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

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

________________________________________________________________________________________

Rubi [A]  time = 0.0574297, 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, Rules used = {77} \[ -\frac{a^2 (A b-a B)}{2 b^4 (a+b x)^2}+\frac{a (2 A b-3 a B)}{b^4 (a+b x)}+\frac{(A b-3 a B) \log (a+b x)}{b^4}+\frac{B x}{b^3} \]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{x^2 (A+B x)}{(a+b x)^3} \, dx &=\int \left (\frac{B}{b^3}-\frac{a^2 (-A b+a B)}{b^3 (a+b x)^3}+\frac{a (-2 A b+3 a B)}{b^3 (a+b x)^2}+\frac{A b-3 a B}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{B x}{b^3}-\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}\\ \end{align*}

Mathematica [A]  time = 0.0268556, 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)*Lo
g[a + b*x])/b^4

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 94, normalized size = 1.3 \begin{align*}{\frac{Bx}{{b}^{3}}}+2\,{\frac{aA}{{b}^{3} \left ( bx+a \right ) }}-3\,{\frac{B{a}^{2}}{{b}^{4} \left ( bx+a \right ) }}-{\frac{{a}^{2}A}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{B{a}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}+{\frac{\ln \left ( bx+a \right ) A}{{b}^{3}}}-3\,{\frac{\ln \left ( bx+a \right ) Ba}{{b}^{4}}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.05091, size = 115, normalized size = 1.62 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(B*x+A)/(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

________________________________________________________________________________________

Fricas [A]  time = 1.94591, size = 278, normalized size = 3.92 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Sympy [A]  time = 0.863201, size = 83, normalized size = 1.17 \begin{align*} \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}} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.19098, size = 97, normalized size = 1.37 \begin{align*} \frac{B x}{b^{3}} - \frac{{\left (3 \, B a - A b\right )} \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b^3 - (3*B*a - A*b)*log(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)