3.355 \(\int \frac{1}{(a+b x)^{5/2}} \, dx\)

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

[Out]

-2/(3*b*(a + b*x)^(3/2))

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Rubi [A]  time = 0.0014716, 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} \[ -\frac{2}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-5/2),x]

[Out]

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

Mathematica [A]  time = 0.0045605, size = 16, normalized size = 1. \[ -\frac{2}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-5/2),x]

[Out]

-2/(3*b*(a + b*x)^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(5/2),x)

[Out]

-2/3/b/(b*x+a)^(3/2)

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Maxima [A]  time = 1.08387, size = 16, normalized size = 1. \begin{align*} -\frac{2}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/((b*x + a)^(3/2)*b)

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Fricas [B]  time = 1.53132, size = 68, normalized size = 4.25 \begin{align*} -\frac{2 \, \sqrt{b x + a}}{3 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(b*x + a)/(b^3*x^2 + 2*a*b^2*x + a^2*b)

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Sympy [A]  time = 0.066401, size = 14, normalized size = 0.88 \begin{align*} - \frac{2}{3 b \left (a + b x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(5/2),x)

[Out]

-2/(3*b*(a + b*x)**(3/2))

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Giac [A]  time = 1.17457, size = 16, normalized size = 1. \begin{align*} -\frac{2}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/((b*x + a)^(3/2)*b)