3.3.35 \(\int \frac {x}{\sqrt {-1+x}+\sqrt {1+x}} \, dx\) [235]

Optimal. Leaf size=45 \[ -\frac {1}{3} (-1+x)^{3/2}-\frac {1}{5} (-1+x)^{5/2}-\frac {1}{3} (1+x)^{3/2}+\frac {1}{5} (1+x)^{5/2} \]

[Out]

-1/3*(x-1)^(3/2)-1/5*(x-1)^(5/2)-1/3*(x+1)^(3/2)+1/5*(x+1)^(5/2)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2129, 45} \begin {gather*} -\frac {1}{5} (x-1)^{5/2}-\frac {1}{3} (x-1)^{3/2}+\frac {1}{5} (x+1)^{5/2}-\frac {1}{3} (x+1)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[-1 + x] + Sqrt[1 + x]),x]

[Out]

-1/3*(-1 + x)^(3/2) - (-1 + x)^(5/2)/5 - (1 + x)^(3/2)/3 + (1 + x)^(5/2)/5

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2129

Int[(u_)/((e_.)*Sqrt[(a_.) + (b_.)*(x_)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[-d/(e*(b*c - a*d
)), Int[u*Sqrt[a + b*x], x], x] + Dist[b/(f*(b*c - a*d)), Int[u*Sqrt[c + d*x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[b*e^2 - d*f^2, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=-\left (\frac {1}{2} \int \sqrt {-1+x} x \, dx\right )+\frac {1}{2} \int x \sqrt {1+x} \, dx\\ &=-\left (\frac {1}{2} \int \left (\sqrt {-1+x}+(-1+x)^{3/2}\right ) \, dx\right )+\frac {1}{2} \int \left (-\sqrt {1+x}+(1+x)^{3/2}\right ) \, dx\\ &=-\frac {1}{3} (-1+x)^{3/2}-\frac {1}{5} (-1+x)^{5/2}-\frac {1}{3} (1+x)^{3/2}+\frac {1}{5} (1+x)^{5/2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.15, size = 34, normalized size = 0.76 \begin {gather*} \frac {1}{15} \left ((1+x)^{3/2} (-2+3 x)+\sqrt {-1+x} \left (2+x-3 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(Sqrt[-1 + x] + Sqrt[1 + x]),x]

[Out]

((1 + x)^(3/2)*(-2 + 3*x) + Sqrt[-1 + x]*(2 + x - 3*x^2))/15

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 30, normalized size = 0.67

method	result	size
default	\(-\frac {\left (x -1\right )^{\frac {3}{2}}}{3}-\frac {\left (x -1\right )^{\frac {5}{2}}}{5}-\frac {\left (x +1\right )^{\frac {3}{2}}}{3}+\frac {\left (x +1\right )^{\frac {5}{2}}}{5}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(x-1)^(3/2)-1/5*(x-1)^(5/2)-1/3*(x+1)^(3/2)+1/5*(x+1)^(5/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(sqrt(x + 1) + sqrt(x - 1)), x)

________________________________________________________________________________________

Fricas [A]
time = 0.57, size = 33, normalized size = 0.73 \begin {gather*} \frac {1}{15} \, {\left (3 \, x^{2} + x - 2\right )} \sqrt {x + 1} - \frac {1}{15} \, {\left (3 \, x^{2} - x - 2\right )} \sqrt {x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*x^2 + x - 2)*sqrt(x + 1) - 1/15*(3*x^2 - x - 2)*sqrt(x - 1)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x - 1} + \sqrt {x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.45, size = 33, normalized size = 0.73 \begin {gather*} \frac {1}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {1}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {1}{15} \, {\left ({\left (3 \, x - 4\right )} {\left (x + 1\right )} + 2\right )} \sqrt {x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(x + 1)^(5/2) - 1/3*(x + 1)^(3/2) - 1/15*((3*x - 4)*(x + 1) + 2)*sqrt(x - 1)

________________________________________________________________________________________

Mupad [B]
time = 0.16, size = 51, normalized size = 1.13 \begin {gather*} \frac {x\,\sqrt {x-1}}{15}+\frac {x\,\sqrt {x+1}}{15}+\frac {2\,\sqrt {x-1}}{15}-\frac {2\,\sqrt {x+1}}{15}-\frac {x^2\,\sqrt {x-1}}{5}+\frac {x^2\,\sqrt {x+1}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((x - 1)^(1/2) + (x + 1)^(1/2)),x)

[Out]

(x*(x - 1)^(1/2))/15 + (x*(x + 1)^(1/2))/15 + (2*(x - 1)^(1/2))/15 - (2*(x + 1)^(1/2))/15 - (x^2*(x - 1)^(1/2)
)/5 + (x^2*(x + 1)^(1/2))/5

________________________________________________________________________________________

Chatgpt [F] Failed to verify
time = 1.00, size = 15, normalized size = 0.33 \begin {gather*} \frac {4 \left (x -1\right )^{\frac {3}{2}}}{3}-2 \sqrt {x -1} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(x/((x-1)^(1/2)+(x+1)^(1/2)),x)

[Out]

4/3*(x-1)^(3/2)-2*(x-1)^(1/2)

________________________________________________________________________________________