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

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 = 0.0688968, 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 \[ \frac{4 \left (a+b \sqrt{c+d x}\right )^{3/2}}{3 b^2 d}-\frac{4 a \sqrt{a+b \sqrt{c+d x}}}{b^2 d} \]

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)

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Rubi in Sympy [A]  time = 4.02939, size = 46, normalized size = 0.85 \[ - \frac{4 a \sqrt{a + b \sqrt{c + d x}}}{b^{2} d} + \frac{4 \left (a + b \sqrt{c + d x}\right )^{\frac{3}{2}}}{3 b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [A]  time = 0.0247305, 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 = 0.008, size = 41, normalized size = 0.8 \[ 4\,{\frac{1/3\, \left ( a+b\sqrt{dx+c} \right ) ^{3/2}-\sqrt{a+b\sqrt{dx+c}}a}{{b}^{2}d}} \]

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

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Maxima [A]  time = 0.691078, size = 57, normalized size = 1.06 \[ \frac{4 \,{\left (\frac{{\left (\sqrt{d x + c} b + a\right )}^{\frac{3}{2}}}{b^{2}} - \frac{3 \, \sqrt{\sqrt{d x + c} b + a} a}{b^{2}}\right )}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(sqrt(d*x + c)*b + a),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 = 0.333463, size = 46, normalized size = 0.85 \[ \frac{4 \, \sqrt{\sqrt{d x + c} b + a}{\left (\sqrt{d x + c} b - 2 \, a\right )}}{3 \, b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(sqrt(d*x + c)*b + a),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. \[ \int \frac{1}{\sqrt{a + b \sqrt{c + d x}}}\, dx \]

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)

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GIAC/XCAS [A]  time = 0.276814, size = 135, normalized size = 2.5 \[ \frac{4 \,{\left (\sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}}{\left (\sqrt{d x + c} b + a\right )}{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right ) - 3 \, \sqrt{{\left (\sqrt{d x + c} b + a\right )} b^{2}} a{\rm sign}\left ({\left (\sqrt{d x + c} b + a\right )} b - a b\right )\right )}}{3 \, b^{2} d{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(sqrt(d*x + c)*b + a),x, algorithm="giac")

[Out]

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