Integrand size = 23, antiderivative size = 48 \[ \int x^2 \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx=\frac {2 x^3}{3}-\frac {1}{4} x \sqrt {1-x^2}+\frac {1}{2} x^3 \sqrt {1-x^2}+\frac {\arcsin (x)}{4} \] Output:
2/3*x^3-1/4*x*(-x^2+1)^(1/2)+1/2*x^3*(-x^2+1)^(1/2)+1/4*arcsin(x)
Time = 0.40 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.40 \[ \int x^2 \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx=\frac {1}{12} \left (8-3 x \sqrt {1-x^2}+x^3 \left (8+6 \sqrt {1-x^2}\right )+12 \arctan \left (\frac {-\sqrt {2}+\sqrt {1+x}}{\sqrt {1-x}}\right )\right ) \] Input:
Integrate[x^2*(Sqrt[1 - x] + Sqrt[1 + x])^2,x]
Output:
(8 - 3*x*Sqrt[1 - x^2] + x^3*(8 + 6*Sqrt[1 - x^2]) + 12*ArcTan[(-Sqrt[2] + Sqrt[1 + x])/Sqrt[1 - x]])/12
Time = 0.44 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {7293, 2009}
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.
\(\displaystyle \int x^2 \left (\sqrt {1-x}+\sqrt {x+1}\right )^2 \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (2 \sqrt {1-x^2} x^2+2 x^2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arcsin (x)}{4}+\frac {2 x^3}{3}-\frac {1}{4} \sqrt {1-x^2} x+\frac {1}{2} \sqrt {1-x^2} x^3\) |
Input:
Int[x^2*(Sqrt[1 - x] + Sqrt[1 + x])^2,x]
Output:
(2*x^3)/3 - (x*Sqrt[1 - x^2])/4 + (x^3*Sqrt[1 - x^2])/2 + ArcSin[x]/4
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.23
method | result | size |
default | \(\frac {2 x^{3}}{3}+\frac {\sqrt {1+x}\, \sqrt {1-x}\, \left (2 x^{3} \sqrt {-x^{2}+1}-x \sqrt {-x^{2}+1}+\arcsin \left (x \right )\right )}{4 \sqrt {-x^{2}+1}}\) | \(59\) |
Input:
int(x^2*((1-x)^(1/2)+(1+x)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
2/3*x^3+1/4*(1+x)^(1/2)*(1-x)^(1/2)*(2*x^3*(-x^2+1)^(1/2)-x*(-x^2+1)^(1/2) +arcsin(x))/(-x^2+1)^(1/2)
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06 \[ \int x^2 \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx=\frac {2}{3} \, x^{3} + \frac {1}{4} \, {\left (2 \, x^{3} - x\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{2} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \] Input:
integrate(x^2*((1-x)^(1/2)+(1+x)^(1/2))^2,x, algorithm="fricas")
Output:
2/3*x^3 + 1/4*(2*x^3 - x)*sqrt(x + 1)*sqrt(-x + 1) - 1/2*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x)
\[ \int x^2 \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx=\int x^{2} \left (\sqrt {1 - x} + \sqrt {x + 1}\right )^{2}\, dx \] Input:
integrate(x**2*((1-x)**(1/2)+(1+x)**(1/2))**2,x)
Output:
Integral(x**2*(sqrt(1 - x) + sqrt(x + 1))**2, x)
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.71 \[ \int x^2 \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx=\frac {2}{3} \, x^{3} - \frac {1}{2} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {1}{4} \, \sqrt {-x^{2} + 1} x + \frac {1}{4} \, \arcsin \left (x\right ) \] Input:
integrate(x^2*((1-x)^(1/2)+(1+x)^(1/2))^2,x, algorithm="maxima")
Output:
2/3*x^3 - 1/2*(-x^2 + 1)^(3/2)*x + 1/4*sqrt(-x^2 + 1)*x + 1/4*arcsin(x)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (36) = 72\).
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.58 \[ \int x^2 \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx=\frac {2}{3} \, x^{3} + \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{2} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \] Input:
integrate(x^2*((1-x)^(1/2)+(1+x)^(1/2))^2,x, algorithm="giac")
Output:
2/3*x^3 + 1/12*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt (-x + 1) + 1/3*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*arcs in(1/2*sqrt(2)*sqrt(x + 1))
Time = 37.40 (sec) , antiderivative size = 563, normalized size of antiderivative = 11.73 \[ \int x^2 \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx=\frac {\frac {4\,\left (\sqrt {1-x}-1\right )}{\sqrt {x+1}-1}-\frac {28\,{\left (\sqrt {1-x}-1\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {28\,{\left (\sqrt {1-x}-1\right )}^5}{{\left (\sqrt {x+1}-1\right )}^5}-\frac {4\,{\left (\sqrt {1-x}-1\right )}^7}{{\left (\sqrt {x+1}-1\right )}^7}}{\frac {4\,{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+\frac {6\,{\left (\sqrt {1-x}-1\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}+\frac {4\,{\left (\sqrt {1-x}-1\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}+\frac {{\left (\sqrt {1-x}-1\right )}^8}{{\left (\sqrt {x+1}-1\right )}^8}+1}-\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right )-\frac {\frac {3\,\left (\sqrt {1-x}-1\right )}{\sqrt {x+1}-1}+\frac {23\,{\left (\sqrt {1-x}-1\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}-\frac {333\,{\left (\sqrt {1-x}-1\right )}^5}{{\left (\sqrt {x+1}-1\right )}^5}+\frac {671\,{\left (\sqrt {1-x}-1\right )}^7}{{\left (\sqrt {x+1}-1\right )}^7}-\frac {671\,{\left (\sqrt {1-x}-1\right )}^9}{{\left (\sqrt {x+1}-1\right )}^9}+\frac {333\,{\left (\sqrt {1-x}-1\right )}^{11}}{{\left (\sqrt {x+1}-1\right )}^{11}}-\frac {23\,{\left (\sqrt {1-x}-1\right )}^{13}}{{\left (\sqrt {x+1}-1\right )}^{13}}-\frac {3\,{\left (\sqrt {1-x}-1\right )}^{15}}{{\left (\sqrt {x+1}-1\right )}^{15}}}{\frac {8\,{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+\frac {28\,{\left (\sqrt {1-x}-1\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}+\frac {56\,{\left (\sqrt {1-x}-1\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {1-x}-1\right )}^8}{{\left (\sqrt {x+1}-1\right )}^8}+\frac {56\,{\left (\sqrt {1-x}-1\right )}^{10}}{{\left (\sqrt {x+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {1-x}-1\right )}^{12}}{{\left (\sqrt {x+1}-1\right )}^{12}}+\frac {8\,{\left (\sqrt {1-x}-1\right )}^{14}}{{\left (\sqrt {x+1}-1\right )}^{14}}+\frac {{\left (\sqrt {1-x}-1\right )}^{16}}{{\left (\sqrt {x+1}-1\right )}^{16}}+1}+\frac {2\,x^3}{3} \] Input:
int(x^2*((x + 1)^(1/2) + (1 - x)^(1/2))^2,x)
Output:
((4*((1 - x)^(1/2) - 1))/((x + 1)^(1/2) - 1) - (28*((1 - x)^(1/2) - 1)^3)/ ((x + 1)^(1/2) - 1)^3 + (28*((1 - x)^(1/2) - 1)^5)/((x + 1)^(1/2) - 1)^5 - (4*((1 - x)^(1/2) - 1)^7)/((x + 1)^(1/2) - 1)^7)/((4*((1 - x)^(1/2) - 1)^ 2)/((x + 1)^(1/2) - 1)^2 + (6*((1 - x)^(1/2) - 1)^4)/((x + 1)^(1/2) - 1)^4 + (4*((1 - x)^(1/2) - 1)^6)/((x + 1)^(1/2) - 1)^6 + ((1 - x)^(1/2) - 1)^8 /((x + 1)^(1/2) - 1)^8 + 1) - atan(((1 - x)^(1/2) - 1)/((x + 1)^(1/2) - 1) ) - ((3*((1 - x)^(1/2) - 1))/((x + 1)^(1/2) - 1) + (23*((1 - x)^(1/2) - 1) ^3)/((x + 1)^(1/2) - 1)^3 - (333*((1 - x)^(1/2) - 1)^5)/((x + 1)^(1/2) - 1 )^5 + (671*((1 - x)^(1/2) - 1)^7)/((x + 1)^(1/2) - 1)^7 - (671*((1 - x)^(1 /2) - 1)^9)/((x + 1)^(1/2) - 1)^9 + (333*((1 - x)^(1/2) - 1)^11)/((x + 1)^ (1/2) - 1)^11 - (23*((1 - x)^(1/2) - 1)^13)/((x + 1)^(1/2) - 1)^13 - (3*(( 1 - x)^(1/2) - 1)^15)/((x + 1)^(1/2) - 1)^15)/((8*((1 - x)^(1/2) - 1)^2)/( (x + 1)^(1/2) - 1)^2 + (28*((1 - x)^(1/2) - 1)^4)/((x + 1)^(1/2) - 1)^4 + (56*((1 - x)^(1/2) - 1)^6)/((x + 1)^(1/2) - 1)^6 + (70*((1 - x)^(1/2) - 1) ^8)/((x + 1)^(1/2) - 1)^8 + (56*((1 - x)^(1/2) - 1)^10)/((x + 1)^(1/2) - 1 )^10 + (28*((1 - x)^(1/2) - 1)^12)/((x + 1)^(1/2) - 1)^12 + (8*((1 - x)^(1 /2) - 1)^14)/((x + 1)^(1/2) - 1)^14 + ((1 - x)^(1/2) - 1)^16/((x + 1)^(1/2 ) - 1)^16 + 1) + (2*x^3)/3
Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int x^2 \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx=-\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}+\frac {\sqrt {x +1}\, \sqrt {1-x}\, x^{3}}{2}-\frac {\sqrt {x +1}\, \sqrt {1-x}\, x}{4}+\frac {2 x^{3}}{3} \] Input:
int(x^2*((1-x)^(1/2)+(1+x)^(1/2))^2,x)
Output:
( - 6*asin(sqrt( - x + 1)/sqrt(2)) + 6*sqrt(x + 1)*sqrt( - x + 1)*x**3 - 3 *sqrt(x + 1)*sqrt( - x + 1)*x + 8*x**3)/12