3.9.41 \(\int \frac {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx\)

Optimal. Leaf size=63 \[ \frac {4 \sqrt {x \left (\sqrt {x^2+x}+x\right )} (4 x-3)}{15 x^2}+\frac {16 \sqrt {x^2+x} \sqrt {x \left (\sqrt {x^2+x}+x\right )}}{15 x^2} \]

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Rubi [F]  time = 1.71, 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 {1}{x \sqrt {x+x^2} \sqrt {x^2+x \sqrt {x+x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.59, size = 87, normalized size = 1.38 \begin {gather*} \frac {4 \sqrt {x \left (x+\sqrt {x (x+1)}\right )} \left (x+\sqrt {x (x+1)}+1\right ) \left (2 x+2 \sqrt {x (x+1)}+1\right ) \left (4 x+4 \sqrt {x (x+1)}-3\right )}{15 \sqrt {x (x+1)} \left (x+\sqrt {x (x+1)}\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[x*(x + Sqrt[x*(1 + x)])]*(1 + x + Sqrt[x*(1 + x)])*(1 + 2*x + 2*Sqrt[x*(1 + x)])*(-3 + 4*x + 4*Sqrt[x*
(1 + x)]))/(15*Sqrt[x*(1 + x)]*(x + Sqrt[x*(1 + x)])^3)

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IntegrateAlgebraic [A]  time = 3.87, size = 63, normalized size = 1.00 \begin {gather*} \frac {4 (-3+4 x) \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{15 x^2}+\frac {16 \sqrt {x+x^2} \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{15 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-3 + 4*x)*Sqrt[x*(x + Sqrt[x + x^2])])/(15*x^2) + (16*Sqrt[x + x^2]*Sqrt[x*(x + Sqrt[x + x^2])])/(15*x^2)

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fricas [A]  time = 0.45, size = 34, normalized size = 0.54 \begin {gather*} \frac {4 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (4 \, x + 4 \, \sqrt {x^{2} + x} - 3\right )}}{15 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\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(1/x/(x^2+x)^(1/2)/(x^2+x*(x^2+x)^(1/2))^(1/2),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{x \sqrt {x^{2}+x}\, \sqrt {x^{2}+x \sqrt {x^{2}+x}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/(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 = -0.02 \begin {gather*} \int \frac {1}{x\,\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(1/(x*(x^2 + x*(x + x^2)^(1/2))^(1/2)*(x + x^2)^(1/2)),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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