Optimal. Leaf size=39 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c e} \]
[Out]
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Rubi [A] time = 0.0724998, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c e} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 18.6183, size = 34, normalized size = 0.87 \[ \frac{\left (d + e x\right )^{3}}{2 e \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.00530468, size = 30, normalized size = 0.77 \[ \frac{x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]
[Out]
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Maple [A] time = 0.003, size = 38, normalized size = 1. \[{\frac{x \left ( ex+2\,d \right ) \left ( ex+d \right ) }{2}{\frac{1}{\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.687438, size = 147, normalized size = 3.77 \[ \frac{c^{2} d^{2} e^{4} \log \left (x + \frac{d}{e}\right )}{\left (c e^{2}\right )^{\frac{5}{2}}} - \frac{c d e^{3} x}{\left (c e^{2}\right )^{\frac{3}{2}}} + \frac{e^{2} x^{2}}{2 \, \sqrt{c e^{2}}} - d^{2} \sqrt{\frac{1}{c e^{2}}} \log \left (x + \frac{d}{e}\right ) + \frac{2 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d}{c e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212994, size = 59, normalized size = 1.51 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x^{2} + 2 \, d x\right )}}{2 \,{\left (c e x + c d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\sqrt{c \left (d + e x\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.275769, size = 50, normalized size = 1.28 \[ \frac{1}{2} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (\frac{d e^{\left (-1\right )}}{c} + \frac{x}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")
[Out]