\(\int \sqrt {-1+x} x \sqrt {1+x} \, dx\) [842]

Optimal result

Integrand size = 16, antiderivative size = 18 \[ \int \sqrt {-1+x} x \sqrt {1+x} \, dx=\frac {1}{3} (-1+x)^{3/2} (1+x)^{3/2} \]

[Out]

1/3*(-1+x)^(3/2)*(1+x)^(3/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {75} \[ \int \sqrt {-1+x} x \sqrt {1+x} \, dx=\frac {1}{3} (x-1)^{3/2} (x+1)^{3/2} \]

[In]

Int[Sqrt[-1 + x]*x*Sqrt[1 + x],x]

[Out]

((-1 + x)^(3/2)*(1 + x)^(3/2))/3

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} (-1+x)^{3/2} (1+x)^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \sqrt {-1+x} x \sqrt {1+x} \, dx=\frac {1}{3} (-1+x)^{3/2} (1+x)^{3/2} \]

[In]

Integrate[Sqrt[-1 + x]*x*Sqrt[1 + x],x]

[Out]

((-1 + x)^(3/2)*(1 + x)^(3/2))/3

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72

method	result	size
gosper	\(\frac {\left (-1+x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}{3}\)	\(13\)
default	\(\frac {\sqrt {-1+x}\, \sqrt {1+x}\, \left (x^{2}-1\right )}{3}\)	\(18\)
risch	\(\frac {\sqrt {-1+x}\, \sqrt {1+x}\, \left (x^{2}-1\right )}{3}\)	\(18\)

[In]

int(x*(-1+x)^(1/2)*(1+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(-1+x)^(3/2)*(1+x)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \sqrt {-1+x} x \sqrt {1+x} \, dx=\frac {1}{3} \, {\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {x - 1} \]

[In]

integrate(x*(-1+x)^(1/2)*(1+x)^(1/2),x, algorithm="fricas")

[Out]

1/3*(x^2 - 1)*sqrt(x + 1)*sqrt(x - 1)

Sympy [F]

\[ \int \sqrt {-1+x} x \sqrt {1+x} \, dx=\int x \sqrt {x - 1} \sqrt {x + 1}\, dx \]

[In]

integrate(x*(-1+x)**(1/2)*(1+x)**(1/2),x)

[Out]

Integral(x*sqrt(x - 1)*sqrt(x + 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.50 \[ \int \sqrt {-1+x} x \sqrt {1+x} \, dx=\frac {1}{3} \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} \]

[In]

integrate(x*(-1+x)^(1/2)*(1+x)^(1/2),x, algorithm="maxima")

[Out]

1/3*(x^2 - 1)^(3/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.17 \[ \int \sqrt {-1+x} x \sqrt {1+x} \, dx=\frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {x - 1} + \frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} {\left (x - 2\right )} \]

[In]

integrate(x*(-1+x)^(1/2)*(1+x)^(1/2),x, algorithm="giac")

[Out]

1/6*((2*x - 5)*(x + 1) + 9)*sqrt(x + 1)*sqrt(x - 1) + 1/2*sqrt(x + 1)*sqrt(x - 1)*(x - 2)

Mupad [B] (verification not implemented)

Time = 1.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \sqrt {-1+x} x \sqrt {1+x} \, dx=\frac {\left (x^2-1\right )\,\sqrt {x-1}\,\sqrt {x+1}}{3} \]

[In]

int(x*(x - 1)^(1/2)*(x + 1)^(1/2),x)

[Out]

((x^2 - 1)*(x - 1)^(1/2)*(x + 1)^(1/2))/3