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