3.649 \(\int \frac{1}{\sqrt{a+b \sqrt{c+d x}}} \, dx\)

Optimal. Leaf size=54 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d}-\frac{4 a \sqrt{a+b \sqrt{c+d x}}}{b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.0314209, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {247, 190, 43} \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d}-\frac{4 a \sqrt{a+b \sqrt{c+d x}}}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

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

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

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

Mathematica [A]  time = 0.0227483, size = 42, normalized size = 0.78 \[ \frac{4 \left (b \sqrt{c+d x}-2 a\right ) \sqrt{a+b \sqrt{c+d x}}}{3 b^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

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

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Maple [A]  time = 0.005, size = 41, normalized size = 0.8 \begin{align*} 4\,{\frac{1/3\, \left ( a+b\sqrt{dx+c} \right ) ^{3/2}-a\sqrt{a+b\sqrt{dx+c}}}{{b}^{2}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.997243, size = 57, normalized size = 1.06 \begin{align*} \frac{4 \,{\left (\frac{{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}}}{b^{2}} - \frac{3 \, \sqrt{\sqrt{d x + c} b + a} a}{b^{2}}\right )}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.29351, size = 85, normalized size = 1.57 \begin{align*} \frac{4 \, \sqrt{\sqrt{d x + c} b + a}{\left (\sqrt{d x + c} b - 2 \, a\right )}}{3 \, b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b \sqrt{c + d x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*sqrt(c + d*x)), x)

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Giac [B]  time = 1.22179, size = 135, normalized size = 2.5 \begin{align*} \frac{4 \,{\left (\sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )} \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 3 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}} a \mathrm{sgn}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )}}{3 \, b^{2} d{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)*sgn((sqrt(d*x + c)*b + a)*b - a*b) - 3*sqrt((sqrt(d
*x + c)*b + a)*b^2)*a*sgn((sqrt(d*x + c)*b + a)*b - a*b))/(b^2*d*abs(b))