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

Optimal. Leaf size=103 \[ \frac{\log (x) \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b B x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.152217, antiderivative size = 103, 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 \[ \frac{\log (x) \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b B x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.3565, size = 99, normalized size = 0.96 \[ - \frac{A \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 a x} + \frac{\left (A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + \frac{\left (A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0387579, size = 44, normalized size = 0.43 \[ \frac{\sqrt{(a+b x)^2} \left (x \log (x) (a B+A b)-a A+b B x^2\right )}{x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.018, size = 42, normalized size = 0.4 \[{\frac{{\it csgn} \left ( bx+a \right ) \left ( A\ln \left ( bx \right ) xb+B\ln \left ( bx \right ) xa+Bb{x}^{2}+aBx-aA \right ) }{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.29888, size = 35, normalized size = 0.34 \[ \frac{B b x^{2} +{\left (B a + A b\right )} x \log \left (x\right ) - A a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.26031, size = 19, normalized size = 0.18 \[ - \frac{A a}{x} + B b x + \left (A b + B a\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a/x + B*b*x + (A*b + B*a)*log(x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.272741, size = 63, normalized size = 0.61 \[ B b x{\rm sign}\left (b x + a\right ) +{\left (B a{\rm sign}\left (b x + a\right ) + A b{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{A a{\rm sign}\left (b x + a\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b*x*sign(b*x + a) + (B*a*sign(b*x + a) + A*b*sign(b*x + a))*ln(abs(x)) - A*a*s
ign(b*x + a)/x