Integrand size = 18, antiderivative size = 47 \[ \int \sqrt {-1+x} x^2 \sqrt {1+x} \, dx=-\frac {1}{8} \sqrt {-1+x} x \sqrt {1+x}+\frac {1}{4} \sqrt {-1+x} x^3 \sqrt {1+x}-\frac {\text {arccosh}(x)}{8} \] Output:
-1/8*(-1+x)^(1/2)*x*(1+x)^(1/2)+1/4*(-1+x)^(1/2)*x^3*(1+x)^(1/2)-1/8*arcco sh(x)
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int \sqrt {-1+x} x^2 \sqrt {1+x} \, dx=\frac {1}{8} \left (x \sqrt {\frac {-1+x}{1+x}} \left (-1-x+2 x^2+2 x^3\right )-2 \text {arctanh}\left (\sqrt {\frac {-1+x}{1+x}}\right )\right ) \] Input:
Integrate[Sqrt[-1 + x]*x^2*Sqrt[1 + x],x]
Output:
(x*Sqrt[(-1 + x)/(1 + x)]*(-1 - x + 2*x^2 + 2*x^3) - 2*ArcTanh[Sqrt[(-1 + x)/(1 + x)]])/8
Time = 0.15 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {101, 40, 43}
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.
\(\displaystyle \int \sqrt {x-1} x^2 \sqrt {x+1} \, dx\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {1}{4} \int \sqrt {x-1} \sqrt {x+1}dx+\frac {1}{4} (x-1)^{3/2} x (x+1)^{3/2}\) |
\(\Big \downarrow \) 40 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \sqrt {x-1} x \sqrt {x+1}-\frac {1}{2} \int \frac {1}{\sqrt {x-1} \sqrt {x+1}}dx\right )+\frac {1}{4} (x-1)^{3/2} x (x+1)^{3/2}\) |
\(\Big \downarrow \) 43 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{2} \sqrt {x-1} x \sqrt {x+1}-\frac {\text {arccosh}(x)}{2}\right )+\frac {1}{4} (x-1)^{3/2} x (x+1)^{3/2}\) |
Input:
Int[Sqrt[-1 + x]*x^2*Sqrt[1 + x],x]
Output:
((-1 + x)^(3/2)*x*(1 + x)^(3/2))/4 + ((Sqrt[-1 + x]*x*Sqrt[1 + x])/2 - Arc Cosh[x]/2)/4
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x* (a + b*x)^m*((c + d*x)^m/(2*m + 1)), x] + Simp[2*a*c*(m/(2*m + 1)) Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ b*c + a*d, 0] && IGtQ[m + 1/2, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.11
method | result | size |
default | \(-\frac {\sqrt {-1+x}\, \sqrt {1+x}\, \left (-2 x^{3} \sqrt {x^{2}-1}+x \sqrt {x^{2}-1}+\ln \left (x +\sqrt {x^{2}-1}\right )\right )}{8 \sqrt {x^{2}-1}}\) | \(52\) |
risch | \(\frac {x \left (2 x^{2}-1\right ) \sqrt {1+x}\, \sqrt {-1+x}}{8}-\frac {\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (1+x \right ) \left (-1+x \right )}}{8 \sqrt {-1+x}\, \sqrt {1+x}}\) | \(53\) |
Input:
int((-1+x)^(1/2)*x^2*(1+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/8*(-1+x)^(1/2)*(1+x)^(1/2)*(-2*x^3*(x^2-1)^(1/2)+x*(x^2-1)^(1/2)+ln(x+( x^2-1)^(1/2)))/(x^2-1)^(1/2)
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.85 \[ \int \sqrt {-1+x} x^2 \sqrt {1+x} \, dx=\frac {1}{8} \, {\left (2 \, x^{3} - x\right )} \sqrt {x + 1} \sqrt {x - 1} + \frac {1}{8} \, \log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) \] Input:
integrate((-1+x)^(1/2)*x^2*(1+x)^(1/2),x, algorithm="fricas")
Output:
1/8*(2*x^3 - x)*sqrt(x + 1)*sqrt(x - 1) + 1/8*log(sqrt(x + 1)*sqrt(x - 1) - x)
\[ \int \sqrt {-1+x} x^2 \sqrt {1+x} \, dx=\int x^{2} \sqrt {x - 1} \sqrt {x + 1}\, dx \] Input:
integrate((-1+x)**(1/2)*x**2*(1+x)**(1/2),x)
Output:
Integral(x**2*sqrt(x - 1)*sqrt(x + 1), x)
Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79 \[ \int \sqrt {-1+x} x^2 \sqrt {1+x} \, dx=\frac {1}{4} \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} x + \frac {1}{8} \, \sqrt {x^{2} - 1} x - \frac {1}{8} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \] Input:
integrate((-1+x)^(1/2)*x^2*(1+x)^(1/2),x, algorithm="maxima")
Output:
1/4*(x^2 - 1)^(3/2)*x + 1/8*sqrt(x^2 - 1)*x - 1/8*log(2*x + 2*sqrt(x^2 - 1 ))
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (33) = 66\).
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.49 \[ \int \sqrt {-1+x} x^2 \sqrt {1+x} \, dx=\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {x - 1} + \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {x - 1} + \frac {1}{4} \, \log \left (\sqrt {x + 1} - \sqrt {x - 1}\right ) \] Input:
integrate((-1+x)^(1/2)*x^2*(1+x)^(1/2),x, algorithm="giac")
Output:
1/24*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(x - 1) + 1/6*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(x - 1) + 1/4*log(sqrt(x + 1) - sqrt(x - 1))
Time = 4.81 (sec) , antiderivative size = 362, normalized size of antiderivative = 7.70 \[ \int \sqrt {-1+x} x^2 \sqrt {1+x} \, dx=-\frac {\mathrm {atanh}\left (\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}\right )}{2}+\frac {\frac {35\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^3}{2\,{\left (\sqrt {x+1}-1\right )}^3}+\frac {273\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {x+1}-1\right )}^5}+\frac {715\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {x+1}-1\right )}^7}+\frac {715\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^9}{2\,{\left (\sqrt {x+1}-1\right )}^9}+\frac {273\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{11}}{2\,{\left (\sqrt {x+1}-1\right )}^{11}}+\frac {35\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{13}}{2\,{\left (\sqrt {x+1}-1\right )}^{13}}+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^{15}}{2\,{\left (\sqrt {x+1}-1\right )}^{15}}+\frac {\sqrt {x-1}-\mathrm {i}}{2\,\left (\sqrt {x+1}-1\right )}}{1+\frac {28\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}-\frac {56\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {x+1}-1\right )}^8}-\frac {56\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {x+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {x+1}-1\right )}^{12}}-\frac {8\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {x+1}-1\right )}^{14}}+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {x+1}-1\right )}^{16}}-\frac {8\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}} \] Input:
int(x^2*(x - 1)^(1/2)*(x + 1)^(1/2),x)
Output:
((35*((x - 1)^(1/2) - 1i)^3)/(2*((x + 1)^(1/2) - 1)^3) + (273*((x - 1)^(1/ 2) - 1i)^5)/(2*((x + 1)^(1/2) - 1)^5) + (715*((x - 1)^(1/2) - 1i)^7)/(2*(( x + 1)^(1/2) - 1)^7) + (715*((x - 1)^(1/2) - 1i)^9)/(2*((x + 1)^(1/2) - 1) ^9) + (273*((x - 1)^(1/2) - 1i)^11)/(2*((x + 1)^(1/2) - 1)^11) + (35*((x - 1)^(1/2) - 1i)^13)/(2*((x + 1)^(1/2) - 1)^13) + ((x - 1)^(1/2) - 1i)^15/( 2*((x + 1)^(1/2) - 1)^15) + ((x - 1)^(1/2) - 1i)/(2*((x + 1)^(1/2) - 1)))/ ((28*((x - 1)^(1/2) - 1i)^4)/((x + 1)^(1/2) - 1)^4 - (8*((x - 1)^(1/2) - 1 i)^2)/((x + 1)^(1/2) - 1)^2 - (56*((x - 1)^(1/2) - 1i)^6)/((x + 1)^(1/2) - 1)^6 + (70*((x - 1)^(1/2) - 1i)^8)/((x + 1)^(1/2) - 1)^8 - (56*((x - 1)^( 1/2) - 1i)^10)/((x + 1)^(1/2) - 1)^10 + (28*((x - 1)^(1/2) - 1i)^12)/((x + 1)^(1/2) - 1)^12 - (8*((x - 1)^(1/2) - 1i)^14)/((x + 1)^(1/2) - 1)^14 + ( (x - 1)^(1/2) - 1i)^16/((x + 1)^(1/2) - 1)^16 + 1) - atanh(((x - 1)^(1/2) - 1i)/((x + 1)^(1/2) - 1))/2
Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89 \[ \int \sqrt {-1+x} x^2 \sqrt {1+x} \, dx=\frac {\sqrt {x +1}\, \sqrt {x -1}\, x^{3}}{4}-\frac {\sqrt {x +1}\, \sqrt {x -1}\, x}{8}-\frac {\mathrm {log}\left (\frac {\sqrt {x -1}+\sqrt {x +1}}{\sqrt {2}}\right )}{4} \] Input:
int((-1+x)^(1/2)*x^2*(1+x)^(1/2),x)
Output:
(2*sqrt(x + 1)*sqrt(x - 1)*x**3 - sqrt(x + 1)*sqrt(x - 1)*x - 2*log((sqrt( x - 1) + sqrt(x + 1))/sqrt(2)))/8