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/(b*x+a)^(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} \[ -\frac {2}{3 b (a+b x)^{3/2}} \]

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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ -\frac {2}{3 b (a+b x)^{3/2}} \]

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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fricas [B]  time = 0.48, size = 31, normalized size = 1.94 \[ -\frac {2 \, \sqrt {b x + a}}{3 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]

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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giac [A]  time = 1.04, size = 12, normalized size = 0.75 \[ -\frac {2}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b} \]

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)

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \[ -\frac {2}{3 \left (b x +a \right )^{\frac {3}{2}} b} \]

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 = 1.31, size = 12, normalized size = 0.75 \[ -\frac {2}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b} \]

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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mupad [B]  time = 0.02, size = 12, normalized size = 0.75 \[ -\frac {2}{3\,b\,{\left (a+b\,x\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^(5/2),x)

[Out]

-2/(3*b*(a + b*x)^(3/2))

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sympy [A]  time = 0.07, size = 14, normalized size = 0.88 \[ - \frac {2}{3 b \left (a + b x\right )^{\frac {3}{2}}} \]

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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