 - 2*x)) + 7*(x^3 - 3*x^2 + 2*x)* log(-(x^2 - sqrt(-x^2 + 2*x)*sqrt(x) - 2*x)/(x^2 - 2*x)))/(x^3 - 3*x^2 + 2 *x)
\[ \int \frac {x^{7/2}}{(1-x)^2 \left (2 x-x^2\right )^{3/2}} \, dx=\int \frac {x^{\frac {7}{2}}}{\left (- x \left (x - 2\right )\right )^{\frac {3}{2}} \left (x - 1\right )^{2}}\, dx \] Input:
integrate(x**(7/2)/(1-x)**2/(-x**2+2*x)**(3/2),x)
Output:
Integral(x**(7/2)/((-x*(x - 2))**(3/2)*(x - 1)**2), x)
\[ \int \frac {x^{7/2}}{(1-x)^2 \left (2 x-x^2\right )^{3/2}} \, dx=\int { \frac {x^{\frac {7}{2}}}{{\left (-x^{2} + 2 \, x\right )}^{\frac {3}{2}} {\left (x - 1\right )}^{2}} \,d x } \] Input:
integrate(x^(7/2)/(1-x)^2/(-x^2+2*x)^(3/2),x, algorithm="maxima")
Output:
integrate(x^(7/2)/((-x^2 + 2*x)^(3/2)*(x - 1)^2), x)
Time = 0.15 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.27 \[ \int \frac {x^{7/2}}{(1-x)^2 \left (2 x-x^2\right )^{3/2}} \, dx=-\frac {9 \, x - 10}{{\left (-x + 2\right )}^{\frac {3}{2}} - \sqrt {-x + 2}} - \frac {7}{2} \, \log \left (\sqrt {-x + 2} + 1\right ) + \frac {7}{2} \, \log \left ({\left | \sqrt {-x + 2} - 1 \right |}\right ) \] Input:
integrate(x^(7/2)/(1-x)^2/(-x^2+2*x)^(3/2),x, algorithm="giac")
Output:
-(9*x - 10)/((-x + 2)^(3/2) - sqrt(-x + 2)) - 7/2*log(sqrt(-x + 2) + 1) + 7/2*log(abs(sqrt(-x + 2) - 1))
Timed out. \[ \int \frac {x^{7/2}}{(1-x)^2 \left (2 x-x^2\right )^{3/2}} \, dx=\int \frac {x^{7/2}}{{\left (2\,x-x^2\right )}^{3/2}\,{\left (x-1\right )}^2} \,d x \] Input:
int(x^(7/2)/((2*x - x^2)^(3/2)*(x - 1)^2),x)
Output:
int(x^(7/2)/((2*x - x^2)^(3/2)*(x - 1)^2), x)
Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.20 \[ \int \frac {x^{7/2}}{(1-x)^2 \left (2 x-x^2\right )^{3/2}} \, dx=\frac {7 \sqrt {-x +2}\, \mathrm {log}\left (\sqrt {-x +2}-1\right ) x -7 \sqrt {-x +2}\, \mathrm {log}\left (\sqrt {-x +2}-1\right )-7 \sqrt {-x +2}\, \mathrm {log}\left (\sqrt {-x +2}+1\right ) x +7 \sqrt {-x +2}\, \mathrm {log}\left (\sqrt {-x +2}+1\right )+18 x -20}{2 \sqrt {-x +2}\, \left (x -1\right )} \] Input:
int(x^(7/2)/(1-x)^2/(-x^2+2*x)^(3/2),x)
Output:
(7*sqrt( - x + 2)*log(sqrt( - x + 2) - 1)*x - 7*sqrt( - x + 2)*log(sqrt( - x + 2) - 1) - 7*sqrt( - x + 2)*log(sqrt( - x + 2) + 1)*x + 7*sqrt( - x + 2)*log(sqrt( - x + 2) + 1) + 18*x - 20)/(2*sqrt( - x + 2)*(x - 1))