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

Optimal. Leaf size=43 \[ \frac{1}{2} \sqrt{x+1} x^{3/2}+\frac{1}{4} \sqrt{x+1} \sqrt{x}-\frac{1}{4} \sinh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0054499, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {50, 54, 215} \[ \frac{1}{2} \sqrt{x+1} x^{3/2}+\frac{1}{4} \sqrt{x+1} \sqrt{x}-\frac{1}{4} \sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*Sqrt[1 + x],x]

[Out]

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0117258, size = 31, normalized size = 0.72 \[ \frac{1}{4} \left (\sqrt{x} \sqrt{x+1} (2 x+1)-\sinh ^{-1}\left (\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*Sqrt[1 + x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 50, normalized size = 1.2 \begin{align*}{\frac{1}{2}\sqrt{x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{1}{4}\sqrt{x}\sqrt{1+x}}-{\frac{1}{8}\sqrt{x \left ( 1+x \right ) }\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^(1/2)*(1+x)^(3/2)-1/4*x^(1/2)*(1+x)^(1/2)-1/8*(x*(1+x))^(1/2)/(1+x)^(1/2)/x^(1/2)*ln(1/2+x+(x^2+x)^(1/2)
)

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Maxima [B]  time = 0.946795, size = 96, normalized size = 2.23 \begin{align*} \frac{\frac{{\left (x + 1\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}} + \frac{\sqrt{x + 1}}{\sqrt{x}}}{4 \,{\left (\frac{{\left (x + 1\right )}^{2}}{x^{2}} - \frac{2 \,{\left (x + 1\right )}}{x} + 1\right )}} - \frac{1}{8} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} + 1\right ) + \frac{1}{8} \, \log \left (\frac{\sqrt{x + 1}}{\sqrt{x}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((x + 1)^(3/2)/x^(3/2) + sqrt(x + 1)/sqrt(x))/((x + 1)^2/x^2 - 2*(x + 1)/x + 1) - 1/8*log(sqrt(x + 1)/sqrt
(x) + 1) + 1/8*log(sqrt(x + 1)/sqrt(x) - 1)

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Fricas [A]  time = 1.71554, size = 105, normalized size = 2.44 \begin{align*} \frac{1}{4} \,{\left (2 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} + \frac{1}{8} \, \log \left (2 \, \sqrt{x + 1} \sqrt{x} - 2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*x + 1)*sqrt(x + 1)*sqrt(x) + 1/8*log(2*sqrt(x + 1)*sqrt(x) - 2*x - 1)

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Sympy [A]  time = 2.59574, size = 119, normalized size = 2.77 \begin{align*} \begin{cases} - \frac{\operatorname{acosh}{\left (\sqrt{x + 1} \right )}}{4} + \frac{\left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{x}} - \frac{3 \left (x + 1\right )^{\frac{3}{2}}}{4 \sqrt{x}} + \frac{\sqrt{x + 1}}{4 \sqrt{x}} & \text{for}\: \left |{x + 1}\right | > 1 \\\frac{i \operatorname{asin}{\left (\sqrt{x + 1} \right )}}{4} - \frac{i \left (x + 1\right )^{\frac{5}{2}}}{2 \sqrt{- x}} + \frac{3 i \left (x + 1\right )^{\frac{3}{2}}}{4 \sqrt{- x}} - \frac{i \sqrt{x + 1}}{4 \sqrt{- x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-acosh(sqrt(x + 1))/4 + (x + 1)**(5/2)/(2*sqrt(x)) - 3*(x + 1)**(3/2)/(4*sqrt(x)) + sqrt(x + 1)/(4*
sqrt(x)), Abs(x + 1) > 1), (I*asin(sqrt(x + 1))/4 - I*(x + 1)**(5/2)/(2*sqrt(-x)) + 3*I*(x + 1)**(3/2)/(4*sqrt
(-x)) - I*sqrt(x + 1)/(4*sqrt(-x)), True))

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Giac [A]  time = 1.1477, size = 42, normalized size = 0.98 \begin{align*} \frac{1}{4} \,{\left (2 \, x + 1\right )} \sqrt{x + 1} \sqrt{x} + \frac{1}{4} \, \log \left ({\left | -\sqrt{x + 1} + \sqrt{x} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*x + 1)*sqrt(x + 1)*sqrt(x) + 1/4*log(abs(-sqrt(x + 1) + sqrt(x)))