3.2235 \(\int \sqrt{a+b \sqrt{x}} \, dx\)

Optimal. Leaf size=42 \[ \frac{4 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^2}-\frac{4 a \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2} \]

[Out]

(-4*a*(a + b*Sqrt[x])^(3/2))/(3*b^2) + (4*(a + b*Sqrt[x])^(5/2))/(5*b^2)

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Rubi [A]  time = 0.0169726, antiderivative size = 42, 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} \[ \frac{4 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^2}-\frac{4 a \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[x]],x]

[Out]

(-4*a*(a + b*Sqrt[x])^(3/2))/(3*b^2) + (4*(a + b*Sqrt[x])^(5/2))/(5*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 \sqrt{a+b \sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int x \sqrt{a+b x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{4 a \left (a+b \sqrt{x}\right )^{3/2}}{3 b^2}+\frac{4 \left (a+b \sqrt{x}\right )^{5/2}}{5 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0159094, size = 32, normalized size = 0.76 \[ \frac{4 \left (a+b \sqrt{x}\right )^{3/2} \left (3 b \sqrt{x}-2 a\right )}{15 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sqrt[x]],x]

[Out]

(4*(a + b*Sqrt[x])^(3/2)*(-2*a + 3*b*Sqrt[x]))/(15*b^2)

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Maple [A]  time = 0.003, size = 30, normalized size = 0.7 \begin{align*} 4\,{\frac{1/5\, \left ( a+b\sqrt{x} \right ) ^{5/2}-1/3\,a \left ( a+b\sqrt{x} \right ) ^{3/2}}{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/b^2*(1/5*(a+b*x^(1/2))^(5/2)-1/3*a*(a+b*x^(1/2))^(3/2))

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Maxima [A]  time = 0.973247, size = 41, normalized size = 0.98 \begin{align*} \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}}}{5 \, b^{2}} - \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(b*sqrt(x) + a)^(5/2)/b^2 - 4/3*(b*sqrt(x) + a)^(3/2)*a/b^2

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Fricas [A]  time = 1.33174, size = 84, normalized size = 2. \begin{align*} \frac{4 \,{\left (3 \, b^{2} x + a b \sqrt{x} - 2 \, a^{2}\right )} \sqrt{b \sqrt{x} + a}}{15 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/15*(3*b^2*x + a*b*sqrt(x) - 2*a^2)*sqrt(b*sqrt(x) + a)/b^2

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Sympy [B]  time = 1.43063, size = 272, normalized size = 6.48 \begin{align*} - \frac{8 a^{\frac{9}{2}} x^{2} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{9}{2}} x^{2}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} - \frac{4 a^{\frac{7}{2}} b x^{\frac{5}{2}} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{8 a^{\frac{7}{2}} b x^{\frac{5}{2}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{16 a^{\frac{5}{2}} b^{2} x^{3} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} + \frac{12 a^{\frac{3}{2}} b^{3} x^{\frac{7}{2}} \sqrt{1 + \frac{b \sqrt{x}}{a}}}{15 a^{2} b^{2} x^{2} + 15 a b^{3} x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*a**(9/2)*x**2*sqrt(1 + b*sqrt(x)/a)/(15*a**2*b**2*x**2 + 15*a*b**3*x**(5/2)) + 8*a**(9/2)*x**2/(15*a**2*b**
2*x**2 + 15*a*b**3*x**(5/2)) - 4*a**(7/2)*b*x**(5/2)*sqrt(1 + b*sqrt(x)/a)/(15*a**2*b**2*x**2 + 15*a*b**3*x**(
5/2)) + 8*a**(7/2)*b*x**(5/2)/(15*a**2*b**2*x**2 + 15*a*b**3*x**(5/2)) + 16*a**(5/2)*b**2*x**3*sqrt(1 + b*sqrt
(x)/a)/(15*a**2*b**2*x**2 + 15*a*b**3*x**(5/2)) + 12*a**(3/2)*b**3*x**(7/2)*sqrt(1 + b*sqrt(x)/a)/(15*a**2*b**
2*x**2 + 15*a*b**3*x**(5/2))

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Giac [A]  time = 1.09986, size = 39, normalized size = 0.93 \begin{align*} \frac{4 \,{\left (3 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a\right )}}{15 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/15*(3*(b*sqrt(x) + a)^(5/2) - 5*(b*sqrt(x) + a)^(3/2)*a)/b^2