3.970 \(\int \sqrt{x} \sqrt{\sqrt{x}+x} \, dx\)

Optimal. Leaf size=82 \[ \frac{1}{2} \sqrt{x} \left (x+\sqrt{x}\right )^{3/2}-\frac{5}{12} \left (x+\sqrt{x}\right )^{3/2}+\frac{5}{32} \left (2 \sqrt{x}+1\right ) \sqrt{x+\sqrt{x}}-\frac{5}{32} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]

[Out]

(5*(1 + 2*Sqrt[x])*Sqrt[Sqrt[x] + x])/32 - (5*(Sqrt[x] + x)^(3/2))/12 + (Sqrt[x]*(Sqrt[x] + x)^(3/2))/2 - (5*A
rcTanh[Sqrt[x]/Sqrt[Sqrt[x] + x]])/32

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Rubi [A]  time = 0.0436464, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {2018, 670, 640, 612, 620, 206} \[ \frac{1}{2} \sqrt{x} \left (x+\sqrt{x}\right )^{3/2}-\frac{5}{12} \left (x+\sqrt{x}\right )^{3/2}+\frac{5}{32} \left (2 \sqrt{x}+1\right ) \sqrt{x+\sqrt{x}}-\frac{5}{32} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+\sqrt{x}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*Sqrt[Sqrt[x] + x],x]

[Out]

(5*(1 + 2*Sqrt[x])*Sqrt[Sqrt[x] + x])/32 - (5*(Sqrt[x] + x)^(3/2))/12 + (Sqrt[x]*(Sqrt[x] + x)^(3/2))/2 - (5*A
rcTanh[Sqrt[x]/Sqrt[Sqrt[x] + x]])/32

Rule 2018

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)
/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 640

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

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{x} \sqrt{\sqrt{x}+x} \, dx &=2 \operatorname{Subst}\left (\int x^2 \sqrt{x+x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{2} \sqrt{x} \left (\sqrt{x}+x\right )^{3/2}-\frac{5}{4} \operatorname{Subst}\left (\int x \sqrt{x+x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{5}{12} \left (\sqrt{x}+x\right )^{3/2}+\frac{1}{2} \sqrt{x} \left (\sqrt{x}+x\right )^{3/2}+\frac{5}{8} \operatorname{Subst}\left (\int \sqrt{x+x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{5}{32} \left (1+2 \sqrt{x}\right ) \sqrt{\sqrt{x}+x}-\frac{5}{12} \left (\sqrt{x}+x\right )^{3/2}+\frac{1}{2} \sqrt{x} \left (\sqrt{x}+x\right )^{3/2}-\frac{5}{64} \operatorname{Subst}\left (\int \frac{1}{\sqrt{x+x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{5}{32} \left (1+2 \sqrt{x}\right ) \sqrt{\sqrt{x}+x}-\frac{5}{12} \left (\sqrt{x}+x\right )^{3/2}+\frac{1}{2} \sqrt{x} \left (\sqrt{x}+x\right )^{3/2}-\frac{5}{32} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{\sqrt{x}+x}}\right )\\ &=\frac{5}{32} \left (1+2 \sqrt{x}\right ) \sqrt{\sqrt{x}+x}-\frac{5}{12} \left (\sqrt{x}+x\right )^{3/2}+\frac{1}{2} \sqrt{x} \left (\sqrt{x}+x\right )^{3/2}-\frac{5}{32} \tanh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{\sqrt{x}+x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0651486, size = 58, normalized size = 0.71 \[ \frac{1}{96} \sqrt{x+\sqrt{x}} \left (48 x^{3/2}+8 x-10 \sqrt{x}-\frac{15 \sinh ^{-1}\left (\sqrt [4]{x}\right )}{\sqrt{\sqrt{x}+1} \sqrt [4]{x}}+15\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Sqrt[x] + x]*(15 - 10*Sqrt[x] + 8*x + 48*x^(3/2) - (15*ArcSinh[x^(1/4)])/(Sqrt[1 + Sqrt[x]]*x^(1/4))))/9
6

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Maple [A]  time = 0.003, size = 54, normalized size = 0.7 \begin{align*}{\frac{1}{2}\sqrt{x} \left ( x+\sqrt{x} \right ) ^{{\frac{3}{2}}}}-{\frac{5}{12} \left ( x+\sqrt{x} \right ) ^{{\frac{3}{2}}}}+{\frac{5}{32} \left ( 1+2\,\sqrt{x} \right ) \sqrt{x+\sqrt{x}}}-{\frac{5}{64}\ln \left ( \sqrt{x}+{\frac{1}{2}}+\sqrt{x+\sqrt{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^(1/2)*(x+x^(1/2))^(3/2)-5/12*(x+x^(1/2))^(3/2)+5/32*(1+2*x^(1/2))*(x+x^(1/2))^(1/2)-5/64*ln(x^(1/2)+1/2+
(x+x^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x + \sqrt{x}} \sqrt{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x + sqrt(x))*sqrt(x), x)

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Fricas [A]  time = 4.61749, size = 174, normalized size = 2.12 \begin{align*} \frac{1}{96} \,{\left (2 \,{\left (24 \, x - 5\right )} \sqrt{x} + 8 \, x + 15\right )} \sqrt{x + \sqrt{x}} + \frac{5}{128} \, \log \left (4 \, \sqrt{x + \sqrt{x}}{\left (2 \, \sqrt{x} + 1\right )} - 8 \, x - 8 \, \sqrt{x} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(2*(24*x - 5)*sqrt(x) + 8*x + 15)*sqrt(x + sqrt(x)) + 5/128*log(4*sqrt(x + sqrt(x))*(2*sqrt(x) + 1) - 8*x
- 8*sqrt(x) - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError