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 = 0.0013755, 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, Rules used = {32} \[ -\frac{2}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-3/2),x]

[Out]

-2/(b*Sqrt[a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^{3/2}} \, dx &=-\frac{2}{b \sqrt{a+b x}}\\ \end{align*}

Mathematica [A]  time = 0.0048989, 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 \begin{align*} -2\,{\frac{1}{b\sqrt{bx+a}}} \end{align*}

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 = 1.0712, size = 16, normalized size = 1.14 \begin{align*} -\frac{2}{\sqrt{b x + a} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

-2/(sqrt(b*x + a)*b)

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Fricas [A]  time = 1.51328, size = 43, normalized size = 3.07 \begin{align*} -\frac{2 \, \sqrt{b x + a}}{b^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2*sqrt(b*x + a)/(b^2*x + a*b)

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Sympy [A]  time = 0.0651, size = 12, normalized size = 0.86 \begin{align*} - \frac{2}{b \sqrt{a + b x}} \end{align*}

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 [A]  time = 1.20343, size = 16, normalized size = 1.14 \begin{align*} -\frac{2}{\sqrt{b x + a} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2/(sqrt(b*x + a)*b)