3.160 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.124481, 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 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.0607, size = 104, normalized size = 0.69 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 a x^{6}} + \frac{b \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{24 a^{2} x^{5}} - \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{60 a^{3} x^{5}} \]

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**7,x)

[Out]

-(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(12*a*x**6) + b*(2*a + 2*b*x)
*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(24*a**2*x**5) - b*(a**2 + 2*a*b*x + b**2*x
**2)**(5/2)/(60*a**3*x**5)

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Mathematica [A]  time = 0.0210546, size = 55, normalized size = 0.36 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^3+36 a^2 b x+45 a b^2 x^2+20 b^3 x^3\right )}{60 x^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^7,x]

[Out]

-(Sqrt[(a + b*x)^2]*(10*a^3 + 36*a^2*b*x + 45*a*b^2*x^2 + 20*b^3*x^3))/(60*x^6*(
a + b*x))

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Maple [A]  time = 0.009, size = 52, normalized size = 0.3 \[ -{\frac{20\,{b}^{3}{x}^{3}+45\,a{b}^{2}{x}^{2}+36\,{a}^{2}bx+10\,{a}^{3}}{60\,{x}^{6} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^7,x)

[Out]

-1/60*(20*b^3*x^3+45*a*b^2*x^2+36*a^2*b*x+10*a^3)*((b*x+a)^2)^(3/2)/x^6/(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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222319, size = 47, normalized size = 0.31 \[ -\frac{20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, x^{6}} \]

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^7,x, algorithm="fricas")

[Out]

-1/60*(20*b^3*x^3 + 45*a*b^2*x^2 + 36*a^2*b*x + 10*a^3)/x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{7}}\, dx \]

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**7,x)

[Out]

Integral(((a + b*x)**2)**(3/2)/x**7, x)

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GIAC/XCAS [A]  time = 0.209102, size = 100, normalized size = 0.66 \[ -\frac{b^{6}{\rm sign}\left (b x + a\right )}{60 \, a^{3}} - \frac{20 \, b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 45 \, a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 36 \, a^{2} b x{\rm sign}\left (b x + a\right ) + 10 \, a^{3}{\rm sign}\left (b x + a\right )}{60 \, x^{6}} \]

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^7,x, algorithm="giac")

[Out]

-1/60*b^6*sign(b*x + a)/a^3 - 1/60*(20*b^3*x^3*sign(b*x + a) + 45*a*b^2*x^2*sign
(b*x + a) + 36*a^2*b*x*sign(b*x + a) + 10*a^3*sign(b*x + a))/x^6