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

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

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Rubi [A]  time = 0.00, antiderivative size = 16, 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}{3 d (c+d x)^{3/2}} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2}{3 d (c+d x)^{3/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2}{3 d (c+d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c + d*x)^(-5/2),x]

[Out]

-2/(3*d*(c + d*x)^(3/2))

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fricas [B]  time = 1.32, size = 31, normalized size = 1.94 \begin {gather*} -\frac {2 \, \sqrt {d x + c}}{3 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end {gather*}

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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giac [A]  time = 0.96, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \end {gather*}

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)

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \begin {gather*} -\frac {2}{3 \left (d x +c \right )^{\frac {3}{2}} d} \end {gather*}

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 = 1.36, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d} \end {gather*}

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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mupad [B]  time = 0.03, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2}{3\,d\,{\left (c+d\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c + d*x)^(5/2),x)

[Out]

-2/(3*d*(c + d*x)^(3/2))

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sympy [A]  time = 0.07, size = 14, normalized size = 0.88 \begin {gather*} - \frac {2}{3 d \left (c + d x\right )^{\frac {3}{2}}} \end {gather*}

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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