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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.02, 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)

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 34, normalized size = 0.63 \[ \frac {4 \, \sqrt {\sqrt {d x + c} b + a} {\left (\sqrt {d x + c} b - 2 \, a\right )}}{3 \, b^{2} d} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.44, size = 38, normalized size = 0.70 \[ \frac {4 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {\sqrt {d x + c} b + a} a\right )}}{3 \, b^{2} d} \]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 41, normalized size = 0.76 \[ \frac {-4 \sqrt {a +\sqrt {d x +c}\, b}\, a +\frac {4 \left (a +\sqrt {d x +c}\, b \right )^{\frac {3}{2}}}{3}}{b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.86, size = 42, normalized size = 0.78 \[ \frac {4 \, {\left (\frac {{\left (\sqrt {d x + c} b + a\right )}^{\frac {3}{2}}}{b^{2}} - \frac {3 \, \sqrt {\sqrt {d x + c} b + a} a}{b^{2}}\right )}}{3 \, d} \]

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

________________________________________________________________________________________

mupad [B]  time = 3.26, size = 44, normalized size = 0.81 \[ \frac {4\,{\left (a+b\,\sqrt {c+d\,x}\right )}^{3/2}}{3\,b^2\,d}-\frac {4\,a\,\sqrt {a+b\,\sqrt {c+d\,x}}}{b^2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \sqrt {c + d x}}}\, dx \]

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)

________________________________________________________________________________________