3.177 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=116 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 a^2 x^7}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 a^3 x^6} \]

[Out]

-((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*a*x^8) + (b*(a + b*x)^5*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(28*a^2*x^7) - (b^2*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(168*a^3*x^6)

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Rubi [A]  time = 0.114544, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 a^2 x^7}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 a^3 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^9,x]

[Out]

-((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*a*x^8) + (b*(a + b*x)^5*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(28*a^2*x^7) - (b^2*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(168*a^3*x^6)

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Rubi in Sympy [A]  time = 12.9711, size = 104, normalized size = 0.9 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{16 a x^{8}} + \frac{b \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{48 a^{2} x^{7}} - \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{168 a^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/x**9,x)

[Out]

-(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(16*a*x**8) + b*(2*a + 2*b*x)
*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(48*a**2*x**7) - b*(a**2 + 2*a*b*x + b**2*x
**2)**(7/2)/(168*a**3*x**7)

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Mathematica [A]  time = 0.0369529, size = 77, normalized size = 0.66 \[ -\frac{\sqrt{(a+b x)^2} \left (21 a^5+120 a^4 b x+280 a^3 b^2 x^2+336 a^2 b^3 x^3+210 a b^4 x^4+56 b^5 x^5\right )}{168 x^8 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(21*a^5 + 120*a^4*b*x + 280*a^3*b^2*x^2 + 336*a^2*b^3*x^3 +
210*a*b^4*x^4 + 56*b^5*x^5))/(168*x^8*(a + b*x))

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Maple [A]  time = 0.009, size = 74, normalized size = 0.6 \[ -{\frac{56\,{b}^{5}{x}^{5}+210\,a{b}^{4}{x}^{4}+336\,{a}^{2}{b}^{3}{x}^{3}+280\,{a}^{3}{b}^{2}{x}^{2}+120\,{a}^{4}bx+21\,{a}^{5}}{168\,{x}^{8} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^9,x)

[Out]

-1/168*(56*b^5*x^5+210*a*b^4*x^4+336*a^2*b^3*x^3+280*a^3*b^2*x^2+120*a^4*b*x+21*
a^5)*((b*x+a)^2)^(5/2)/x^8/(b*x+a)^5

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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)^(5/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222135, size = 77, normalized size = 0.66 \[ -\frac{56 \, b^{5} x^{5} + 210 \, a b^{4} x^{4} + 336 \, a^{2} b^{3} x^{3} + 280 \, a^{3} b^{2} x^{2} + 120 \, a^{4} b x + 21 \, a^{5}}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/x^9,x, algorithm="fricas")

[Out]

-1/168*(56*b^5*x^5 + 210*a*b^4*x^4 + 336*a^2*b^3*x^3 + 280*a^3*b^2*x^2 + 120*a^4
*b*x + 21*a^5)/x^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/x**9,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**9, x)

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GIAC/XCAS [A]  time = 0.209205, size = 146, normalized size = 1.26 \[ -\frac{b^{8}{\rm sign}\left (b x + a\right )}{168 \, a^{3}} - \frac{56 \, b^{5} x^{5}{\rm sign}\left (b x + a\right ) + 210 \, a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + 336 \, a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 280 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 120 \, a^{4} b x{\rm sign}\left (b x + a\right ) + 21 \, a^{5}{\rm sign}\left (b x + a\right )}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/x^9,x, algorithm="giac")

[Out]

-1/168*b^8*sign(b*x + a)/a^3 - 1/168*(56*b^5*x^5*sign(b*x + a) + 210*a*b^4*x^4*s
ign(b*x + a) + 336*a^2*b^3*x^3*sign(b*x + a) + 280*a^3*b^2*x^2*sign(b*x + a) + 1
20*a^4*b*x*sign(b*x + a) + 21*a^5*sign(b*x + a))/x^8