Optimal. Leaf size=39 \[ -\frac{a+b x}{2 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2} \]
[Out]
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Rubi [A] time = 0.113516, antiderivative size = 39, 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 \[ -\frac{a+b x}{2 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/((d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 16.9688, size = 36, normalized size = 0.92 \[ - \frac{a + b x}{2 e \left (d + e x\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(e*x+d)**3/((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0204092, size = 30, normalized size = 0.77 \[ -\frac{a+b x}{2 e \sqrt{(a+b x)^2} (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/((d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
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Maple [A] time = 0.004, size = 27, normalized size = 0.7 \[ -{\frac{bx+a}{2\,e \left ( ex+d \right ) ^{2}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(e*x+d)^3/((b*x+a)^2)^(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)/(sqrt((b*x + a)^2)*(e*x + d)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.27426, size = 32, normalized size = 0.82 \[ -\frac{1}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.3913, size = 26, normalized size = 0.67 \[ - \frac{1}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(e*x+d)**3/((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.290976, size = 24, normalized size = 0.62 \[ -\frac{e^{\left (-1\right )}{\rm sign}\left (b x + a\right )}{2 \,{\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^3),x, algorithm="giac")
[Out]