3.176 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx\)

Optimal. Leaf size=76 \[ \frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{42 a^2 x^6}-\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{7 a x^7} \]

[Out]

-((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*a*x^7) + (b*(a + b*x)^5*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(42*a^2*x^6)

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Rubi [A]  time = 0.0828154, 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 \[ \frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{42 a^2 x^6}-\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^8,x]

[Out]

-((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*a*x^7) + (b*(a + b*x)^5*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(42*a^2*x^6)

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Rubi in Sympy [A]  time = 6.86558, size = 63, normalized size = 0.83 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 a x^{7}} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{42 a^{2} x^{7}} \]

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**8,x)

[Out]

-(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(12*a*x**7) + (a**2 + 2*a*b*x
+ b**2*x**2)**(7/2)/(42*a**2*x**7)

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Mathematica [A]  time = 0.02809, size = 77, normalized size = 1.01 \[ -\frac{\sqrt{(a+b x)^2} \left (6 a^5+35 a^4 b x+84 a^3 b^2 x^2+105 a^2 b^3 x^3+70 a b^4 x^4+21 b^5 x^5\right )}{42 x^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^8,x]

[Out]

-(Sqrt[(a + b*x)^2]*(6*a^5 + 35*a^4*b*x + 84*a^3*b^2*x^2 + 105*a^2*b^3*x^3 + 70*
a*b^4*x^4 + 21*b^5*x^5))/(42*x^7*(a + b*x))

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Maple [A]  time = 0.008, size = 74, normalized size = 1. \[ -{\frac{21\,{b}^{5}{x}^{5}+70\,a{b}^{4}{x}^{4}+105\,{a}^{2}{b}^{3}{x}^{3}+84\,{a}^{3}{b}^{2}{x}^{2}+35\,{a}^{4}bx+6\,{a}^{5}}{42\,{x}^{7} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^8,x)

[Out]

-1/42*(21*b^5*x^5+70*a*b^4*x^4+105*a^2*b^3*x^3+84*a^3*b^2*x^2+35*a^4*b*x+6*a^5)*
((b*x+a)^2)^(5/2)/x^7/(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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243653, size = 77, normalized size = 1.01 \[ -\frac{21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \]

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^8,x, algorithm="fricas")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x
+ 6*a^5)/x^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{8}}\, dx \]

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**8,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**8, x)

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GIAC/XCAS [A]  time = 0.210124, size = 146, normalized size = 1.92 \[ \frac{b^{7}{\rm sign}\left (b x + a\right )}{42 \, a^{2}} - \frac{21 \, b^{5} x^{5}{\rm sign}\left (b x + a\right ) + 70 \, a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + 105 \, a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 84 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 35 \, a^{4} b x{\rm sign}\left (b x + a\right ) + 6 \, a^{5}{\rm sign}\left (b x + a\right )}{42 \, x^{7}} \]

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^8,x, algorithm="giac")

[Out]

1/42*b^7*sign(b*x + a)/a^2 - 1/42*(21*b^5*x^5*sign(b*x + a) + 70*a*b^4*x^4*sign(
b*x + a) + 105*a^2*b^3*x^3*sign(b*x + a) + 84*a^3*b^2*x^2*sign(b*x + a) + 35*a^4
*b*x*sign(b*x + a) + 6*a^5*sign(b*x + a))/x^7