Optimal. Leaf size=47 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}} \]
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Rubi [A] time = 0.052506, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)),x]
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Rubi in Sympy [A] time = 21.7754, size = 41, normalized size = 0.87 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\sqrt{b} \sqrt{a e - b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0325717, size = 47, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)),x]
[Out]
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Maple [A] time = 0.008, size = 37, normalized size = 0.8 \[ 2\,{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(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((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(e*x + d)),x, algorithm="maxima")
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Fricas [A] time = 0.294633, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{\sqrt{b^{2} d - a b e}}, -\frac{2 \, \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{\sqrt{-b^{2} d + a b e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.21392, size = 189, normalized size = 4.02 \[ - 2 \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{b}{a e - b d}} \sqrt{d + e x}} \right )}}{\sqrt{\frac{b}{a e - b d}} \left (a e - b d\right )} & \text{for}\: \frac{b}{a e - b d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{b}{a e - b d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{b}{a e - b d}} \left (a e - b d\right )} & \text{for}\: \frac{1}{d + e x} > - \frac{b}{a e - b d} \wedge \frac{b}{a e - b d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{b}{a e - b d}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{b}{a e - b d}} \left (a e - b d\right )} & \text{for}\: \frac{b}{a e - b d} < 0 \wedge \frac{1}{d + e x} < - \frac{b}{a e - b d} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.283704, size = 55, normalized size = 1.17 \[ \frac{2 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(e*x + d)),x, algorithm="giac")
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