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 = 0.0015836, 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} \[ -\frac{2}{d \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(-3/2),x]

[Out]

-2/(d*Sqrt[c + d*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}{(c+d x)^{3/2}} \, dx &=-\frac{2}{d \sqrt{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.0037861, 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 = 0.002, size = 13, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{d\sqrt{dx+c}}} \end{align*}

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 = 0.931832, size = 16, normalized size = 1.14 \begin{align*} -\frac{2}{\sqrt{d x + c} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(d*x + c)*d)

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Fricas [A]  time = 2.10794, size = 43, normalized size = 3.07 \begin{align*} -\frac{2 \, \sqrt{d x + c}}{d^{2} x + c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(d*x + c)/(d^2*x + c*d)

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Sympy [A]  time = 0.056988, size = 12, normalized size = 0.86 \begin{align*} - \frac{2}{d \sqrt{c + d x}} \end{align*}

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 [A]  time = 1.06052, size = 16, normalized size = 1.14 \begin{align*} -\frac{2}{\sqrt{d x + c} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(d*x + c)*d)