3.18.77 \(\int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx\)

Optimal. Leaf size=119 \[ \sqrt {x+1} \left (\frac {2 (x+1)}{3}-\frac {4}{3} \sqrt {\sqrt {x+1}+1}\right )-\frac {4}{3} \sqrt {\sqrt {x+1}+1}-\frac {2}{5} \left (\sqrt {5}-5\right ) \log \left (-2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}-1\right )+\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right ) \]

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Rubi [A]  time = 0.69, antiderivative size = 127, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1628, 632, 31} \begin {gather*} \frac {2}{3} \left (\sqrt {x+1}+1\right )^3-2 \left (\sqrt {x+1}+1\right )^2-\frac {4}{3} \left (\sqrt {x+1}+1\right )^{3/2}+2 \sqrt {x+1}+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}-\sqrt {5}+1\right )+\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[1 + x] - (4*(1 + Sqrt[1 + x])^(3/2))/3 - 2*(1 + Sqrt[1 + x])^2 + (2*(1 + Sqrt[1 + x])^3)/3 + (2*(5 - Sq
rt[5])*Log[1 - Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/5 + (2*(5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 +
x]]])/5

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {x \sqrt {1+x}}{x+\sqrt {1+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )}{-1+x^2+\sqrt {1+x}} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {(-1+x) x^2 (1+x)^2 \left (-2+x^2\right )}{-1+x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (x-x^2-2 x^3+x^5+\frac {x}{-1+x+x^2}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}-\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3+4 \operatorname {Subst}\left (\int \frac {x}{-1+x+x^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}-\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3+\frac {1}{5} \left (2 \left (5-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+\frac {1}{5} \left (2 \left (5+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=2 \sqrt {1+x}-\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-2 \left (1+\sqrt {1+x}\right )^2+\frac {2}{3} \left (1+\sqrt {1+x}\right )^3+\frac {2}{5} \left (5-\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 128, normalized size = 1.08 \begin {gather*} \frac {1}{15} \left (10 \left (\sqrt {x+1} x+\sqrt {x+1}-2 \sqrt {x+1} \sqrt {\sqrt {x+1}+1}-2 \sqrt {\sqrt {x+1}+1}-2\right )-6 \left (\sqrt {5}-5\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}-\sqrt {5}+1\right )+6 \left (5+\sqrt {5}\right ) \log \left (2 \sqrt {\sqrt {x+1}+1}+\sqrt {5}+1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(-2 + Sqrt[1 + x] + x*Sqrt[1 + x] - 2*Sqrt[1 + Sqrt[1 + x]] - 2*Sqrt[1 + x]*Sqrt[1 + Sqrt[1 + x]]) - 6*(-5
+ Sqrt[5])*Log[1 - Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]] + 6*(5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 +
x]]])/15

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IntegrateAlgebraic [A]  time = 0.14, size = 99, normalized size = 0.83 \begin {gather*} -\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {2}{3} \left (1+(1+x)^{3/2}\right )-\frac {2}{5} \left (-5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 \sqrt {1+\sqrt {1+x}}\right )+\frac {2}{5} \left (5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+\sqrt {1+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(1 + Sqrt[1 + x])^(3/2))/3 + (2*(1 + (1 + x)^(3/2)))/3 - (2*(-5 + Sqrt[5])*Log[-1 + Sqrt[5] - 2*Sqrt[1 + S
qrt[1 + x]]])/5 + (2*(5 + Sqrt[5])*Log[1 + Sqrt[5] + 2*Sqrt[1 + Sqrt[1 + x]]])/5

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fricas [A]  time = 0.60, size = 130, normalized size = 1.09 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} + \frac {2}{5} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (3 \, x + 1\right )} - {\left (\sqrt {5} {\left (x + 2\right )} - 5 \, x\right )} \sqrt {x + 1} + {\left (\sqrt {5} {\left (x + 2\right )} + {\left (\sqrt {5} {\left (2 \, x - 1\right )} - 5\right )} \sqrt {x + 1} - 5 \, x\right )} \sqrt {\sqrt {x + 1} + 1} + 3 \, x + 3}{x^{2} - x - 1}\right ) + 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - 4/3*(sqrt(x + 1) + 1)^(3/2) + 2/5*sqrt(5)*log((2*x^2 - sqrt(5)*(3*x + 1) - (sqrt(5)*(x + 2
) - 5*x)*sqrt(x + 1) + (sqrt(5)*(x + 2) + (sqrt(5)*(2*x - 1) - 5)*sqrt(x + 1) - 5*x)*sqrt(sqrt(x + 1) + 1) + 3
*x + 3)/(x^2 - x - 1)) + 2*log(sqrt(x + 1) + sqrt(sqrt(x + 1) + 1))

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giac [A]  time = 0.16, size = 102, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{3} - 2 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} - \frac {2}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) + 2 \, \sqrt {x + 1} + 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + 2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(sqrt(x + 1) + 1)^3 - 2*(sqrt(x + 1) + 1)^2 - 4/3*(sqrt(x + 1) + 1)^(3/2) - 2/5*sqrt(5)*log(-(sqrt(5) - 2*
sqrt(sqrt(x + 1) + 1) - 1)/(sqrt(5) + 2*sqrt(sqrt(x + 1) + 1) + 1)) + 2*sqrt(x + 1) + 2*log(sqrt(x + 1) + sqrt
(sqrt(x + 1) + 1)) + 2

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maple [A]  time = 0.08, size = 85, normalized size = 0.71

method	result	size
derivativedivides	\(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+2 \sqrt {1+x}+2+2 \ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )+\frac {4 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}\)	\(85\)
default	\(\frac {2 \left (1+\sqrt {1+x}\right )^{3}}{3}-2 \left (1+\sqrt {1+x}\right )^{2}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+2 \sqrt {1+x}+2+2 \ln \left (\sqrt {1+x}+\sqrt {1+\sqrt {1+x}}\right )+\frac {4 \sqrt {5}\, \arctanh \left (\frac {\left (2 \sqrt {1+\sqrt {1+x}}+1\right ) \sqrt {5}}{5}\right )}{5}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(1+x)^(1/2)/(x+(1+(1+x)^(1/2))^(1/2)),x,method=_RETURNVERBOSE)

[Out]

2/3*(1+(1+x)^(1/2))^3-2*(1+(1+x)^(1/2))^2-4/3*(1+(1+x)^(1/2))^(3/2)+2*(1+x)^(1/2)+2+2*ln((1+x)^(1/2)+(1+(1+x)^
(1/2))^(1/2))+4/5*5^(1/2)*arctanh(1/5*(2*(1+(1+x)^(1/2))^(1/2)+1)*5^(1/2))

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maxima [A]  time = 0.86, size = 102, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{3} - 2 \, {\left (\sqrt {x + 1} + 1\right )}^{2} - \frac {4}{3} \, {\left (\sqrt {x + 1} + 1\right )}^{\frac {3}{2}} - \frac {2}{5} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, \sqrt {\sqrt {x + 1} + 1} - 1}{\sqrt {5} + 2 \, \sqrt {\sqrt {x + 1} + 1} + 1}\right ) + 2 \, \sqrt {x + 1} + 2 \, \log \left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1}\right ) + 2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(sqrt(x + 1) + 1)^3 - 2*(sqrt(x + 1) + 1)^2 - 4/3*(sqrt(x + 1) + 1)^(3/2) - 2/5*sqrt(5)*log(-(sqrt(5) - 2*
sqrt(sqrt(x + 1) + 1) - 1)/(sqrt(5) + 2*sqrt(sqrt(x + 1) + 1) + 1)) + 2*sqrt(x + 1) + 2*log(sqrt(x + 1) + sqrt
(sqrt(x + 1) + 1)) + 2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 138.85, size = 175, normalized size = 1.47 \begin {gather*} 2 \sqrt {x + 1} - \frac {4 \left (\sqrt {x + 1} + 1\right )^{\frac {3}{2}}}{3} + \frac {2 \left (\sqrt {x + 1} + 1\right )^{3}}{3} - 2 \left (\sqrt {x + 1} + 1\right )^{2} - 8 \left (\begin {cases} - \frac {\sqrt {5} \operatorname {acoth}{\left (\frac {2 \sqrt {5} \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )^{2} > \frac {5}{4} \\- \frac {\sqrt {5} \operatorname {atanh}{\left (\frac {2 \sqrt {5} \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )}{5} \right )}}{10} & \text {for}\: \left (\sqrt {\sqrt {x + 1} + 1} + \frac {1}{2}\right )^{2} < \frac {5}{4} \end {cases}\right ) + 2 \log {\left (\sqrt {x + 1} + \sqrt {\sqrt {x + 1} + 1} \right )} + 2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1) - 4*(sqrt(x + 1) + 1)**(3/2)/3 + 2*(sqrt(x + 1) + 1)**3/3 - 2*(sqrt(x + 1) + 1)**2 - 8*Piecewise
((-sqrt(5)*acoth(2*sqrt(5)*(sqrt(sqrt(x + 1) + 1) + 1/2)/5)/10, (sqrt(sqrt(x + 1) + 1) + 1/2)**2 > 5/4), (-sqr
t(5)*atanh(2*sqrt(5)*(sqrt(sqrt(x + 1) + 1) + 1/2)/5)/10, (sqrt(sqrt(x + 1) + 1) + 1/2)**2 < 5/4)) + 2*log(sqr
t(x + 1) + sqrt(sqrt(x + 1) + 1)) + 2

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