3.164 \(\int \frac{x^2 (A+B x)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=71 \[ -\frac{a^2 (A b-a B)}{2 b^4 (a+b x)^2}+\frac{a (2 A b-3 a B)}{b^4 (a+b x)}+\frac{(A b-3 a B) \log (a+b x)}{b^4}+\frac{B x}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.148667, antiderivative size = 71, 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 \[ -\frac{a^2 (A b-a B)}{2 b^4 (a+b x)^2}+\frac{a (2 A b-3 a B)}{b^4 (a+b x)}+\frac{(A b-3 a B) \log (a+b x)}{b^4}+\frac{B x}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

(B*x)/b^3 - (a^2*(A*b - a*B))/(2*b^4*(a + b*x)^2) + (a*(2*A*b - 3*a*B))/(b^4*(a
+ b*x)) + ((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. \[ - \frac{a^{2} \left (A b - B a\right )}{2 b^{4} \left (a + b x\right )^{2}} + \frac{a \left (2 A b - 3 B a\right )}{b^{4} \left (a + b x\right )} + \frac{\int B\, dx}{b^{3}} + \frac{\left (A b - 3 B a\right ) \log{\left (a + b x \right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0451, size = 75, normalized size = 1.06 \[ \frac{2 a A b-3 a^2 B}{b^4 (a+b x)}+\frac{a^3 B-a^2 A b}{2 b^4 (a+b x)^2}+\frac{(A b-3 a B) \log (a+b x)}{b^4}+\frac{B x}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 94, normalized size = 1.3 \[{\frac{Bx}{{b}^{3}}}+{\frac{\ln \left ( bx+a \right ) A}{{b}^{3}}}-3\,{\frac{\ln \left ( bx+a \right ) Ba}{{b}^{4}}}+2\,{\frac{aA}{ \left ( bx+a \right ){b}^{3}}}-3\,{\frac{{a}^{2}B}{ \left ( bx+a \right ){b}^{4}}}-{\frac{{a}^{2}A}{2\, \left ( bx+a \right ) ^{2}{b}^{3}}}+{\frac{{a}^{3}B}{2\, \left ( bx+a \right ) ^{2}{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(B*x+A)/(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.34258, size = 115, normalized size = 1.62 \[ -\frac{5 \, B a^{3} - 3 \, A a^{2} b + 2 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac{B x}{b^{3}} - \frac{{\left (3 \, B a - A b\right )} \log \left (b x + a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^2/(b*x + a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.205285, size = 181, normalized size = 2.55 \[ \frac{2 \, B b^{3} x^{3} + 4 \, B a b^{2} x^{2} - 5 \, B a^{3} + 3 \, A a^{2} b - 4 \,{\left (B a^{2} b - A a b^{2}\right )} x - 2 \,{\left (3 \, B a^{3} - A a^{2} b +{\left (3 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} b - A a b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*B*b^3*x^3 + 4*B*a*b^2*x^2 - 5*B*a^3 + 3*A*a^2*b - 4*(B*a^2*b - A*a*b^2)*x
- 2*(3*B*a^3 - A*a^2*b + (3*B*a*b^2 - A*b^3)*x^2 + 2*(3*B*a^2*b - A*a*b^2)*x)*l
og(b*x + a))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)

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Sympy [A]  time = 4.42159, size = 83, normalized size = 1.17 \[ \frac{B x}{b^{3}} - \frac{- 3 A a^{2} b + 5 B a^{3} + x \left (- 4 A a b^{2} + 6 B a^{2} b\right )}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} - \frac{\left (- A b + 3 B a\right ) \log{\left (a + b x \right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.253638, size = 97, normalized size = 1.37 \[ \frac{B x}{b^{3}} - \frac{{\left (3 \, B a - A b\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{5 \, B a^{3} - 3 \, A a^{2} b + 2 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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