3.719 \(\int \sqrt{1+\sqrt{1+\sqrt{-1+x}}} x \, dx\)

Optimal. Leaf size=160 \[ \frac{8}{17} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{17/2}-\frac{56}{15} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{15/2}+\frac{144}{13} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{13/2}-\frac{160}{11} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{11/2}+8 \left (\sqrt{\sqrt{x-1}+1}+1\right )^{9/2}-\frac{24}{7} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{7/2}+\frac{16}{5} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{5/2} \]

[Out]

(16*(1 + Sqrt[1 + Sqrt[-1 + x]])^(5/2))/5 - (24*(1 + Sqrt[1 + Sqrt[-1 + x]])^(7/2))/7 + 8*(1 + Sqrt[1 + Sqrt[-
1 + x]])^(9/2) - (160*(1 + Sqrt[1 + Sqrt[-1 + x]])^(11/2))/11 + (144*(1 + Sqrt[1 + Sqrt[-1 + x]])^(13/2))/13 -
(56*(1 + Sqrt[1 + Sqrt[-1 + x]])^(15/2))/15 + (8*(1 + Sqrt[1 + Sqrt[-1 + x]])^(17/2))/17

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Rubi [A]  time = 0.275713, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {1618, 1620} \[ \frac{8}{17} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{17/2}-\frac{56}{15} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{15/2}+\frac{144}{13} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{13/2}-\frac{160}{11} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{11/2}+8 \left (\sqrt{\sqrt{x-1}+1}+1\right )^{9/2}-\frac{24}{7} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{7/2}+\frac{16}{5} \left (\sqrt{\sqrt{x-1}+1}+1\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Sqrt[1 + Sqrt[-1 + x]]]*x,x]

[Out]

(16*(1 + Sqrt[1 + Sqrt[-1 + x]])^(5/2))/5 - (24*(1 + Sqrt[1 + Sqrt[-1 + x]])^(7/2))/7 + 8*(1 + Sqrt[1 + Sqrt[-
1 + x]])^(9/2) - (160*(1 + Sqrt[1 + Sqrt[-1 + x]])^(11/2))/11 + (144*(1 + Sqrt[1 + Sqrt[-1 + x]])^(13/2))/13 -
(56*(1 + Sqrt[1 + Sqrt[-1 + x]])^(15/2))/15 + (8*(1 + Sqrt[1 + Sqrt[-1 + x]])^(17/2))/17

Rule 1618

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[PolynomialQuotient[Px, a + b
*x, x]*(a + b*x)^(m + 1)*(c + d*x)^n, x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && EqQ[PolynomialRema
inder[Px, a + b*x, x], 0]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \sqrt{1+\sqrt{1+\sqrt{-1+x}}} x \, dx &=2 \operatorname{Subst}\left (\int x \left (1+x^2\right ) \sqrt{1+\sqrt{1+x}} \, dx,x,\sqrt{-1+x}\right )\\ &=4 \operatorname{Subst}\left (\int x \sqrt{1+x} \left (-1+x^2\right ) \left (1+\left (-1+x^2\right )^2\right ) \, dx,x,\sqrt{1+\sqrt{-1+x}}\right )\\ &=4 \operatorname{Subst}\left (\int x (1+x)^{3/2} \left (-2+2 x+2 x^2-2 x^3-x^4+x^5\right ) \, dx,x,\sqrt{1+\sqrt{-1+x}}\right )\\ &=4 \operatorname{Subst}\left (\int \left (2 (1+x)^{3/2}-3 (1+x)^{5/2}+9 (1+x)^{7/2}-20 (1+x)^{9/2}+18 (1+x)^{11/2}-7 (1+x)^{13/2}+(1+x)^{15/2}\right ) \, dx,x,\sqrt{1+\sqrt{-1+x}}\right )\\ &=\frac{16}{5} \left (1+\sqrt{1+\sqrt{-1+x}}\right )^{5/2}-\frac{24}{7} \left (1+\sqrt{1+\sqrt{-1+x}}\right )^{7/2}+8 \left (1+\sqrt{1+\sqrt{-1+x}}\right )^{9/2}-\frac{160}{11} \left (1+\sqrt{1+\sqrt{-1+x}}\right )^{11/2}+\frac{144}{13} \left (1+\sqrt{1+\sqrt{-1+x}}\right )^{13/2}-\frac{56}{15} \left (1+\sqrt{1+\sqrt{-1+x}}\right )^{15/2}+\frac{8}{17} \left (1+\sqrt{1+\sqrt{-1+x}}\right )^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0896872, size = 103, normalized size = 0.64 \[ \frac{8 \left (\sqrt{\sqrt{x-1}+1}+1\right )^{5/2} \left (8 \left (84 \sqrt{x-1} \sqrt{\sqrt{x-1}+1}-3030 \sqrt{\sqrt{x-1}+1}+1715 \sqrt{x-1}+2591\right )+77 \left (-377 \sqrt{\sqrt{x-1}+1}+195 \sqrt{x-1}+365\right ) x\right )}{255255} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sqrt[1 + Sqrt[-1 + x]]]*x,x]

[Out]

(8*(1 + Sqrt[1 + Sqrt[-1 + x]])^(5/2)*(8*(2591 - 3030*Sqrt[1 + Sqrt[-1 + x]] + 1715*Sqrt[-1 + x] + 84*Sqrt[1 +
Sqrt[-1 + x]]*Sqrt[-1 + x]) + 77*(365 - 377*Sqrt[1 + Sqrt[-1 + x]] + 195*Sqrt[-1 + x])*x))/255255

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Maple [A]  time = 0.006, size = 107, normalized size = 0.7 \begin{align*}{\frac{16}{5} \left ( 1+\sqrt{1+\sqrt{x-1}} \right ) ^{{\frac{5}{2}}}}-{\frac{24}{7} \left ( 1+\sqrt{1+\sqrt{x-1}} \right ) ^{{\frac{7}{2}}}}+8\, \left ( 1+\sqrt{1+\sqrt{x-1}} \right ) ^{9/2}-{\frac{160}{11} \left ( 1+\sqrt{1+\sqrt{x-1}} \right ) ^{{\frac{11}{2}}}}+{\frac{144}{13} \left ( 1+\sqrt{1+\sqrt{x-1}} \right ) ^{{\frac{13}{2}}}}-{\frac{56}{15} \left ( 1+\sqrt{1+\sqrt{x-1}} \right ) ^{{\frac{15}{2}}}}+{\frac{8}{17} \left ( 1+\sqrt{1+\sqrt{x-1}} \right ) ^{{\frac{17}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16/5*(1+(1+(x-1)^(1/2))^(1/2))^(5/2)-24/7*(1+(1+(x-1)^(1/2))^(1/2))^(7/2)+8*(1+(1+(x-1)^(1/2))^(1/2))^(9/2)-16
0/11*(1+(1+(x-1)^(1/2))^(1/2))^(11/2)+144/13*(1+(1+(x-1)^(1/2))^(1/2))^(13/2)-56/15*(1+(1+(x-1)^(1/2))^(1/2))^
(15/2)+8/17*(1+(1+(x-1)^(1/2))^(1/2))^(17/2)

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Maxima [A]  time = 1.00873, size = 143, normalized size = 0.89 \begin{align*} \frac{8}{17} \,{\left (\sqrt{\sqrt{x - 1} + 1} + 1\right )}^{\frac{17}{2}} - \frac{56}{15} \,{\left (\sqrt{\sqrt{x - 1} + 1} + 1\right )}^{\frac{15}{2}} + \frac{144}{13} \,{\left (\sqrt{\sqrt{x - 1} + 1} + 1\right )}^{\frac{13}{2}} - \frac{160}{11} \,{\left (\sqrt{\sqrt{x - 1} + 1} + 1\right )}^{\frac{11}{2}} + 8 \,{\left (\sqrt{\sqrt{x - 1} + 1} + 1\right )}^{\frac{9}{2}} - \frac{24}{7} \,{\left (\sqrt{\sqrt{x - 1} + 1} + 1\right )}^{\frac{7}{2}} + \frac{16}{5} \,{\left (\sqrt{\sqrt{x - 1} + 1} + 1\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/17*(sqrt(sqrt(x - 1) + 1) + 1)^(17/2) - 56/15*(sqrt(sqrt(x - 1) + 1) + 1)^(15/2) + 144/13*(sqrt(sqrt(x - 1)
+ 1) + 1)^(13/2) - 160/11*(sqrt(sqrt(x - 1) + 1) + 1)^(11/2) + 8*(sqrt(sqrt(x - 1) + 1) + 1)^(9/2) - 24/7*(sqr
t(sqrt(x - 1) + 1) + 1)^(7/2) + 16/5*(sqrt(sqrt(x - 1) + 1) + 1)^(5/2)

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Fricas [A]  time = 1.51213, size = 228, normalized size = 1.42 \begin{align*} \frac{8}{255255} \,{\left (15015 \, x^{2} +{\left (77 \, x + 1032\right )} \sqrt{x - 1} +{\left ({\left (1001 \, x + 4544\right )} \sqrt{x - 1} - 1176 \, x - 7696\right )} \sqrt{\sqrt{x - 1} + 1} - 1799 \, x - 22088\right )} \sqrt{\sqrt{\sqrt{x - 1} + 1} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/255255*(15015*x^2 + (77*x + 1032)*sqrt(x - 1) + ((1001*x + 4544)*sqrt(x - 1) - 1176*x - 7696)*sqrt(sqrt(x -
1) + 1) - 1799*x - 22088)*sqrt(sqrt(sqrt(x - 1) + 1) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\sqrt{\sqrt{x - 1} + 1} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(sqrt(sqrt(x - 1) + 1) + 1), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError