3.180 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=231 \[ -\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^4*b*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(2*x^10*(a + b*x)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(9*x^9*(a + b*x)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^8*(a + b*x))
- (5*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^5*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(6*x^6*(a + b*x))

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Rubi [A]  time = 0.169113, antiderivative size = 231, 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 \[ -\frac{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{5 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^12,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^4*b*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(2*x^10*(a + b*x)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(9*x^9*(a + b*x)) - (5*a^2*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^8*(a + b*x))
- (5*a*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^5*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(6*x^6*(a + b*x))

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Rubi in Sympy [A]  time = 23.0747, size = 185, normalized size = 0.8 \[ \frac{a b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2772 x^{7} \left (a + b x\right )} - \frac{b^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{396 x^{7}} - \frac{b^{3} \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{132 x^{8}} - \frac{2 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{99 x^{9}} - \frac{b \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{22 x^{10}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{11 x^{11}} \]

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**12,x)

[Out]

a*b**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2772*x**7*(a + b*x)) - b**4*sqrt(a**2 +
2*a*b*x + b**2*x**2)/(396*x**7) - b**3*(a + b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**
2)/(132*x**8) - 2*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(99*x**9) - b*(a + b*
x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(22*x**10) - (a**2 + 2*a*b*x + b**2*x**2)
**(5/2)/(11*x**11)

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Mathematica [A]  time = 0.0272475, size = 77, normalized size = 0.33 \[ -\frac{\sqrt{(a+b x)^2} \left (252 a^5+1386 a^4 b x+3080 a^3 b^2 x^2+3465 a^2 b^3 x^3+1980 a b^4 x^4+462 b^5 x^5\right )}{2772 x^{11} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^12,x]

[Out]

-(Sqrt[(a + b*x)^2]*(252*a^5 + 1386*a^4*b*x + 3080*a^3*b^2*x^2 + 3465*a^2*b^3*x^
3 + 1980*a*b^4*x^4 + 462*b^5*x^5))/(2772*x^11*(a + b*x))

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Maple [A]  time = 0.009, size = 74, normalized size = 0.3 \[ -{\frac{462\,{b}^{5}{x}^{5}+1980\,a{b}^{4}{x}^{4}+3465\,{a}^{2}{b}^{3}{x}^{3}+3080\,{a}^{3}{b}^{2}{x}^{2}+1386\,{a}^{4}bx+252\,{a}^{5}}{2772\,{x}^{11} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^12,x)

[Out]

-1/2772*(462*b^5*x^5+1980*a*b^4*x^4+3465*a^2*b^3*x^3+3080*a^3*b^2*x^2+1386*a^4*b
*x+252*a^5)*((b*x+a)^2)^(5/2)/x^11/(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^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231106, size = 77, normalized size = 0.33 \[ -\frac{462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \]

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^12,x, algorithm="fricas")

[Out]

-1/2772*(462*b^5*x^5 + 1980*a*b^4*x^4 + 3465*a^2*b^3*x^3 + 3080*a^3*b^2*x^2 + 13
86*a^4*b*x + 252*a^5)/x^11

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{12}}\, dx \]

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**12,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**12, x)

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GIAC/XCAS [A]  time = 0.210877, size = 146, normalized size = 0.63 \[ \frac{b^{11}{\rm sign}\left (b x + a\right )}{2772 \, a^{6}} - \frac{462 \, b^{5} x^{5}{\rm sign}\left (b x + a\right ) + 1980 \, a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + 3465 \, a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 3080 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 1386 \, a^{4} b x{\rm sign}\left (b x + a\right ) + 252 \, a^{5}{\rm sign}\left (b x + a\right )}{2772 \, x^{11}} \]

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^12,x, algorithm="giac")

[Out]

1/2772*b^11*sign(b*x + a)/a^6 - 1/2772*(462*b^5*x^5*sign(b*x + a) + 1980*a*b^4*x
^4*sign(b*x + a) + 3465*a^2*b^3*x^3*sign(b*x + a) + 3080*a^3*b^2*x^2*sign(b*x +
a) + 1386*a^4*b*x*sign(b*x + a) + 252*a^5*sign(b*x + a))/x^11