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

Optimal. Leaf size=120 \[ \frac{x (a+b x) (A b-a B)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a (a+b x) (A b-a B) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^2 (a+b x)}{2 b \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.181921, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{x (a+b x) (A b-a B)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a (a+b x) (A b-a B) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^2 (a+b x)}{2 b \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.3957, size = 110, normalized size = 0.92 \[ \frac{B x^{2} \left (2 a + 2 b x\right )}{4 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{a \left (a + b x\right ) \left (A b - B a\right ) \log{\left (a + b x \right )}}{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{\left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**2*(2*a + 2*b*x)/(4*b*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - a*(a + b*x)*(A*b -
B*a)*log(a + b*x)/(b**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + (A*b - B*a)*sqrt(a*
*2 + 2*a*b*x + b**2*x**2)/b**3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0535741, size = 57, normalized size = 0.48 \[ \frac{(a+b x) (b x (-2 a B+2 A b+b B x)+2 a (a B-A b) \log (a+b x))}{2 b^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 66, normalized size = 0.6 \[ -{\frac{ \left ( bx+a \right ) \left ( -{b}^{2}B{x}^{2}+2\,A\ln \left ( bx+a \right ) ab-2\,Ax{b}^{2}-2\,B\ln \left ( bx+a \right ){a}^{2}+2\,Bxab \right ) }{2\,{b}^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b*x+a)*(-b^2*B*x^2+2*A*ln(b*x+a)*a*b-2*A*x*b^2-2*B*ln(b*x+a)*a^2+2*B*x*a*b
)/((b*x+a)^2)^(1/2)/b^3

_______________________________________________________________________________________

Maxima [A]  time = 0.683531, size = 117, normalized size = 0.98 \[ \frac{B a^{2} b^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{B a b x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{B x^{2}}{2 \, \sqrt{b^{2}}} - \frac{A a \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} A}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*a^2*b^2*log(x + a/b)/(b^2)^(5/2) - B*a*b*x/(b^2)^(3/2) + 1/2*B*x^2/sqrt(b^2) -
A*a*sqrt(b^(-2))*log(x + a/b)/b + sqrt(b^2*x^2 + 2*a*b*x + a^2)*A/b^2

_______________________________________________________________________________________

Fricas [A]  time = 0.307069, size = 63, normalized size = 0.52 \[ \frac{B b^{2} x^{2} - 2 \,{\left (B a b - A b^{2}\right )} x + 2 \,{\left (B a^{2} - A a b\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(B*b^2*x^2 - 2*(B*a*b - A*b^2)*x + 2*(B*a^2 - A*a*b)*log(b*x + a))/b^3

_______________________________________________________________________________________

Sympy [A]  time = 1.41605, size = 37, normalized size = 0.31 \[ \frac{B x^{2}}{2 b} + \frac{a \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{3}} - \frac{x \left (- A b + B a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**2/(2*b) + a*(-A*b + B*a)*log(a + b*x)/b**3 - x*(-A*b + B*a)/b**2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269534, size = 101, normalized size = 0.84 \[ \frac{B b x^{2}{\rm sign}\left (b x + a\right ) - 2 \, B a x{\rm sign}\left (b x + a\right ) + 2 \, A b x{\rm sign}\left (b x + a\right )}{2 \, b^{2}} + \frac{{\left (B a^{2}{\rm sign}\left (b x + a\right ) - A a b{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(B*b*x^2*sign(b*x + a) - 2*B*a*x*sign(b*x + a) + 2*A*b*x*sign(b*x + a))/b^2
+ (B*a^2*sign(b*x + a) - A*a*b*sign(b*x + a))*ln(abs(b*x + a))/b^3