3.178 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=229 \[ -\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{x^5 (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^6 (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (5*a^4*b*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(8*x^8*(a + b*x)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(7*x^7*(a + b*x)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^6*(a + b*x))
- (a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x^5*(a + b*x)) - (b^5*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(4*x^4*(a + b*x))

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Rubi [A]  time = 0.164327, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{x^5 (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^6 (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{5 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^10,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (5*a^4*b*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(8*x^8*(a + b*x)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(7*x^7*(a + b*x)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^6*(a + b*x))
- (a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x^5*(a + b*x)) - (b^5*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(4*x^4*(a + b*x))

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Rubi in Sympy [A]  time = 18.4029, size = 146, normalized size = 0.64 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{18 a x^{9}} + \frac{b \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{48 a^{2} x^{8}} - \frac{b^{2} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{144 a^{3} x^{7}} + \frac{b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{504 a^{4} x^{7}} \]

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**10,x)

[Out]

-(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(18*a*x**9) + b*(2*a + 2*b*x)
*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(48*a**2*x**8) - b**2*(2*a + 2*b*x)*(a**2 +
2*a*b*x + b**2*x**2)**(5/2)/(144*a**3*x**7) + b**2*(a**2 + 2*a*b*x + b**2*x**2)
**(7/2)/(504*a**4*x**7)

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Mathematica [A]  time = 0.0395118, size = 77, normalized size = 0.34 \[ -\frac{\sqrt{(a+b x)^2} \left (56 a^5+315 a^4 b x+720 a^3 b^2 x^2+840 a^2 b^3 x^3+504 a b^4 x^4+126 b^5 x^5\right )}{504 x^9 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^10,x]

[Out]

-(Sqrt[(a + b*x)^2]*(56*a^5 + 315*a^4*b*x + 720*a^3*b^2*x^2 + 840*a^2*b^3*x^3 +
504*a*b^4*x^4 + 126*b^5*x^5))/(504*x^9*(a + b*x))

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Maple [A]  time = 0.009, size = 74, normalized size = 0.3 \[ -{\frac{126\,{b}^{5}{x}^{5}+504\,a{b}^{4}{x}^{4}+840\,{a}^{2}{b}^{3}{x}^{3}+720\,{a}^{3}{b}^{2}{x}^{2}+315\,{a}^{4}bx+56\,{a}^{5}}{504\,{x}^{9} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^10,x)

[Out]

-1/504*(126*b^5*x^5+504*a*b^4*x^4+840*a^2*b^3*x^3+720*a^3*b^2*x^2+315*a^4*b*x+56
*a^5)*((b*x+a)^2)^(5/2)/x^9/(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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220318, size = 77, normalized size = 0.34 \[ -\frac{126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \]

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^10,x, algorithm="fricas")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^
4*b*x + 56*a^5)/x^9

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{10}}\, dx \]

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**10,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**10, x)

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GIAC/XCAS [A]  time = 0.209387, size = 146, normalized size = 0.64 \[ \frac{b^{9}{\rm sign}\left (b x + a\right )}{504 \, a^{4}} - \frac{126 \, b^{5} x^{5}{\rm sign}\left (b x + a\right ) + 504 \, a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + 840 \, a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 720 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 315 \, a^{4} b x{\rm sign}\left (b x + a\right ) + 56 \, a^{5}{\rm sign}\left (b x + a\right )}{504 \, x^{9}} \]

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^10,x, algorithm="giac")

[Out]

1/504*b^9*sign(b*x + a)/a^4 - 1/504*(126*b^5*x^5*sign(b*x + a) + 504*a*b^4*x^4*s
ign(b*x + a) + 840*a^2*b^3*x^3*sign(b*x + a) + 720*a^3*b^2*x^2*sign(b*x + a) + 3
15*a^4*b*x*sign(b*x + a) + 56*a^5*sign(b*x + a))/x^9