3.201 \(\int \frac{x^3}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{x^4}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0603818, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{x^4}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.45057, size = 34, normalized size = 0.92 \[ \frac{x^{4} \left (2 a + 2 b x\right )}{8 a \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

x**4*(2*a + 2*b*x)/(8*a*(a**2 + 2*a*b*x + b**2*x**2)**(5/2))

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Mathematica [A]  time = 0.0248611, size = 55, normalized size = 1.49 \[ \frac{-a^3-4 a^2 b x-6 a b^2 x^2-4 b^3 x^3}{4 b^4 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-a^3 - 4*a^2*b*x - 6*a*b^2*x^2 - 4*b^3*x^3)/(4*b^4*(a + b*x)^3*Sqrt[(a + b*x)^2
])

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Maple [A]  time = 0.009, size = 48, normalized size = 1.3 \[ -{\frac{ \left ( bx+a \right ) \left ( 4\,{b}^{3}{x}^{3}+6\,a{b}^{2}{x}^{2}+4\,{a}^{2}bx+{a}^{3} \right ) }{4\,{b}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.798552, size = 181, normalized size = 4.89 \[ -\frac{x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{2 \, a^{2}}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{4}} - \frac{a^{3} b}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{2 \, a^{2}}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} - \frac{a}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} + \frac{a^{3}}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}} b^{3}{\left (x + \frac{a}{b}\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")

[Out]

-x^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 2/3*a^2/((b^2*x^2 + 2*a*b*x + a^2)^
(3/2)*b^4) - 1/4*a^3*b/((b^2)^(9/2)*(x + a/b)^4) + 2/3*a^2/((b^2)^(7/2)*(x + a/b
)^3) - 1/2*a/((b^2)^(5/2)*b*(x + a/b)^2) + 1/2*a^3/((b^2)^(5/2)*b^3*(x + a/b)^4)

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Fricas [A]  time = 0.224518, size = 103, normalized size = 2.78 \[ -\frac{4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/4*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)/(b^8*x^4 + 4*a*b^7*x^3 + 6*a^2*
b^6*x^2 + 4*a^3*b^5*x + a^4*b^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(x**3/((a + b*x)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.56971, size = 4, normalized size = 0.11 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x