3.10.96 \(\int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx\) [996]

Optimal. Leaf size=66 \[ \frac {1}{2} \left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x+\sqrt {x} \sqrt {1+x}}-\frac {3 \sin ^{-1}\left (\sqrt {x}-\sqrt {1+x}\right )}{2 \sqrt {2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [F]
time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[-x + Sqrt[x]*Sqrt[1 + x]]/Sqrt[1 + x],x]

[Out]

2*Defer[Subst][Defer[Int][Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]], x], x, Sqrt[1 + x]]

Rubi steps

\begin {align*} \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx &=2 \text {Subst}\left (\int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx,x,\sqrt {1+x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.64, size = 91, normalized size = 1.38 \begin {gather*} \frac {1}{4} \left (2 \left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x+\sqrt {x} \sqrt {1+x}}-3 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {-2 x+2 \sqrt {x} \sqrt {1+x}}}{-\sqrt {x}+\sqrt {1+x}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(Sqrt[x] + 3*Sqrt[1 + x])*Sqrt[-x + Sqrt[x]*Sqrt[1 + x]] - 3*Sqrt[2]*ArcTan[Sqrt[-2*x + 2*Sqrt[x]*Sqrt[1 +
x]]/(-Sqrt[x] + Sqrt[1 + x])])/4

________________________________________________________________________________________

Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {-x +\sqrt {x}\, \sqrt {1+x}}}{\sqrt {1+x}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (46) = 92\).
time = 190.19, size = 127, normalized size = 1.92 \begin {gather*} \frac {1}{2} \, \sqrt {\sqrt {x + 1} \sqrt {x} - x} {\left (3 \, \sqrt {x + 1} + \sqrt {x}\right )} + \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {2 \, {\left (13230146471646497941753920 \, \sqrt {2} {\left (4 \, x + 1\right )} \sqrt {x + 1} \sqrt {x} + 472818412040 \, \sqrt {2} {\left (111925814517792 \, x^{2} + 83944360888344 \, x + 621904154881\right )} + 3497681703681 \, {\left (\sqrt {2} {\left (50987674019848 \, x - 621904154881\right )} \sqrt {x + 1} + \sqrt {2} {\left (60938140497944 \, x - 24871932855043\right )} \sqrt {x}\right )} \sqrt {\sqrt {x + 1} \sqrt {x} - x}\right )}}{139214258174109988596285504 \, x^{2} - 678551053586366160645570576 \, x + 386764777858250836124161}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(sqrt(x + 1)*sqrt(x) - x)*(3*sqrt(x + 1) + sqrt(x)) + 3/8*sqrt(2)*arctan(2*(13230146471646497941753920
*sqrt(2)*(4*x + 1)*sqrt(x + 1)*sqrt(x) + 472818412040*sqrt(2)*(111925814517792*x^2 + 83944360888344*x + 621904
154881) + 3497681703681*(sqrt(2)*(50987674019848*x - 621904154881)*sqrt(x + 1) + sqrt(2)*(60938140497944*x - 2
4871932855043)*sqrt(x))*sqrt(sqrt(x + 1)*sqrt(x) - x))/(139214258174109988596285504*x^2 - 67855105358636616064
5570576*x + 386764777858250836124161))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {\sqrt {x}\,\sqrt {x+1}-x}}{\sqrt {x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________