3.1539 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=46 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 (d+e x)^2 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.07097, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 (d+e x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.485, size = 44, normalized size = 0.96 \[ - \frac{\left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 \left (d + e x\right )^{2} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x+a)**2)**(1/2)/(e*x+d)**3,x)

[Out]

-(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(4*(d + e*x)**2*(a*e - b*d))

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Mathematica [A]  time = 0.032325, size = 44, normalized size = 0.96 \[ -\frac{\sqrt{(a+b x)^2} (a e+b (d+2 e x))}{2 e^2 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^3,x]

[Out]

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

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Maple [A]  time = 0.006, size = 41, normalized size = 0.9 \[ -{\frac{2\,bex+ae+bd}{2\, \left ( ex+d \right ) ^{2}{e}^{2} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)/(e*x+d)^3,x)

[Out]

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

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.204197, size = 51, normalized size = 1.11 \[ -\frac{2 \, b e x + b d + a e}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/(e*x + d)^3,x, algorithm="fricas")

[Out]

-1/2*(2*b*e*x + b*d + a*e)/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)

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Sympy [A]  time = 1.75761, size = 39, normalized size = 0.85 \[ - \frac{a e + b d + 2 b e x}{2 d^{2} e^{2} + 4 d e^{3} x + 2 e^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x+a)**2)**(1/2)/(e*x+d)**3,x)

[Out]

-(a*e + b*d + 2*b*e*x)/(2*d**2*e**2 + 4*d*e**3*x + 2*e**4*x**2)

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GIAC/XCAS [A]  time = 0.212605, size = 59, normalized size = 1.28 \[ -\frac{{\left (2 \, b x e{\rm sign}\left (b x + a\right ) + b d{\rm sign}\left (b x + a\right ) + a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/(e*x + d)^3,x, algorithm="giac")

[Out]

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