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]

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

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 1.18, size = 23, normalized size = 0.58 \[ \frac {4 \, \sqrt {b \sqrt {x} + a} {\left (b \sqrt {x} - 2 \, a\right )}}{3 \, b^{2}} \]

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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giac [A]  time = 0.16, size = 27, normalized size = 0.68 \[ \frac {4 \, {\left ({\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b \sqrt {x} + a} a\right )}}{3 \, b^{2}} \]

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

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maple [A]  time = 0.00, size = 30, normalized size = 0.75 \[ \frac {-4 \sqrt {b \sqrt {x}+a}\, a +\frac {4 \left (b \sqrt {x}+a \right )^{\frac {3}{2}}}{3}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.81, size = 30, normalized size = 0.75 \[ \frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}}}{3 \, b^{2}} - \frac {4 \, \sqrt {b \sqrt {x} + a} a}{b^{2}} \]

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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mupad [B]  time = 1.20, size = 37, normalized size = 0.92 \[ \frac {x\,\sqrt {\frac {b\,\sqrt {x}}{a}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},2;\ 3;\ -\frac {b\,\sqrt {x}}{a}\right )}{\sqrt {a+b\,\sqrt {x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^(1/2))^(1/2),x)

[Out]

(x*((b*x^(1/2))/a + 1)^(1/2)*hypergeom([1/2, 2], 3, -(b*x^(1/2))/a))/(a + b*x^(1/2))^(1/2)

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sympy [B]  time = 1.19, size = 219, normalized size = 5.48 \[ - \frac {8 a^{\frac {7}{2}} x^{2} \sqrt {1 + \frac {b \sqrt {x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac {5}{2}}} + \frac {8 a^{\frac {7}{2}} x^{2}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac {5}{2}}} - \frac {4 a^{\frac {5}{2}} b x^{\frac {5}{2}} \sqrt {1 + \frac {b \sqrt {x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac {5}{2}}} + \frac {8 a^{\frac {5}{2}} b x^{\frac {5}{2}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac {5}{2}}} + \frac {4 a^{\frac {3}{2}} b^{2} x^{3} \sqrt {1 + \frac {b \sqrt {x}}{a}}}{3 a^{2} b^{2} x^{2} + 3 a b^{3} x^{\frac {5}{2}}} \]

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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