3.203 \(\int \frac{x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{a}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{1}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0573502, antiderivative size = 63, 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 \[ \frac{a}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{1}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.26539, size = 61, normalized size = 0.97 \[ \frac{a \left (2 a + 2 b x\right )}{8 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} - \frac{1}{3 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

a*(2*a + 2*b*x)/(8*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)) - 1/(3*b**2*(a**2 +
2*a*b*x + b**2*x**2)**(3/2))

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Mathematica [A]  time = 0.0181309, size = 33, normalized size = 0.52 \[ \frac{-a-4 b x}{12 b^2 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-a - 4*b*x)/(12*b^2*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.006, size = 26, normalized size = 0.4 \[ -{\frac{ \left ( bx+a \right ) \left ( 4\,bx+a \right ) }{12\,{b}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/12*(b*x+a)*(4*b*x+a)/b^2/((b*x+a)^2)^(5/2)

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Maxima [A]  time = 0.787486, size = 59, normalized size = 0.94 \[ -\frac{1}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} + \frac{a}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

-1/3/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) + 1/4*a/((b^2)^(5/2)*b*(x + a/b)^4)

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Fricas [A]  time = 0.230257, size = 73, normalized size = 1.16 \[ -\frac{4 \, b x + a}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/12*(4*b*x + a)/(b^6*x^4 + 4*a*b^5*x^3 + 6*a^2*b^4*x^2 + 4*a^3*b^3*x + a^4*b^2
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(x/((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.548091, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x