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

Optimal. Leaf size=61 \[ \frac{a}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0540339, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a}{2 b^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + a/(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 = 5.97952, size = 34, normalized size = 0.56 \[ \frac{x^{2} \left (2 a + 2 b x\right )}{4 a \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(2*a + 2*b*x)/(4*a*(a**2 + 2*a*b*x + b**2*x**2)**(3/2))

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Mathematica [A]  time = 0.0223092, size = 33, normalized size = 0.54 \[ \frac{-a-2 b x}{2 b^2 (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 26, normalized size = 0.4 \[ -{\frac{ \left ( bx+a \right ) \left ( 2\,bx+a \right ) }{2\,{b}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.70958, size = 59, normalized size = 0.97 \[ -\frac{1}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{a}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227579, size = 43, normalized size = 0.7 \[ -\frac{2 \, b x + a}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*b*x + a)/(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. \[ \int \frac{x}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/((a + b*x)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.582766, size = 4, normalized size = 0.07 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x