Optimal. Leaf size=69 \[ \frac{(a+b x) (b d-a e) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2} \]
[Out]
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Rubi [A] time = 0.079359, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{(a+b x) (b d-a e) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2}}{b^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 11.0295, size = 66, normalized size = 0.96 \[ \frac{e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{2}} - \frac{\left (a + b x\right ) \left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0311113, size = 40, normalized size = 0.58 \[ \frac{(a+b x) ((b d-a e) \log (a+b x)+b e x)}{b^2 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.007, size = 45, normalized size = 0.7 \[ -{\frac{ \left ( bx+a \right ) \left ( \ln \left ( bx+a \right ) ae-\ln \left ( bx+a \right ) bd-bex \right ) }{{b}^{2}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/((b*x+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.688302, size = 80, normalized size = 1.16 \[ \sqrt{\frac{1}{b^{2}}} d \log \left (x + \frac{a}{b}\right ) - \frac{a \sqrt{\frac{1}{b^{2}}} e \log \left (x + \frac{a}{b}\right )}{b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/sqrt((b*x + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.204243, size = 32, normalized size = 0.46 \[ \frac{b e x +{\left (b d - a e\right )} \log \left (b x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/sqrt((b*x + a)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.27463, size = 20, normalized size = 0.29 \[ \frac{e x}{b} - \frac{\left (a e - b d\right ) \log{\left (a + b x \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212786, size = 62, normalized size = 0.9 \[ \frac{x e{\rm sign}\left (b x + a\right )}{b} + \frac{{\left (b d{\rm sign}\left (b x + a\right ) - a e{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/sqrt((b*x + a)^2),x, algorithm="giac")
[Out]