3.15.29 \(\int \frac {1}{(c+d x)^{3/2}} \, dx\) [1429]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2}{d \sqrt {c+d x}} \end {gather*}

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*}

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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {2}{d \sqrt {c+d x}} \end {gather*}

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 = 0.14, size = 13, normalized size = 0.93

method	result	size
gosper	\(-\frac {2}{d \sqrt {d x +c}}\)	\(13\)
derivativedivides	\(-\frac {2}{d \sqrt {d x +c}}\)	\(13\)
default	\(-\frac {2}{d \sqrt {d x +c}}\)	\(13\)
trager	\(-\frac {2}{d \sqrt {d x +c}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2}{\sqrt {d x + c} d} \end {gather*}

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 = 1.09, size = 20, normalized size = 1.43 \begin {gather*} -\frac {2 \, \sqrt {d x + c}}{d^{2} x + c d} \end {gather*}

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 = 0.01, size = 12, normalized size = 0.86 \begin {gather*} - \frac {2}{d \sqrt {c + d x}} \end {gather*}

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 = 1.26, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2}{\sqrt {d x + c} d} \end {gather*}

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)

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Mupad [B]
time = 0.02, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2}{d\,\sqrt {c+d\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c + d*x)^(3/2),x)

[Out]

-2/(d*(c + d*x)^(1/2))

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