3.636 \(\int \frac{x^2 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=72 \[ -\frac{a^2 B}{2 b^4 (a+b x)^2}+\frac{x^3 (A b-a B)}{3 a b (a+b x)^3}+\frac{2 a B}{b^4 (a+b x)}+\frac{B \log (a+b x)}{b^4} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0912492, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{a^2 B}{2 b^4 (a+b x)^2}+\frac{x^3 (A b-a B)}{3 a b (a+b x)^3}+\frac{2 a B}{b^4 (a+b x)}+\frac{B \log (a+b x)}{b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.1829, size = 63, normalized size = 0.88 \[ - \frac{B a^{2}}{2 b^{4} \left (a + b x\right )^{2}} + \frac{2 B a}{b^{4} \left (a + b x\right )} + \frac{B \log{\left (a + b x \right )}}{b^{4}} + \frac{x^{3} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.044444, size = 73, normalized size = 1.01 \[ \frac{11 a^3 B+a^2 (27 b B x-2 A b)-6 a b^2 x (A-3 B x)+6 B (a+b x)^3 \log (a+b x)-6 A b^3 x^2}{6 b^4 (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(11*a^3*B - 6*A*b^3*x^2 - 6*a*b^2*x*(A - 3*B*x) + a^2*(-2*A*b + 27*b*B*x) + 6*B*
(a + b*x)^3*Log[a + b*x])/(6*b^4*(a + b*x)^3)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 101, normalized size = 1.4 \[{\frac{aA}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{2}B}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{A{a}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{B{a}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}+{\frac{B\ln \left ( bx+a \right ) }{{b}^{4}}}-{\frac{A}{{b}^{3} \left ( bx+a \right ) }}+3\,{\frac{Ba}{{b}^{4} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a/b^3/(b*x+a)^2*A-3/2*a^2*B/b^4/(b*x+a)^2-1/3*a^2/b^3/(b*x+a)^3*A+1/3*a^3/b^4/(b
*x+a)^3*B+B*ln(b*x+a)/b^4-1/b^3/(b*x+a)*A+3*a*B/b^4/(b*x+a)

_______________________________________________________________________________________

Maxima [A]  time = 0.685471, size = 135, normalized size = 1.88 \[ \frac{11 \, B a^{3} - 2 \, A a^{2} b + 6 \,{\left (3 \, B a b^{2} - A b^{3}\right )} x^{2} + 3 \,{\left (9 \, B a^{2} b - 2 \, A a b^{2}\right )} x}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac{B \log \left (b x + a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(11*B*a^3 - 2*A*a^2*b + 6*(3*B*a*b^2 - A*b^3)*x^2 + 3*(9*B*a^2*b - 2*A*a*b^2
)*x)/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4) + B*log(b*x + a)/b^4

_______________________________________________________________________________________

Fricas [A]  time = 0.283095, size = 173, normalized size = 2.4 \[ \frac{11 \, B a^{3} - 2 \, A a^{2} b + 6 \,{\left (3 \, B a b^{2} - A b^{3}\right )} x^{2} + 3 \,{\left (9 \, B a^{2} b - 2 \, A a b^{2}\right )} x + 6 \,{\left (B b^{3} x^{3} + 3 \, B a b^{2} x^{2} + 3 \, B a^{2} b x + B a^{3}\right )} \log \left (b x + a\right )}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(11*B*a^3 - 2*A*a^2*b + 6*(3*B*a*b^2 - A*b^3)*x^2 + 3*(9*B*a^2*b - 2*A*a*b^2
)*x + 6*(B*b^3*x^3 + 3*B*a*b^2*x^2 + 3*B*a^2*b*x + B*a^3)*log(b*x + a))/(b^7*x^3
+ 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4)

_______________________________________________________________________________________

Sympy [A]  time = 3.28232, size = 100, normalized size = 1.39 \[ \frac{B \log{\left (a + b x \right )}}{b^{4}} + \frac{- 2 A a^{2} b + 11 B a^{3} + x^{2} \left (- 6 A b^{3} + 18 B a b^{2}\right ) + x \left (- 6 A a b^{2} + 27 B a^{2} b\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*log(a + b*x)/b**4 + (-2*A*a**2*b + 11*B*a**3 + x**2*(-6*A*b**3 + 18*B*a*b**2)
+ x*(-6*A*a*b**2 + 27*B*a**2*b))/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2
+ 6*b**7*x**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269587, size = 103, normalized size = 1.43 \[ \frac{B{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} + \frac{6 \,{\left (3 \, B a b - A b^{2}\right )} x^{2} + 3 \,{\left (9 \, B a^{2} - 2 \, A a b\right )} x + \frac{11 \, B a^{3} - 2 \, A a^{2} b}{b}}{6 \,{\left (b x + a\right )}^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*ln(abs(b*x + a))/b^4 + 1/6*(6*(3*B*a*b - A*b^2)*x^2 + 3*(9*B*a^2 - 2*A*a*b)*x
+ (11*B*a^3 - 2*A*a^2*b)/b)/((b*x + a)^3*b^3)