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

Optimal. Leaf size=69 \[ -\frac{a^2 (A b-a B)}{b^4 (a+b x)}-\frac{a (2 A b-3 a B) \log (a+b x)}{b^4}+\frac{x (A b-2 a B)}{b^3}+\frac{B x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.133818, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a^2 (A b-a B)}{b^4 (a+b x)}-\frac{a (2 A b-3 a B) \log (a+b x)}{b^4}+\frac{x (A b-2 a B)}{b^3}+\frac{B x^2}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B \int x\, dx}{b^{2}} - \frac{a^{2} \left (A b - B a\right )}{b^{4} \left (a + b x\right )} - \frac{a \left (2 A b - 3 B a\right ) \log{\left (a + b x \right )}}{b^{4}} + \left (A b - 2 B a\right ) \int \frac{1}{b^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*Integral(x, x)/b**2 - a**2*(A*b - B*a)/(b**4*(a + b*x)) - a*(2*A*b - 3*B*a)*lo
g(a + b*x)/b**4 + (A*b - 2*B*a)*Integral(b**(-3), x)

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Mathematica [A]  time = 0.0908134, size = 66, normalized size = 0.96 \[ \frac{\frac{2 a^2 (a B-A b)}{a+b x}+2 b x (A b-2 a B)+2 a (3 a B-2 A b) \log (a+b x)+b^2 B x^2}{2 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 84, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,{b}^{2}}}+{\frac{Ax}{{b}^{2}}}-2\,{\frac{Bax}{{b}^{3}}}-2\,{\frac{a\ln \left ( bx+a \right ) A}{{b}^{3}}}+3\,{\frac{{a}^{2}\ln \left ( bx+a \right ) B}{{b}^{4}}}-{\frac{{a}^{2}A}{ \left ( bx+a \right ){b}^{3}}}+{\frac{{a}^{3}B}{ \left ( bx+a \right ){b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(B*x+A)/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.35206, size = 100, normalized size = 1.45 \[ \frac{B a^{3} - A a^{2} b}{b^{5} x + a b^{4}} + \frac{B b x^{2} - 2 \,{\left (2 \, B a - A b\right )} x}{2 \, b^{3}} + \frac{{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left (b x + a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.20149, size = 153, normalized size = 2.22 \[ \frac{B b^{3} x^{3} + 2 \, B a^{3} - 2 \, A a^{2} b -{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} - 2 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x + 2 \,{\left (3 \, B a^{3} - 2 \, A a^{2} b +{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.30673, size = 66, normalized size = 0.96 \[ \frac{B x^{2}}{2 b^{2}} + \frac{a \left (- 2 A b + 3 B a\right ) \log{\left (a + b x \right )}}{b^{4}} + \frac{- A a^{2} b + B a^{3}}{a b^{4} + b^{5} x} - \frac{x \left (- A b + 2 B a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**2/(2*b**2) + a*(-2*A*b + 3*B*a)*log(a + b*x)/b**4 + (-A*a**2*b + B*a**3)/(a
*b**4 + b**5*x) - x*(-A*b + 2*B*a)/b**3

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GIAC/XCAS [A]  time = 0.258722, size = 150, normalized size = 2.17 \[ \frac{{\left (b x + a\right )}^{2}{\left (B - \frac{2 \,{\left (3 \, B a b - A b^{2}\right )}}{{\left (b x + a\right )} b}\right )}}{2 \, b^{4}} - \frac{{\left (3 \, B a^{2} - 2 \, A a b\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{\frac{B a^{3} b^{2}}{b x + a} - \frac{A a^{2} b^{3}}{b x + a}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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