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

Optimal. Leaf size=109 \[ \frac {3 \sqrt {x^2+x} \sqrt {x \left (\sqrt {x^2+x}+x\right )}}{2 x}+\sqrt {x \left (\sqrt {x^2+x}+x\right )} \left (-\frac {3 \sqrt {\sqrt {x^2+x}-x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2+x}-x}\right )}{2 \sqrt {2} x}-\frac {1}{2}\right ) \]

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Rubi [F]  time = 0.77, 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}}}{\sqrt {x+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[x^2 + x*Sqrt[x + x^2]]/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]]/Sqrt[1 + x^2], x], x, Sqrt[x]])
/Sqrt[x + x^2]

Rubi steps

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

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Mathematica [A]  time = 0.36, size = 140, normalized size = 1.28 \begin {gather*} \frac {(x+1) \sqrt {x \left (x+\sqrt {x (x+1)}\right )} \left (2 \sqrt {x+\sqrt {x (x+1)}} \left (2 x+2 \sqrt {x (x+1)}+3\right )-3 \sqrt {4 x+4 \sqrt {x (x+1)}+2} \sinh ^{-1}\left (\sqrt {2} \sqrt {x+\sqrt {x (x+1)}}\right )\right )}{4 \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]]/Sqrt[x + x^2],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 4.38, size = 109, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt {x+x^2} \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{2 x}+\sqrt {x \left (x+\sqrt {x+x^2}\right )} \left (-\frac {1}{2}-\frac {3 \sqrt {-x+\sqrt {x+x^2}} \tanh ^{-1}\left (\sqrt {2} \sqrt {-x+\sqrt {x+x^2}}\right )}{2 \sqrt {2} x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.60, size = 96, normalized size = 0.88 \begin {gather*} \frac {3 \, \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 - 3 \, \sqrt {x^{2} + x}\right )}}{8 \, 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^2+x)^(1/2),x, algorithm="fricas")

[Out]

1/8*(3*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 - 3*sqrt(x^2 + 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}}\,{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^2+x)^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^2+x*(x^2+x)^(1/2))^(1/2)/(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}}\,{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^2+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^2 + sqrt(x^2 + x)*x)/sqrt(x^2 + 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}}}{\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^2)^(1/2),x)

[Out]

int((x^2 + x*(x + x^2)^(1/2))^(1/2)/(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 )}}{\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**2+x)**(1/2),x)

[Out]

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

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