3.162 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=151 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.119339, antiderivative size = 151, 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 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^9,x]

[Out]

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

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Rubi in Sympy [A]  time = 14.8646, size = 124, normalized size = 0.82 \[ \frac{a b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{280 x^{6} \left (a + b x\right )} - \frac{3 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{140 x^{6}} - \frac{3 b \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{56 x^{7}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 x^{8}} \]

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**9,x)

[Out]

a*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(280*x**6*(a + b*x)) - 3*b**2*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/(140*x**6) - 3*b*(a + b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**
2)/(56*x**7) - (a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(8*x**8)

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Mathematica [A]  time = 0.0265336, size = 55, normalized size = 0.36 \[ -\frac{\sqrt{(a+b x)^2} \left (35 a^3+120 a^2 b x+140 a b^2 x^2+56 b^3 x^3\right )}{280 x^8 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(35*a^3 + 120*a^2*b*x + 140*a*b^2*x^2 + 56*b^3*x^3))/(280*x^
8*(a + b*x))

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Maple [A]  time = 0.009, size = 52, normalized size = 0.3 \[ -{\frac{56\,{b}^{3}{x}^{3}+140\,a{b}^{2}{x}^{2}+120\,{a}^{2}bx+35\,{a}^{3}}{280\,{x}^{8} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^9,x)

[Out]

-1/280*(56*b^3*x^3+140*a*b^2*x^2+120*a^2*b*x+35*a^3)*((b*x+a)^2)^(3/2)/x^8/(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^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221194, size = 47, normalized size = 0.31 \[ -\frac{56 \, b^{3} x^{3} + 140 \, a b^{2} x^{2} + 120 \, a^{2} b x + 35 \, a^{3}}{280 \, x^{8}} \]

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^9,x, algorithm="fricas")

[Out]

-1/280*(56*b^3*x^3 + 140*a*b^2*x^2 + 120*a^2*b*x + 35*a^3)/x^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/x**9,x)

[Out]

Integral(((a + b*x)**2)**(3/2)/x**9, x)

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GIAC/XCAS [A]  time = 0.210715, size = 100, normalized size = 0.66 \[ -\frac{b^{8}{\rm sign}\left (b x + a\right )}{280 \, a^{5}} - \frac{56 \, b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 140 \, a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 120 \, a^{2} b x{\rm sign}\left (b x + a\right ) + 35 \, a^{3}{\rm sign}\left (b x + a\right )}{280 \, x^{8}} \]

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^9,x, algorithm="giac")

[Out]

-1/280*b^8*sign(b*x + a)/a^5 - 1/280*(56*b^3*x^3*sign(b*x + a) + 140*a*b^2*x^2*s
ign(b*x + a) + 120*a^2*b*x*sign(b*x + a) + 35*a^3*sign(b*x + a))/x^8