3.467 \(\int \sqrt{a+b \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=56 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^2 d}-\frac{4 a \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0679516, antiderivative size = 56, 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 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{5/2}}{5 b^2 d}-\frac{4 a \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c + d*x]],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.01873, size = 48, normalized size = 0.86 \[ - \frac{4 a \left (a + b \sqrt{c + d x}\right )^{\frac{3}{2}}}{3 b^{2} d} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{5}{2}}}{5 b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(d*x+c)**(1/2))**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0493423, size = 55, normalized size = 0.98 \[ \frac{4 \sqrt{a+b \sqrt{c+d x}} \left (-2 a^2+a b \sqrt{c+d x}+3 b^2 (c+d x)\right )}{15 b^2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 41, normalized size = 0.7 \[ 4\,{\frac{1/5\, \left ( a+b\sqrt{dx+c} \right ) ^{5/2}-1/3\, \left ( a+b\sqrt{dx+c} \right ) ^{3/2}a}{{b}^{2}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.693783, size = 58, normalized size = 1.04 \[ \frac{4 \,{\left (\frac{3 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{15 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.341986, size = 68, normalized size = 1.21 \[ \frac{4 \,{\left (3 \, b^{2} d x + 3 \, b^{2} c + \sqrt{d x + c} a b - 2 \, a^{2}\right )} \sqrt{\sqrt{d x + c} b + a}}{15 \, b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b \sqrt{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.28698, size = 159, normalized size = 2.84 \[ \frac{4 \,{\left (3 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )}^{2} b^{2}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 5 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )} a b^{2}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )}{\left | b \right |}}{15 \, b^{6} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(d*x + c)*b + a),x, algorithm="giac")

[Out]

4/15*(3*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a)^2*b^2*sign((sqrt(d
*x + c)*b + a)*b - a*b) - 5*sqrt((sqrt(d*x + c)*b + a)*b^2)*(sqrt(d*x + c)*b + a
)*a*b^2*sign((sqrt(d*x + c)*b + a)*b - a*b))*abs(b)/(b^6*d)