3.660 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{x^3} \, dx\)

Optimal. Leaf size=108 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}+\frac{b B \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.147336, antiderivative size = 108, 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 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}+\frac{b B \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.1176, size = 99, normalized size = 0.92 \[ - \frac{A \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 a x^{2}} - \frac{B a \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{x \left (a + b x\right )} + \frac{B b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**3,x)

[Out]

-A*(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(4*a*x**2) - B*a*sqrt(a**2 + 2
*a*b*x + b**2*x**2)/(x*(a + b*x)) + B*b*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/
(a + b*x)

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Mathematica [A]  time = 0.0366787, size = 48, normalized size = 0.44 \[ -\frac{\sqrt{(a+b x)^2} \left (a (A+2 B x)+2 A b x-2 b B x^2 \log (x)\right )}{2 x^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.018, size = 37, normalized size = 0.3 \[ -{\frac{{\it csgn} \left ( bx+a \right ) \left ( -2\,B\ln \left ( bx \right ){x}^{2}b+2\,Abx+2\,aBx+aA \right ) }{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*((b*x+a)^2)^(1/2)/x^3,x)

[Out]

-1/2*csgn(b*x+a)*(-2*B*ln(b*x)*x^2*b+2*A*b*x+2*a*B*x+a*A)/x^2

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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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266968, size = 39, normalized size = 0.36 \[ \frac{2 \, B b x^{2} \log \left (x\right ) - A a - 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/x^3,x, algorithm="fricas")

[Out]

1/2*(2*B*b*x^2*log(x) - A*a - 2*(B*a + A*b)*x)/x^2

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Sympy [A]  time = 1.5745, size = 26, normalized size = 0.24 \[ B b \log{\left (x \right )} - \frac{A a + x \left (2 A b + 2 B a\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**3,x)

[Out]

B*b*log(x) - (A*a + x*(2*A*b + 2*B*a))/(2*x**2)

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GIAC/XCAS [A]  time = 0.268304, size = 68, normalized size = 0.63 \[ B b{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{A a{\rm sign}\left (b x + a\right ) + 2 \,{\left (B a{\rm sign}\left (b x + a\right ) + A b{\rm sign}\left (b x + a\right )\right )} x}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/x^3,x, algorithm="giac")

[Out]

B*b*ln(abs(x))*sign(b*x + a) - 1/2*(A*a*sign(b*x + a) + 2*(B*a*sign(b*x + a) + A
*b*sign(b*x + a))*x)/x^2