3.1429 \(\int \frac{1}{(c+d x)^{3/2}} \, dx\)

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

[Out]

-2/(d*Sqrt[c + d*x])

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Rubi [A]  time = 0.00708826, 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 \[ -\frac{2}{d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(-3/2),x]

[Out]

-2/(d*Sqrt[c + d*x])

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Rubi in Sympy [A]  time = 1.35269, size = 12, normalized size = 0.86 \[ - \frac{2}{d \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x+c)**(3/2),x)

[Out]

-2/(d*sqrt(c + d*x))

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Mathematica [A]  time = 0.00438601, size = 14, normalized size = 1. \[ -\frac{2}{d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(-3/2),x]

[Out]

-2/(d*Sqrt[c + d*x])

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Maple [A]  time = 0.003, size = 13, normalized size = 0.9 \[ -2\,{\frac{1}{d\sqrt{dx+c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x+c)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.34711, 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((d*x + c)^(-3/2),x, algorithm="maxima")

[Out]

-2/(sqrt(d*x + c)*d)

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Fricas [A]  time = 0.206818, 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((d*x + c)^(-3/2),x, algorithm="fricas")

[Out]

-2/(sqrt(d*x + c)*d)

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Sympy [A]  time = 0.039206, size = 12, normalized size = 0.86 \[ - \frac{2}{d \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x+c)**(3/2),x)

[Out]

-2/(d*sqrt(c + d*x))

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GIAC/XCAS [A]  time = 0.21695, 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((d*x + c)^(-3/2),x, algorithm="giac")

[Out]

-2/(sqrt(d*x + c)*d)