Optimal. Leaf size=85 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) (d+e x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)} \]
[Out]
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Rubi [A] time = 0.120931, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) (d+e x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 22.8254, size = 76, normalized size = 0.89 \[ \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{2} \left (a + b x\right )} - \frac{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{2} \left (a + b x\right ) \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x+a)**2)**(1/2)/(e*x+d)**2,x)
[Out]
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Mathematica [A] time = 0.0369926, size = 50, normalized size = 0.59 \[ \frac{\sqrt{(a+b x)^2} (-a e+b (d+e x) \log (d+e x)+b d)}{e^2 (a+b x) (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^2,x]
[Out]
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Maple [C] time = 0.015, size = 51, normalized size = 0.6 \[{\frac{{\it csgn} \left ( bx+a \right ) \left ( \ln \left ( bex+bd \right ) xbe+\ln \left ( bex+bd \right ) bd-ae+bd \right ) }{{e}^{2} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x+a)^2)^(1/2)/(e*x+d)^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(sqrt((b*x + a)^2)/(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206633, size = 50, normalized size = 0.59 \[ \frac{b d - a e +{\left (b e x + b d\right )} \log \left (e x + d\right )}{e^{3} x + d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)/(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.47269, size = 27, normalized size = 0.32 \[ \frac{b \log{\left (d + e x \right )}}{e^{2}} - \frac{a e - b d}{d e^{2} + e^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x+a)**2)**(1/2)/(e*x+d)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.211564, size = 69, normalized size = 0.81 \[ b e^{\left (-2\right )}{\rm ln}\left ({\left | x e + d \right |}\right ){\rm sign}\left (b x + a\right ) + \frac{{\left (b d{\rm sign}\left (b x + a\right ) - a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}}{x e + d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)/(e*x + d)^2,x, algorithm="giac")
[Out]