3.347 \(\int \frac{1}{(a+b x)^{3/2}} \, dx\)

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

[Out]

-2/(b*Sqrt[a + b*x])

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Rubi [A]  time = 0.00670844, 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 \[ -\frac{2}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(-3/2),x]

[Out]

-2/(b*Sqrt[a + b*x])

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Rubi in Sympy [A]  time = 1.28241, size = 12, normalized size = 0.86 \[ - \frac{2}{b \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(3/2),x)

[Out]

-2/(b*sqrt(a + b*x))

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Mathematica [A]  time = 0.00421834, size = 14, normalized size = 1. \[ -\frac{2}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(-3/2),x]

[Out]

-2/(b*Sqrt[a + b*x])

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Maple [A]  time = 0.003, size = 13, normalized size = 0.9 \[ -2\,{\frac{1}{b\sqrt{bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^(3/2),x)

[Out]

-2/b/(b*x+a)^(1/2)

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Maxima [A]  time = 1.3345, size = 16, normalized size = 1.14 \[ -\frac{2}{\sqrt{b x + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(-3/2),x, algorithm="maxima")

[Out]

-2/(sqrt(b*x + a)*b)

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Fricas [A]  time = 0.216789, size = 16, normalized size = 1.14 \[ -\frac{2}{\sqrt{b x + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(-3/2),x, algorithm="fricas")

[Out]

-2/(sqrt(b*x + a)*b)

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Sympy [A]  time = 0.077346, size = 12, normalized size = 0.86 \[ - \frac{2}{b \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(3/2),x)

[Out]

-2/(b*sqrt(a + b*x))

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GIAC/XCAS [A]  time = 0.204708, size = 16, normalized size = 1.14 \[ -\frac{2}{\sqrt{b x + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(-3/2),x, algorithm="giac")

[Out]

-2/(sqrt(b*x + a)*b)