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]

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

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.41, size = 34, normalized size = 0.79 \[ \frac {1}{4} \, {\left (2 \, x + 1\right )} \sqrt {x + 1} \sqrt {x} + \frac {1}{8} \, \log \left (2 \, \sqrt {x + 1} \sqrt {x} - 2 \, x - 1\right ) \]

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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giac [A]  time = 1.03, size = 39, normalized size = 0.91 \[ \frac {1}{4} \, {\left (2 \, x - 3\right )} \sqrt {x + 1} \sqrt {x} + \sqrt {x + 1} \sqrt {x} + \frac {1}{4} \, \log \left (\sqrt {x + 1} - \sqrt {x}\right ) \]

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

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maple [A]  time = 0.00, size = 50, normalized size = 1.16 \[ \frac {\left (x +1\right )^{\frac {3}{2}} \sqrt {x}}{2}-\frac {\sqrt {x +1}\, \sqrt {x}}{4}-\frac {\sqrt {\left (x +1\right ) x}\, \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{8 \sqrt {x +1}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.42, size = 71, normalized size = 1.65 \[ \frac {\frac {{\left (x + 1\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}} + \frac {\sqrt {x + 1}}{\sqrt {x}}}{4 \, {\left (\frac {{\left (x + 1\right )}^{2}}{x^{2}} - \frac {2 \, {\left (x + 1\right )}}{x} + 1\right )}} - \frac {1}{8} \, \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} + 1\right ) + \frac {1}{8} \, \log \left (\frac {\sqrt {x + 1}}{\sqrt {x}} - 1\right ) \]

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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mupad [B]  time = 0.22, size = 30, normalized size = 0.70 \[ \sqrt {x}\,\left (\frac {x}{2}+\frac {1}{4}\right )\,\sqrt {x+1}-\frac {\ln \left (x+\sqrt {x}\,\sqrt {x+1}+\frac {1}{2}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(x + 1)^(1/2),x)

[Out]

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

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sympy [A]  time = 2.57, size = 119, normalized size = 2.77 \[ \begin {cases} - \frac {\operatorname {acosh}{\left (\sqrt {x + 1} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x}} - \frac {3 \left (x + 1\right )^{\frac {3}{2}}}{4 \sqrt {x}} + \frac {\sqrt {x + 1}}{4 \sqrt {x}} & \text {for}\: \left |{x + 1}\right | > 1 \\\frac {i \operatorname {asin}{\left (\sqrt {x + 1} \right )}}{4} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {- x}} + \frac {3 i \left (x + 1\right )^{\frac {3}{2}}}{4 \sqrt {- x}} - \frac {i \sqrt {x + 1}}{4 \sqrt {- x}} & \text {otherwise} \end {cases} \]

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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