3.1418 \(\int \frac{1}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=14 \[ \frac{2 \sqrt{c+d x}}{d} \]

[Out]

(2*Sqrt[c + d*x])/d

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Rubi [A]  time = 0.00709146, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2 \sqrt{c+d x}}{d} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x])/d

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Rubi in Sympy [A]  time = 1.36657, size = 10, normalized size = 0.71 \[ \frac{2 \sqrt{c + d x}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c + d*x)/d

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Mathematica [A]  time = 0.00289841, size = 14, normalized size = 1. \[ \frac{2 \sqrt{c+d x}}{d} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x])/d

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Maple [A]  time = 0.003, size = 13, normalized size = 0.9 \[ 2\,{\frac{\sqrt{dx+c}}{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x+c)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.38129, size = 16, normalized size = 1.14 \[ \frac{2 \, \sqrt{d x + c}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(d*x + c)/d

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Fricas [A]  time = 0.217544, size = 16, normalized size = 1.14 \[ \frac{2 \, \sqrt{d x + c}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(d*x + c)/d

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Sympy [A]  time = 0.033776, size = 10, normalized size = 0.71 \[ \frac{2 \sqrt{c + d x}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x+c)**(1/2),x)

[Out]

2*sqrt(c + d*x)/d

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GIAC/XCAS [A]  time = 0.219224, size = 16, normalized size = 1.14 \[ \frac{2 \, \sqrt{d x + c}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(d*x + c)/d