Optimal. Leaf size=47 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}} \]
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Rubi [A] time = 0.060896, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 8.75618, size = 41, normalized size = 0.87 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{\sqrt{b} \sqrt{a d - b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0352816, size = 47, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)*Sqrt[c + d*x]),x]
[Out]
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Maple [A] time = 0.009, size = 37, normalized size = 0.8 \[ 2\,{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)/(d*x+c)^(1/2),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(1/((b*x + a)*sqrt(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228043, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{\sqrt{b^{2} c - a b d}{\left (b d x + 2 \, b c - a d\right )} - 2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x + c}}{b x + a}\right )}{\sqrt{b^{2} c - a b d}}, -\frac{2 \, \arctan \left (-\frac{b c - a d}{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}\right )}{\sqrt{-b^{2} c + a b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*sqrt(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.18598, size = 189, normalized size = 4.02 \[ - 2 \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{b}{a d - b c}} \sqrt{c + d x}} \right )}}{\sqrt{\frac{b}{a d - b c}} \left (a d - b c\right )} & \text{for}\: \frac{b}{a d - b c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{b}{a d - b c}} \sqrt{c + d x}} \right )}}{\sqrt{- \frac{b}{a d - b c}} \left (a d - b c\right )} & \text{for}\: \frac{1}{c + d x} > - \frac{b}{a d - b c} \wedge \frac{b}{a d - b c} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{b}{a d - b c}} \sqrt{c + d x}} \right )}}{\sqrt{- \frac{b}{a d - b c}} \left (a d - b c\right )} & \text{for}\: \frac{b}{a d - b c} < 0 \wedge \frac{1}{c + d x} < - \frac{b}{a d - b c} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217547, size = 51, normalized size = 1.09 \[ \frac{2 \, \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)*sqrt(d*x + c)),x, algorithm="giac")
[Out]