3.1439 \(\int \frac{1}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{2}{3 d (c+d x)^{3/2}} \]

[Out]

-2/(3*d*(c + d*x)^(3/2))

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Rubi [A]  time = 0.0013823, antiderivative size = 16, 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} \[ -\frac{2}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(-5/2),x]

[Out]

-2/(3*d*(c + d*x)^(3/2))

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)^{5/2}} \, dx &=-\frac{2}{3 d (c+d x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0067318, size = 16, normalized size = 1. \[ -\frac{2}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(-5/2),x]

[Out]

-2/(3*d*(c + d*x)^(3/2))

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Maple [A]  time = 0.002, size = 13, normalized size = 0.8 \begin{align*} -{\frac{2}{3\,d} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^(5/2),x)

[Out]

-2/3/d/(d*x+c)^(3/2)

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Maxima [A]  time = 0.959591, size = 16, normalized size = 1. \begin{align*} -\frac{2}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

-2/3/((d*x + c)^(3/2)*d)

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Fricas [B]  time = 2.04605, size = 68, normalized size = 4.25 \begin{align*} -\frac{2 \, \sqrt{d x + c}}{3 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(d*x + c)/(d^3*x^2 + 2*c*d^2*x + c^2*d)

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Sympy [A]  time = 0.06058, size = 14, normalized size = 0.88 \begin{align*} - \frac{2}{3 d \left (c + d x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**(5/2),x)

[Out]

-2/(3*d*(c + d*x)**(3/2))

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Giac [A]  time = 1.08114, size = 16, normalized size = 1. \begin{align*} -\frac{2}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^(5/2),x, algorithm="giac")

[Out]

-2/3/((d*x + c)^(3/2)*d)