3.2021 \(\int \frac{a+b x}{(d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x) (b d-a e)} \]

[Out]

Sqrt[a^2 + 2*a*b*x + b^2*x^2]/((b*d - a*e)*(d + e*x))

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Rubi [A]  time = 0.106803, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x) (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/((d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

Sqrt[a^2 + 2*a*b*x + b^2*x^2]/((b*d - a*e)*(d + e*x))

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Rubi in Sympy [A]  time = 16.7389, size = 32, normalized size = 0.84 \[ - \frac{a + b x}{e \left (d + e x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(e*x+d)**2/((b*x+a)**2)**(1/2),x)

[Out]

-(a + b*x)/(e*(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0193203, size = 28, normalized size = 0.74 \[ -\frac{a+b x}{e \sqrt{(a+b x)^2} (d+e x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/((d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-((a + b*x)/(e*Sqrt[(a + b*x)^2]*(d + e*x)))

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Maple [A]  time = 0.005, size = 27, normalized size = 0.7 \[ -{\frac{bx+a}{e \left ( ex+d \right ) }{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(e*x+d)^2/((b*x+a)^2)^(1/2),x)

[Out]

-1/(e*x+d)/e*(b*x+a)/((b*x+a)^2)^(1/2)

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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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274574, size = 18, normalized size = 0.47 \[ -\frac{1}{e^{2} x + d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

-1/(e^2*x + d*e)

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Sympy [A]  time = 1.21681, size = 10, normalized size = 0.26 \[ - \frac{1}{d e + e^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(e*x+d)**2/((b*x+a)**2)**(1/2),x)

[Out]

-1/(d*e + e**2*x)

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GIAC/XCAS [A]  time = 0.288953, size = 24, normalized size = 0.63 \[ -\frac{e^{\left (-1\right )}{\rm sign}\left (b x + a\right )}{x e + d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^2),x, algorithm="giac")

[Out]

-e^(-1)*sign(b*x + a)/(x*e + d)