3.2241 \(\int \frac{1}{\sqrt{a+b \sqrt{x}}} \, dx\)

Optimal. Leaf size=40 \[ \frac{4 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2}-\frac{4 a \sqrt{a+b \sqrt{x}}}{b^2} \]

[Out]

(-4*a*Sqrt[a + b*Sqrt[x]])/b^2 + (4*(a + b*Sqrt[x])^(3/2))/(3*b^2)

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Rubi [A]  time = 0.0170122, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {190, 43} \[ \frac{4 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2}-\frac{4 a \sqrt{a+b \sqrt{x}}}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Sqrt[x]],x]

[Out]

(-4*a*Sqrt[a + b*Sqrt[x]])/b^2 + (4*(a + b*Sqrt[x])^(3/2))/(3*b^2)

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0134767, size = 31, normalized size = 0.78 \[ \frac{4 \left (b \sqrt{x}-2 a\right ) \sqrt{a+b \sqrt{x}}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Sqrt[x]],x]

[Out]

(4*(-2*a + b*Sqrt[x])*Sqrt[a + b*Sqrt[x]])/(3*b^2)

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Maple [A]  time = 0.003, size = 30, normalized size = 0.8 \begin{align*} 4\,{\frac{1/3\, \left ( a+b\sqrt{x} \right ) ^{3/2}-a\sqrt{a+b\sqrt{x}}}{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.948876, size = 41, normalized size = 1.02 \begin{align*} \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}}}{3 \, b^{2}} - \frac{4 \, \sqrt{b \sqrt{x} + a} a}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(b*sqrt(x) + a)^(3/2)/b^2 - 4*sqrt(b*sqrt(x) + a)*a/b^2

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Fricas [A]  time = 1.31325, size = 63, normalized size = 1.58 \begin{align*} \frac{4 \, \sqrt{b \sqrt{x} + a}{\left (b \sqrt{x} - 2 \, a\right )}}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*sqrt(b*sqrt(x) + a)*(b*sqrt(x) - 2*a)/b^2

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Sympy [B]  time = 0.98405, size = 219, normalized size = 5.48 \begin{align*} - \frac{8 a^{\frac{7}{2}} x^{2} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{7}{2}} x^{2}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} - \frac{4 a^{\frac{5}{2}} b x^{\frac{5}{2}} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{5}{2}} b x^{\frac{5}{2}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} + \frac{4 a^{\frac{3}{2}} b^{2} x^{3} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*a**(7/2)*x**2*sqrt(1 + b*sqrt(x)/a)/(3*a**2*b**2*x**2 + 3*a*b**3*x**(5/2)) + 8*a**(7/2)*x**2/(3*a**2*b**2*x
**2 + 3*a*b**3*x**(5/2)) - 4*a**(5/2)*b*x**(5/2)*sqrt(1 + b*sqrt(x)/a)/(3*a**2*b**2*x**2 + 3*a*b**3*x**(5/2))
+ 8*a**(5/2)*b*x**(5/2)/(3*a**2*b**2*x**2 + 3*a*b**3*x**(5/2)) + 4*a**(3/2)*b**2*x**3*sqrt(1 + b*sqrt(x)/a)/(3
*a**2*b**2*x**2 + 3*a*b**3*x**(5/2))

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Giac [A]  time = 1.11293, size = 36, normalized size = 0.9 \begin{align*} \frac{4 \,{\left ({\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b \sqrt{x} + a} a\right )}}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*((b*sqrt(x) + a)^(3/2) - 3*sqrt(b*sqrt(x) + a)*a)/b^2