3.159 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=76 \[ \frac{b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}-\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 a x^5} \]

[Out]

-((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*a*x^5) + (b*(a + b*x)^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(20*a^2*x^4)

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Rubi [A]  time = 0.0858729, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}-\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^6,x]

[Out]

-((a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*a*x^5) + (b*(a + b*x)^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(20*a^2*x^4)

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Rubi in Sympy [A]  time = 6.97431, size = 63, normalized size = 0.83 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 a x^{5}} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{20 a^{2} x^{5}} \]

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**6,x)

[Out]

-(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(8*a*x**5) + (a**2 + 2*a*b*x
+ b**2*x**2)**(5/2)/(20*a**2*x**5)

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Mathematica [A]  time = 0.0254399, size = 55, normalized size = 0.72 \[ -\frac{\sqrt{(a+b x)^2} \left (4 a^3+15 a^2 b x+20 a b^2 x^2+10 b^3 x^3\right )}{20 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(4*a^3 + 15*a^2*b*x + 20*a*b^2*x^2 + 10*b^3*x^3))/(20*x^5*(a
+ b*x))

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Maple [A]  time = 0.009, size = 52, normalized size = 0.7 \[ -{\frac{10\,{b}^{3}{x}^{3}+20\,a{b}^{2}{x}^{2}+15\,{a}^{2}bx+4\,{a}^{3}}{20\,{x}^{5} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^6,x)

[Out]

-1/20*(10*b^3*x^3+20*a*b^2*x^2+15*a^2*b*x+4*a^3)*((b*x+a)^2)^(3/2)/x^5/(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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237097, size = 47, normalized size = 0.62 \[ -\frac{10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \]

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^6,x, algorithm="fricas")

[Out]

-1/20*(10*b^3*x^3 + 20*a*b^2*x^2 + 15*a^2*b*x + 4*a^3)/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{6}}\, dx \]

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**6,x)

[Out]

Integral(((a + b*x)**2)**(3/2)/x**6, x)

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GIAC/XCAS [A]  time = 0.209807, size = 100, normalized size = 1.32 \[ \frac{b^{5}{\rm sign}\left (b x + a\right )}{20 \, a^{2}} - \frac{10 \, b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 20 \, a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 15 \, a^{2} b x{\rm sign}\left (b x + a\right ) + 4 \, a^{3}{\rm sign}\left (b x + a\right )}{20 \, x^{5}} \]

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^6,x, algorithm="giac")

[Out]

1/20*b^5*sign(b*x + a)/a^2 - 1/20*(10*b^3*x^3*sign(b*x + a) + 20*a*b^2*x^2*sign(
b*x + a) + 15*a^2*b*x*sign(b*x + a) + 4*a^3*sign(b*x + a))/x^5