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

Optimal. Leaf size=68 \[ \frac {2 \sqrt {2} \sqrt {\sqrt {x^2+x}-x} \sqrt {x \left (\sqrt {x^2+x}+x\right )} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+x}-x}\right )}{x} \]

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Rubi [F]  time = 1.92, 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^2+x \sqrt {x+x^2}}}{x \sqrt {x+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt {x^2+x \sqrt {x+x^2}}}{x \sqrt {x+x^2}} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x}\right ) \int \frac {\sqrt {x^2+x \sqrt {x+x^2}}}{x^{3/2} \sqrt {1+x}} \, dx}{\sqrt {x+x^2}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x^4+x^2 \sqrt {x^2+x^4}}}{x^2 \sqrt {1+x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.47, size = 107, normalized size = 1.57 \begin {gather*} \frac {2 \sqrt {2} (x+1) \sqrt {x \left (x+\sqrt {x (x+1)}\right )} \sqrt {2 x+2 \sqrt {x (x+1)}+1} \sinh ^{-1}\left (\sqrt {2} \sqrt {x+\sqrt {x (x+1)}}\right )}{\sqrt {x (x+1)} \sqrt {x+\sqrt {x (x+1)}} \left (x+\sqrt {x (x+1)}+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[2]*(1 + x)*Sqrt[x*(x + Sqrt[x*(1 + x)])]*Sqrt[1 + 2*x + 2*Sqrt[x*(1 + x)]]*ArcSinh[Sqrt[2]*Sqrt[x + Sq
rt[x*(1 + x)]]])/(Sqrt[x*(1 + x)]*Sqrt[x + Sqrt[x*(1 + x)]]*(1 + x + Sqrt[x*(1 + x)]))

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IntegrateAlgebraic [A]  time = 4.04, size = 68, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {2} \sqrt {-x+\sqrt {x+x^2}} \sqrt {x \left (x+\sqrt {x+x^2}\right )} \tanh ^{-1}\left (\sqrt {2} \sqrt {-x+\sqrt {x+x^2}}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 60, normalized size = 0.88 \begin {gather*} \sqrt {2} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+x \sqrt {x^{2}+x}}}{x \sqrt {x^{2}+x}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^2+x*(x^2+x)^(1/2))^(1/2)/x/(x^2+x)^(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} + \sqrt {x^{2} + x} x}}{\sqrt {x^{2} + x} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^2 + sqrt(x^2 + x)*x)/(sqrt(x^2 + 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\,\sqrt {x^2+x}}}{x\,\sqrt {x^2+x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^2 + x*(x + x^2)^(1/2))^(1/2)/(x*(x + x^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 + \sqrt {x^{2} + x}\right )}}{x \sqrt {x \left (x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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