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

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 = 0.0511365, 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 \[ \frac{2 \sqrt{c+d x}}{b d}-\frac{2 a \log \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]

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)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a \log{\left (a + b \sqrt{c + d x} \right )}}{b^{2} d} + \frac{2 \int ^{\sqrt{c + d x}} \frac{1}{b}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*(d*x+c)**(1/2)),x)

[Out]

-2*a*log(a + b*sqrt(c + d*x))/(b**2*d) + 2*Integral(1/b, (x, sqrt(c + d*x)))/d

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Mathematica [A]  time = 0.0176275, size = 37, normalized size = 0.9 \[ \frac{2 b \sqrt{c+d x}-2 a \log \left (a+b \sqrt{c+d x}\right )}{b^2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 87, normalized size = 2.1 \[ 2\,{\frac{\sqrt{dx+c}}{bd}}-{\frac{a}{{b}^{2}d}\ln \left ( a+b\sqrt{dx+c} \right ) }+{\frac{a}{{b}^{2}d}\ln \left ( -a+b\sqrt{dx+c} \right ) }-{\frac{a\ln \left ({b}^{2}dx+{b}^{2}c-{a}^{2} \right ) }{{b}^{2}d}} \]

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

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Maxima [A]  time = 0.696424, size = 47, normalized size = 1.15 \[ -\frac{2 \,{\left (\frac{a \log \left (\sqrt{d x + c} b + a\right )}{b^{2}} - \frac{\sqrt{d x + c}}{b}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x + c)*b + a),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 = 0.285814, size = 45, normalized size = 1.1 \[ -\frac{2 \,{\left (a \log \left (\sqrt{d x + c} b + a\right ) - \sqrt{d x + c} b\right )}}{b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x + c)*b + a),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 = 1.92524, size = 49, normalized size = 1.2 \[ \begin{cases} \frac{x}{a} & \text{for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac{x}{a + b \sqrt{c}} & \text{for}\: d = 0 \\- \frac{2 a \log{\left (\frac{a}{b} + \sqrt{c + d x} \right )}}{b^{2} d} + \frac{2 \sqrt{c + d x}}{b d} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.277209, size = 68, normalized size = 1.66 \[ -\frac{2 \, a{\rm ln}\left ({\left | \sqrt{d x + c} b + a \right |}\right )}{b^{2} d} + \frac{2 \, a{\rm ln}\left ({\left | a \right |}\right )}{b^{2} d} + \frac{2 \, \sqrt{d x + c}}{b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(d*x + c)*b + a),x, algorithm="giac")

[Out]

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