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

Optimal. Leaf size=21 \[ -\frac{2}{b d \sqrt{d (a+b x)+c}} \]

[Out]

-2/(b*d*Sqrt[c + d*(a + b*x)])

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Rubi [A]  time = 0.0105183, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {33, 32} \[ -\frac{2}{b d \sqrt{d (a+b x)+c}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*(a + b*x))^(-3/2),x]

[Out]

-2/(b*d*Sqrt[c + d*(a + b*x)])

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, 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}{(c+d (a+b x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(c+d x)^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{2}{b d \sqrt{c+d (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0130418, size = 21, normalized size = 1. \[ -\frac{2}{b d \sqrt{d (a+b x)+c}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*(a + b*x))^(-3/2),x]

[Out]

-2/(b*d*Sqrt[c + d*(a + b*x)])

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Maple [A]  time = 0.001, size = 20, normalized size = 1. \begin{align*} -2\,{\frac{1}{\sqrt{bdx+ad+c}bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09262, size = 26, normalized size = 1.24 \begin{align*} -\frac{2}{\sqrt{{\left (b x + a\right )} d + c} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt((b*x + a)*d + c)*b*d)

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Fricas [A]  time = 1.48526, size = 76, normalized size = 3.62 \begin{align*} -\frac{2 \, \sqrt{b d x + a d + c}}{b^{2} d^{2} x + a b d^{2} + b c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.39265, size = 58, normalized size = 2.76 \begin{align*} \begin{cases} \frac{x}{c^{\frac{3}{2}}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x}{\left (a d + c\right )^{\frac{3}{2}}} & \text{for}\: b = 0 \\\frac{x}{c^{\frac{3}{2}}} & \text{for}\: d = 0 \\- \frac{2 \sqrt{a d + b d x + c}}{a b d^{2} + b^{2} d^{2} x + b c d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/c**(3/2), Eq(b, 0) & Eq(d, 0)), (x/(a*d + c)**(3/2), Eq(b, 0)), (x/c**(3/2), Eq(d, 0)), (-2*sqrt(
a*d + b*d*x + c)/(a*b*d**2 + b**2*d**2*x + b*c*d), True))

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Giac [A]  time = 1.16669, size = 26, normalized size = 1.24 \begin{align*} -\frac{2}{\sqrt{{\left (b x + a\right )} d + c} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt((b*x + a)*d + c)*b*d)