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

Optimal. Leaf size=97 \[ \frac{a^3 (A b-a B)}{3 b^5 (a+b x)^3}-\frac{a^2 (3 A b-4 a B)}{2 b^5 (a+b x)^2}+\frac{3 a (A b-2 a B)}{b^5 (a+b x)}+\frac{(A b-4 a B) \log (a+b x)}{b^5}+\frac{B x}{b^4} \]

[Out]

(B*x)/b^4 + (a^3*(A*b - a*B))/(3*b^5*(a + b*x)^3) - (a^2*(3*A*b - 4*a*B))/(2*b^5*(a + b*x)^2) + (3*a*(A*b - 2*
a*B))/(b^5*(a + b*x)) + ((A*b - 4*a*B)*Log[a + b*x])/b^5

________________________________________________________________________________________

Rubi [A]  time = 0.088243, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 77} \[ \frac{a^3 (A b-a B)}{3 b^5 (a+b x)^3}-\frac{a^2 (3 A b-4 a B)}{2 b^5 (a+b x)^2}+\frac{3 a (A b-2 a B)}{b^5 (a+b x)}+\frac{(A b-4 a B) \log (a+b x)}{b^5}+\frac{B x}{b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*x)/b^4 + (a^3*(A*b - a*B))/(3*b^5*(a + b*x)^3) - (a^2*(3*A*b - 4*a*B))/(2*b^5*(a + b*x)^2) + (3*a*(A*b - 2*
a*B))/(b^5*(a + b*x)) + ((A*b - 4*a*B)*Log[a + b*x])/b^5

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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^3 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{x^3 (A+B x)}{(a+b x)^4} \, dx\\ &=\int \left (\frac{B}{b^4}+\frac{a^3 (-A b+a B)}{b^4 (a+b x)^4}-\frac{a^2 (-3 A b+4 a B)}{b^4 (a+b x)^3}+\frac{3 a (-A b+2 a B)}{b^4 (a+b x)^2}+\frac{A b-4 a B}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{B x}{b^4}+\frac{a^3 (A b-a B)}{3 b^5 (a+b x)^3}-\frac{a^2 (3 A b-4 a B)}{2 b^5 (a+b x)^2}+\frac{3 a (A b-2 a B)}{b^5 (a+b x)}+\frac{(A b-4 a B) \log (a+b x)}{b^5}\\ \end{align*}

Mathematica [A]  time = 0.0374205, size = 97, normalized size = 1. \[ \frac{9 a^2 b^2 x (3 A-2 B x)+a^3 b (11 A-54 B x)-26 a^4 B+18 a b^3 x^2 (A+B x)+6 (a+b x)^3 (A b-4 a B) \log (a+b x)+6 b^4 B x^4}{6 b^5 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-26*a^4*B + 6*b^4*B*x^4 + a^3*b*(11*A - 54*B*x) + 9*a^2*b^2*x*(3*A - 2*B*x) + 18*a*b^3*x^2*(A + B*x) + 6*(A*b
- 4*a*B)*(a + b*x)^3*Log[a + b*x])/(6*b^5*(a + b*x)^3)

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 126, normalized size = 1.3 \begin{align*}{\frac{Bx}{{b}^{4}}}+3\,{\frac{aA}{{b}^{4} \left ( bx+a \right ) }}-6\,{\frac{B{a}^{2}}{{b}^{5} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) A}{{b}^{4}}}-4\,{\frac{\ln \left ( bx+a \right ) aB}{{b}^{5}}}-{\frac{3\,A{a}^{2}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}+2\,{\frac{B{a}^{3}}{{b}^{5} \left ( bx+a \right ) ^{2}}}+{\frac{A{a}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{B{a}^{4}}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.04908, size = 162, normalized size = 1.67 \begin{align*} -\frac{26 \, B a^{4} - 11 \, A a^{3} b + 18 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 3 \,{\left (20 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x}{6 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac{B x}{b^{4}} - \frac{{\left (4 \, B a - A b\right )} \log \left (b x + a\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(26*B*a^4 - 11*A*a^3*b + 18*(2*B*a^2*b^2 - A*a*b^3)*x^2 + 3*(20*B*a^3*b - 9*A*a^2*b^2)*x)/(b^8*x^3 + 3*a*
b^7*x^2 + 3*a^2*b^6*x + a^3*b^5) + B*x/b^4 - (4*B*a - A*b)*log(b*x + a)/b^5

________________________________________________________________________________________

Fricas [B]  time = 1.29585, size = 398, normalized size = 4.1 \begin{align*} \frac{6 \, B b^{4} x^{4} + 18 \, B a b^{3} x^{3} - 26 \, B a^{4} + 11 \, A a^{3} b - 18 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 27 \,{\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x - 6 \,{\left (4 \, B a^{4} - A a^{3} b +{\left (4 \, B a b^{3} - A b^{4}\right )} x^{3} + 3 \,{\left (4 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 3 \,{\left (4 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*B*b^4*x^4 + 18*B*a*b^3*x^3 - 26*B*a^4 + 11*A*a^3*b - 18*(B*a^2*b^2 - A*a*b^3)*x^2 - 27*(2*B*a^3*b - A*a
^2*b^2)*x - 6*(4*B*a^4 - A*a^3*b + (4*B*a*b^3 - A*b^4)*x^3 + 3*(4*B*a^2*b^2 - A*a*b^3)*x^2 + 3*(4*B*a^3*b - A*
a^2*b^2)*x)*log(b*x + a))/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5)

________________________________________________________________________________________

Sympy [A]  time = 1.19437, size = 119, normalized size = 1.23 \begin{align*} \frac{B x}{b^{4}} - \frac{- 11 A a^{3} b + 26 B a^{4} + x^{2} \left (- 18 A a b^{3} + 36 B a^{2} b^{2}\right ) + x \left (- 27 A a^{2} b^{2} + 60 B a^{3} b\right )}{6 a^{3} b^{5} + 18 a^{2} b^{6} x + 18 a b^{7} x^{2} + 6 b^{8} x^{3}} - \frac{\left (- A b + 4 B a\right ) \log{\left (a + b x \right )}}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b**4 - (-11*A*a**3*b + 26*B*a**4 + x**2*(-18*A*a*b**3 + 36*B*a**2*b**2) + x*(-27*A*a**2*b**2 + 60*B*a**3*b
))/(6*a**3*b**5 + 18*a**2*b**6*x + 18*a*b**7*x**2 + 6*b**8*x**3) - (-A*b + 4*B*a)*log(a + b*x)/b**5

________________________________________________________________________________________

Giac [A]  time = 1.14473, size = 130, normalized size = 1.34 \begin{align*} \frac{B x}{b^{4}} - \frac{{\left (4 \, B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac{26 \, B a^{4} - 11 \, A a^{3} b + 18 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 3 \,{\left (20 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x}{6 \,{\left (b x + a\right )}^{3} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*x/b^4 - (4*B*a - A*b)*log(abs(b*x + a))/b^5 - 1/6*(26*B*a^4 - 11*A*a^3*b + 18*(2*B*a^2*b^2 - A*a*b^3)*x^2 +
3*(20*B*a^3*b - 9*A*a^2*b^2)*x)/((b*x + a)^3*b^5)