Optimal. Leaf size=62 \[ \frac{2 \sqrt{c+d x}}{b}-\frac{2 \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}} \]
[Out]
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Rubi [A] time = 0.115765, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{2 \sqrt{c+d x}}{b}-\frac{2 \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x]/(a + b*x),x]
[Out]
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Rubi in Sympy [A] time = 12.7231, size = 53, normalized size = 0.85 \[ \frac{2 \sqrt{c + d x}}{b} - \frac{2 \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(1/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0584219, size = 62, normalized size = 1. \[ \frac{2 \sqrt{c+d x}}{b}-\frac{2 \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x]/(a + b*x),x]
[Out]
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Maple [A] time = 0.017, size = 92, normalized size = 1.5 \[ 2\,{\frac{\sqrt{dx+c}}{b}}-2\,{\frac{ad}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{c}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(1/2)/(b*x+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221315, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) + 2 \, \sqrt{d x + c}}{b}, -\frac{2 \,{\left (\sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) - \sqrt{d x + c}\right )}}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.72679, size = 178, normalized size = 2.87 \[ \frac{2 \left (\frac{d \sqrt{c + d x}}{b} - \frac{d \left (a d - b c\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b \sqrt{\frac{a d - b c}{b}}} & \text{for}\: \frac{a d - b c}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: c + d x > \frac{- a d + b c}{b} \wedge \frac{a d - b c}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{- a d + b c}{b}}} \right )}}{b \sqrt{\frac{- a d + b c}{b}}} & \text{for}\: \frac{a d - b c}{b} < 0 \wedge c + d x < \frac{- a d + b c}{b} \end{cases}\right )}{b}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(1/2)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.219063, size = 84, normalized size = 1.35 \[ \frac{2 \,{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b} + \frac{2 \, \sqrt{d x + c}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(b*x + a),x, algorithm="giac")
[Out]