3.161 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=151 \[ -\frac{a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.132317, 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 \[ -\frac{a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^8,x]

[Out]

-(a^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (a^2*b*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(2*x^6*(a + b*x)) - (3*a*b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^
5*(a + b*x)) - (b^3*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 = 14.5282, size = 121, normalized size = 0.8 \[ \frac{a b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{140 x^{5} \left (a + b x\right )} - \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{28 x^{5}} - \frac{b \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{14 x^{6}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/x**8,x)

[Out]

a*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(140*x**5*(a + b*x)) - b**2*sqrt(a**2 +
2*a*b*x + b**2*x**2)/(28*x**5) - b*(a + b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(1
4*x**6) - (a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(7*x**7)

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Mathematica [A]  time = 0.0219406, size = 55, normalized size = 0.36 \[ -\frac{\sqrt{(a+b x)^2} \left (20 a^3+70 a^2 b x+84 a b^2 x^2+35 b^3 x^3\right )}{140 x^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^8,x]

[Out]

-(Sqrt[(a + b*x)^2]*(20*a^3 + 70*a^2*b*x + 84*a*b^2*x^2 + 35*b^3*x^3))/(140*x^7*
(a + b*x))

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Maple [A]  time = 0.008, size = 52, normalized size = 0.3 \[ -{\frac{35\,{b}^{3}{x}^{3}+84\,a{b}^{2}{x}^{2}+70\,{a}^{2}bx+20\,{a}^{3}}{140\,{x}^{7} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^8,x)

[Out]

-1/140*(35*b^3*x^3+84*a*b^2*x^2+70*a^2*b*x+20*a^3)*((b*x+a)^2)^(3/2)/x^7/(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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225095, size = 47, normalized size = 0.31 \[ -\frac{35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \]

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^8,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*x^3 + 84*a*b^2*x^2 + 70*a^2*b*x + 20*a^3)/x^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/x**8,x)

[Out]

Integral(((a + b*x)**2)**(3/2)/x**8, x)

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GIAC/XCAS [A]  time = 0.208405, size = 100, normalized size = 0.66 \[ \frac{b^{7}{\rm sign}\left (b x + a\right )}{140 \, a^{4}} - \frac{35 \, b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 84 \, a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 70 \, a^{2} b x{\rm sign}\left (b x + a\right ) + 20 \, a^{3}{\rm sign}\left (b x + a\right )}{140 \, x^{7}} \]

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^8,x, algorithm="giac")

[Out]

1/140*b^7*sign(b*x + a)/a^4 - 1/140*(35*b^3*x^3*sign(b*x + a) + 84*a*b^2*x^2*sig
n(b*x + a) + 70*a^2*b*x*sign(b*x + a) + 20*a^3*sign(b*x + a))/x^7