3.4.47 \(\int \frac {1}{(a+b x)^{3/2}} \, dx\)

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

________________________________________________________________________________________

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}{b \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-3/2),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {2}{b \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-3/2),x]

[Out]

-2/(b*Sqrt[a + b*x])

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(-3/2),x]

[Out]

-2/(b*Sqrt[a + b*x])

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 20, normalized size = 1.43 \begin {gather*} -\frac {2 \, \sqrt {b x + a}}{b^{2} x + a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2*sqrt(b*x + a)/(b^2*x + a*b)

________________________________________________________________________________________

giac [A]  time = 1.24, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2}{\sqrt {b x + a} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

-2/(sqrt(b*x + a)*b)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 13, normalized size = 0.93 \begin {gather*} -\frac {2}{\sqrt {b x +a}\, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(3/2),x)

[Out]

-2/b/(b*x+a)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 12, normalized size = 0.86 \begin {gather*} -\frac {2}{\sqrt {b x + a} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

-2/(sqrt(b*x + a)*b)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^(3/2),x)

[Out]

-2/(b*(a + b*x)^(1/2))

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 12, normalized size = 0.86 \begin {gather*} - \frac {2}{b \sqrt {a + b x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**(3/2),x)

[Out]

-2/(b*sqrt(a + b*x))

________________________________________________________________________________________