3.17.2 \(\int \frac {\sqrt {x+x^2}}{x \sqrt {x^2+x \sqrt {x+x^2}}} \, dx\)

Optimal. Leaf size=109 \[ \sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (\frac {\sqrt {2} \sqrt {\sqrt {x^2+x}-x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+x}-x}\right )}{x}+\frac {2 (x-2)}{x}\right )-\frac {2 \sqrt {x^2+x} \sqrt {x \left (\sqrt {x^2+x}+x\right )}}{x} \]

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Rubi [F]  time = 1.23, 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+x^2}}{x \sqrt {x^2+x \sqrt {x+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[x + x^2]/(x*Sqrt[x^2 + x*Sqrt[x + x^2]]),x]

[Out]

(2*Sqrt[x + x^2]*Defer[Subst][Defer[Int][Sqrt[1 + x^2]/Sqrt[x^4 + x^2*Sqrt[x^2 + x^4]], x], x, Sqrt[x]])/(Sqrt
[x]*Sqrt[1 + x])

Rubi steps

\begin {align*} \int \frac {\sqrt {x+x^2}}{x \sqrt {x^2+x \sqrt {x+x^2}}} \, dx &=\frac {\sqrt {x+x^2} \int \frac {\sqrt {1+x}}{\sqrt {x} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx}{\sqrt {x} \sqrt {1+x}}\\ &=\frac {\left (2 \sqrt {x+x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {x^4+x^2 \sqrt {x^2+x^4}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x}}\\ \end {align*}

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Mathematica [C]  time = 0.48, size = 97, normalized size = 0.89 \begin {gather*} -\frac {2 \sqrt {x (x+1)} \left (\left (x+\sqrt {x (x+1)}\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {1}{2 \left (x+\sqrt {x (x+1)}\right )}\right )+4 x+4 \sqrt {x (x+1)}+2\right )}{\sqrt {x \left (x+\sqrt {x (x+1)}\right )} \left (x+\sqrt {x (x+1)}+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x + x^2]/(x*Sqrt[x^2 + x*Sqrt[x + x^2]]),x]

[Out]

(-2*Sqrt[x*(1 + x)]*(2 + 4*x + 4*Sqrt[x*(1 + x)] + (x + Sqrt[x*(1 + x)])*Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1
/(2*(x + Sqrt[x*(1 + x)]))]))/(Sqrt[x*(x + Sqrt[x*(1 + x)])]*(1 + x + Sqrt[x*(1 + x)]))

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IntegrateAlgebraic [A]  time = 4.09, size = 109, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x+x^2} \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{x}+\sqrt {x \left (x+\sqrt {x+x^2}\right )} \left (\frac {2 (-2+x)}{x}+\frac {\sqrt {2} \sqrt {-x+\sqrt {x+x^2}} \tanh ^{-1}\left (\sqrt {2} \sqrt {-x+\sqrt {x+x^2}}\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[x + x^2]/(x*Sqrt[x^2 + x*Sqrt[x + x^2]]),x]

[Out]

(-2*Sqrt[x + x^2]*Sqrt[x*(x + Sqrt[x + x^2])])/x + Sqrt[x*(x + Sqrt[x + x^2])]*((2*(-2 + x))/x + (Sqrt[2]*Sqrt
[-x + Sqrt[x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[-x + Sqrt[x + x^2]]])/x)

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fricas [A]  time = 0.60, size = 96, normalized size = 0.88 \begin {gather*} \frac {\sqrt {2} x \log \left (\frac {4 \, x^{2} + 2 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + x}\right )} + 4 \, \sqrt {x^{2} + x} x + x}{x}\right ) + 4 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (x - \sqrt {x^{2} + x} - 2\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(2)*x*log((4*x^2 + 2*sqrt(x^2 + sqrt(x^2 + x)*x)*(sqrt(2)*x + sqrt(2)*sqrt(x^2 + x)) + 4*sqrt(x^2 + x
)*x + x)/x) + 4*sqrt(x^2 + sqrt(x^2 + x)*x)*(x - sqrt(x^2 + x) - 2))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + x}}{\sqrt {x^{2} + \sqrt {x^{2} + x} x} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^2 + x)/(sqrt(x^2 + sqrt(x^2 + x)*x)*x), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+x}}{x \sqrt {x^{2}+x \sqrt {x^{2}+x}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + x}}{\sqrt {x^{2} + \sqrt {x^{2} + x} x} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^2 + x)/(sqrt(x^2 + sqrt(x^2 + x)*x)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x^2+x}}{x\,\sqrt {x^2+x\,\sqrt {x^2+x}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x + 1\right )}}{x \sqrt {x \left (x + \sqrt {x^{2} + x}\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(x + 1))/(x*sqrt(x*(x + sqrt(x**2 + x)))), x)

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