3.1588 \(\int \frac{d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{b d-a e}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0741141, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{b d-a e}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.50859, size = 42, normalized size = 0.61 \[ \frac{\left (2 a + 2 b x\right ) \left (d + e x\right )^{2}}{4 \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0300736, size = 39, normalized size = 0.57 \[ \frac{-a e-b (d+2 e x)}{2 b^2 (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 32, normalized size = 0.5 \[ -{\frac{ \left ( bx+a \right ) \left ( 2\,bex+ae+bd \right ) }{2\,{b}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.733911, size = 85, normalized size = 1.23 \[ -\frac{e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{d}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{a e}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.204903, size = 51, normalized size = 0.74 \[ -\frac{2 \, b e x + b d + a e}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)/((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.574097, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x