3.40 \(\int \frac{1}{\sqrt{c+d (a+b x)}} \, dx\)

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

[Out]

(2*Sqrt[c + d*(a + b*x)])/(b*d)

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Rubi [A]  time = 0.0103886, 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} \[ \frac{2 \sqrt{d (a+b x)+c}}{b d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c + d*(a + b*x)],x]

[Out]

(2*Sqrt[c + d*(a + b*x)])/(b*d)

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}{\sqrt{c+d (a+b x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x}} \, dx,x,a+b x\right )}{b}\\ &=\frac{2 \sqrt{c+d (a+b x)}}{b d}\\ \end{align*}

Mathematica [A]  time = 0.0127891, size = 21, normalized size = 1. \[ \frac{2 \sqrt{d (a+b x)+c}}{b d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c + d*(a + b*x)],x]

[Out]

(2*Sqrt[c + d*(a + b*x)])/(b*d)

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Maple [A]  time = 0.002, size = 20, normalized size = 1. \begin{align*} 2\,{\frac{\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))^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.03585, 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))^(1/2),x, algorithm="maxima")

[Out]

2*sqrt((b*x + a)*d + c)/(b*d)

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Fricas [A]  time = 1.49795, size = 42, normalized size = 2. \begin{align*} \frac{2 \, \sqrt{b d x + a d + c}}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*d*x + a*d + c)/(b*d)

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Sympy [A]  time = 1.19525, size = 31, normalized size = 1.48 \begin{align*} \begin{cases} \frac{x}{\sqrt{a d + c}} & \text{for}\: b = 0 \\\frac{x}{\sqrt{c}} & \text{for}\: d = 0 \\\frac{2 \sqrt{c + d \left (a + b x\right )}}{b 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))**(1/2),x)

[Out]

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

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Giac [A]  time = 1.13198, size = 26, normalized size = 1.24 \begin{align*} \frac{2 \, \sqrt{b d x + a d + c}}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*d*x + a*d + c)/(b*d)