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

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

[Out]

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

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Rubi [A]  time = 0.0517182, 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 \[ -\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.26876, size = 36, normalized size = 0.97 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(8*a*x**4)

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Mathematica [A]  time = 0.0199465, size = 53, normalized size = 1.43 \[ -\frac{\sqrt{(a+b x)^2} \left (a^3+4 a^2 b x+6 a b^2 x^2+4 b^3 x^3\right )}{4 x^4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.009, size = 50, normalized size = 1.4 \[ -{\frac{4\,{b}^{3}{x}^{3}+6\,a{b}^{2}{x}^{2}+4\,{a}^{2}bx+{a}^{3}}{4\,{x}^{4} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222574, size = 45, normalized size = 1.22 \[ -\frac{4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209546, size = 99, normalized size = 2.68 \[ -\frac{b^{4}{\rm sign}\left (b x + a\right )}{4 \, a} - \frac{4 \, b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 6 \, a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 4 \, a^{2} b x{\rm sign}\left (b x + a\right ) + a^{3}{\rm sign}\left (b x + a\right )}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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