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

Optimal. Leaf size=41 \[ \frac{2 \sqrt{c+d x}}{b d}-\frac{2 a \log \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]

[Out]

(2*Sqrt[c + d*x])/(b*d) - (2*a*Log[a + b*Sqrt[c + d*x]])/(b^2*d)

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Rubi [A]  time = 0.0241772, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {247, 190, 43} \[ \frac{2 \sqrt{c+d x}}{b d}-\frac{2 a \log \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[c + d*x])^(-1),x]

[Out]

(2*Sqrt[c + d*x])/(b*d) - (2*a*Log[a + b*Sqrt[c + d*x]])/(b^2*d)

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0188489, size = 39, normalized size = 0.95 \[ \frac{2 \left (\frac{\sqrt{c+d x}}{b}-\frac{a \log \left (a+b \sqrt{c+d x}\right )}{b^2}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[c + d*x])^(-1),x]

[Out]

(2*(Sqrt[c + d*x]/b - (a*Log[a + b*Sqrt[c + d*x]])/b^2))/d

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Maple [B]  time = 0.007, size = 87, normalized size = 2.1 \begin{align*} 2\,{\frac{\sqrt{dx+c}}{bd}}+{\frac{a}{{b}^{2}d}\ln \left ( -a+b\sqrt{dx+c} \right ) }-{\frac{a}{{b}^{2}d}\ln \left ( a+b\sqrt{dx+c} \right ) }-{\frac{a\ln \left ({b}^{2}dx+{b}^{2}c-{a}^{2} \right ) }{{b}^{2}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.976455, size = 47, normalized size = 1.15 \begin{align*} -\frac{2 \,{\left (\frac{a \log \left (\sqrt{d x + c} b + a\right )}{b^{2}} - \frac{\sqrt{d x + c}}{b}\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73875, size = 80, normalized size = 1.95 \begin{align*} -\frac{2 \,{\left (a \log \left (\sqrt{d x + c} b + a\right ) - \sqrt{d x + c} b\right )}}{b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.496423, size = 49, normalized size = 1.2 \begin{align*} \begin{cases} \frac{x}{a} & \text{for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac{x}{a + b \sqrt{c}} & \text{for}\: d = 0 \\- \frac{2 a \log{\left (\frac{a}{b} + \sqrt{c + d x} \right )}}{b^{2} d} + \frac{2 \sqrt{c + d x}}{b d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17932, size = 68, normalized size = 1.66 \begin{align*} -\frac{2 \, a \log \left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{2} d} + \frac{2 \, a \log \left ({\left | a \right |}\right )}{b^{2} d} + \frac{2 \, \sqrt{d x + c}}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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