3.663 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{x^6} \, dx\)

Optimal. Leaf size=114 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.156647, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^4 (a+b x)}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^6,x]

[Out]

-(a*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a + b*x)) - ((A*b + a*B)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x)) - (b*B*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 = 18.7451, size = 112, normalized size = 0.98 \[ - \frac{A \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{10 a x^{5}} - \frac{\left (\frac{A b}{20} - \frac{B a}{12}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{x^{4} \left (a + b x\right )} + \frac{\left (3 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**6,x)

[Out]

-A*(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(10*a*x**5) - (A*b/20 - B*a/12
)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(x**4*(a + b*x)) + (3*A*b - 5*B*a)*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/(15*a*x**4)

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Mathematica [A]  time = 0.0271963, size = 49, normalized size = 0.43 \[ -\frac{\sqrt{(a+b x)^2} (3 a (4 A+5 B x)+5 b x (3 A+4 B x))}{60 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(5*b*x*(3*A + 4*B*x) + 3*a*(4*A + 5*B*x)))/(60*x^5*(a + b*x)
)

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Maple [A]  time = 0.008, size = 44, normalized size = 0.4 \[ -{\frac{20\,Bb{x}^{2}+15\,Abx+15\,aBx+12\,aA}{60\,{x}^{5} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*((b*x+a)^2)^(1/2)/x^6,x)

[Out]

-1/60*(20*B*b*x^2+15*A*b*x+15*B*a*x+12*A*a)*((b*x+a)^2)^(1/2)/x^5/(b*x+a)

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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(sqrt((b*x + a)^2)*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.27782, size = 36, normalized size = 0.32 \[ -\frac{20 \, B b x^{2} + 12 \, A a + 15 \,{\left (B a + A b\right )} x}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

-1/60*(20*B*b*x^2 + 12*A*a + 15*(B*a + A*b)*x)/x^5

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Sympy [A]  time = 2.34367, size = 31, normalized size = 0.27 \[ - \frac{12 A a + 20 B b x^{2} + x \left (15 A b + 15 B a\right )}{60 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**6,x)

[Out]

-(12*A*a + 20*B*b*x**2 + x*(15*A*b + 15*B*a))/(60*x**5)

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GIAC/XCAS [A]  time = 0.268931, size = 104, normalized size = 0.91 \[ -\frac{{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )}{\rm sign}\left (b x + a\right )}{60 \, a^{4}} - \frac{20 \, B b x^{2}{\rm sign}\left (b x + a\right ) + 15 \, B a x{\rm sign}\left (b x + a\right ) + 15 \, A b x{\rm sign}\left (b x + a\right ) + 12 \, A a{\rm sign}\left (b x + a\right )}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/x^6,x, algorithm="giac")

[Out]

-1/60*(5*B*a*b^4 - 3*A*b^5)*sign(b*x + a)/a^4 - 1/60*(20*B*b*x^2*sign(b*x + a) +
15*B*a*x*sign(b*x + a) + 15*A*b*x*sign(b*x + a) + 12*A*a*sign(b*x + a))/x^5