3.196 \(\int \frac{1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=165 \[ -\frac{b}{2 a^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b \log (x) (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b (a+b x) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.181626, antiderivative size = 165, 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 \[ -\frac{b}{2 a^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b \log (x) (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b (a+b x) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.887, size = 163, normalized size = 0.99 \[ \frac{2 a + 2 b x}{4 a x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{3}{2 a^{2} x \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{3 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{4} \left (a + b x\right )} + \frac{3 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{4} \left (a + b x\right )} - \frac{3 \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0608851, size = 81, normalized size = 0.49 \[ \frac{-a \left (2 a^2+9 a b x+6 b^2 x^2\right )-6 b x \log (x) (a+b x)^2+6 b x (a+b x)^2 \log (a+b x)}{2 a^4 x (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-(a*(2*a^2 + 9*a*b*x + 6*b^2*x^2)) - 6*b*x*(a + b*x)^2*Log[x] + 6*b*x*(a + b*x)
^2*Log[a + b*x])/(2*a^4*x*(a + b*x)*Sqrt[(a + b*x)^2])

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 117, normalized size = 0.7 \[ -{\frac{ \left ( 6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+12\,a{b}^{2}\ln \left ( x \right ){x}^{2}-12\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}+6\,{a}^{2}b\ln \left ( x \right ) x-6\,\ln \left ( bx+a \right ) x{a}^{2}b+6\,a{b}^{2}{x}^{2}+9\,{a}^{2}bx+2\,{a}^{3} \right ) \left ( bx+a \right ) }{2\,{a}^{4}x} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(6*b^3*ln(x)*x^3-6*b^3*ln(b*x+a)*x^3+12*a*b^2*ln(x)*x^2-12*ln(b*x+a)*x^2*a*
b^2+6*a^2*b*ln(x)*x-6*ln(b*x+a)*x*a^2*b+6*a*b^2*x^2+9*a^2*b*x+2*a^3)*(b*x+a)/a^4
/x/((b*x+a)^2)^(3/2)

_______________________________________________________________________________________

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(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.230551, size = 147, normalized size = 0.89 \[ -\frac{6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3} - 6 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (x\right )}{2 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^2),x, algorithm="fricas")

[Out]

-1/2*(6*a*b^2*x^2 + 9*a^2*b*x + 2*a^3 - 6*(b^3*x^3 + 2*a*b^2*x^2 + a^2*b*x)*log(
b*x + a) + 6*(b^3*x^3 + 2*a*b^2*x^2 + a^2*b*x)*log(x))/(a^4*b^2*x^3 + 2*a^5*b*x^
2 + a^6*x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**2*((a + b*x)**2)**(3/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.569149, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^2),x, algorithm="giac")

[Out]

sage0*x