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

Optimal. Leaf size=106 \[ -\frac{a x (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.115377, antiderivative size = 106, 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 \[ -\frac{a x (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^2 (a+b x) \log (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

-((a*x*(a + b*x))/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + (x^2*(a + b*x))/(2*b*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) + (a^2*(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 = 15.9445, size = 99, normalized size = 0.93 \[ \frac{a^{2} \left (a + b x\right ) \log{\left (a + b x \right )}}{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{a \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{3}} + \frac{x^{2} \left (2 a + 2 b x\right )}{4 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0271643, size = 45, normalized size = 0.42 \[ \frac{(a+b x) \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 44, normalized size = 0.4 \[{\frac{ \left ( bx+a \right ) \left ({b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -2\,abx \right ) }{2\,{b}^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/((b*x+a)^2)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.713959, size = 55, normalized size = 0.52 \[ \frac{a^{2} b^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{a b x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{x^{2}}{2 \, \sqrt{b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.216941, size = 39, normalized size = 0.37 \[ \frac{b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.13109, size = 26, normalized size = 0.25 \[ \frac{a^{2} \log{\left (a + b x \right )}}{b^{3}} - \frac{a x}{b^{2}} + \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207935, size = 65, normalized size = 0.61 \[ \frac{a^{2}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (b x + a\right )}{b^{3}} + \frac{b x^{2}{\rm sign}\left (b x + a\right ) - 2 \, a x{\rm sign}\left (b x + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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