3.634 \(\int \frac{x^4 (A+B x)}{(a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=121 \[ -\frac{a^4 (A b-a B)}{3 b^6 (a+b x)^3}+\frac{a^3 (4 A b-5 a B)}{2 b^6 (a+b x)^2}-\frac{2 a^2 (3 A b-5 a B)}{b^6 (a+b x)}+\frac{x (A b-4 a B)}{b^5}-\frac{2 a (2 A b-5 a B) \log (a+b x)}{b^6}+\frac{B x^2}{2 b^4} \]

[Out]

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

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Rubi [A]  time = 0.123403, antiderivative size = 121, 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} \[ -\frac{a^4 (A b-a B)}{3 b^6 (a+b x)^3}+\frac{a^3 (4 A b-5 a B)}{2 b^6 (a+b x)^2}-\frac{2 a^2 (3 A b-5 a B)}{b^6 (a+b x)}+\frac{x (A b-4 a B)}{b^5}-\frac{2 a (2 A b-5 a B) \log (a+b x)}{b^6}+\frac{B x^2}{2 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

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^4 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{x^4 (A+B x)}{(a+b x)^4} \, dx\\ &=\int \left (\frac{A b-4 a B}{b^5}+\frac{B x}{b^4}-\frac{a^4 (-A b+a B)}{b^5 (a+b x)^4}+\frac{a^3 (-4 A b+5 a B)}{b^5 (a+b x)^3}-\frac{2 a^2 (-3 A b+5 a B)}{b^5 (a+b x)^2}+\frac{2 a (-2 A b+5 a B)}{b^5 (a+b x)}\right ) \, dx\\ &=\frac{(A b-4 a B) x}{b^5}+\frac{B x^2}{2 b^4}-\frac{a^4 (A b-a B)}{3 b^6 (a+b x)^3}+\frac{a^3 (4 A b-5 a B)}{2 b^6 (a+b x)^2}-\frac{2 a^2 (3 A b-5 a B)}{b^6 (a+b x)}-\frac{2 a (2 A b-5 a B) \log (a+b x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0461896, size = 129, normalized size = 1.07 \[ \frac{2 \left (5 a^3 B-3 a^2 A b\right )}{b^6 (a+b x)}+\frac{4 a^3 A b-5 a^4 B}{2 b^6 (a+b x)^2}+\frac{a^5 B-a^4 A b}{3 b^6 (a+b x)^3}+\frac{2 \left (5 a^2 B-2 a A b\right ) \log (a+b x)}{b^6}+\frac{x (A b-4 a B)}{b^5}+\frac{B x^2}{2 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.01, size = 149, normalized size = 1.2 \begin{align*}{\frac{B{x}^{2}}{2\,{b}^{4}}}+{\frac{Ax}{{b}^{4}}}-4\,{\frac{aBx}{{b}^{5}}}-6\,{\frac{A{a}^{2}}{{b}^{5} \left ( bx+a \right ) }}+10\,{\frac{B{a}^{3}}{{b}^{6} \left ( bx+a \right ) }}+2\,{\frac{A{a}^{3}}{{b}^{5} \left ( bx+a \right ) ^{2}}}-{\frac{5\,B{a}^{4}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}-4\,{\frac{a\ln \left ( bx+a \right ) A}{{b}^{5}}}+10\,{\frac{{a}^{2}\ln \left ( bx+a \right ) B}{{b}^{6}}}-{\frac{{a}^{4}A}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{B{a}^{5}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/2*B*x^2/b^4+1/b^4*A*x-4/b^5*a*B*x-6*a^2/b^5/(b*x+a)*A+10*a^3/b^6/(b*x+a)*B+2*a^3/b^5/(b*x+a)^2*A-5/2*a^4/b^6
/(b*x+a)^2*B-4*a/b^5*ln(b*x+a)*A+10*a^2/b^6*ln(b*x+a)*B-1/3*a^4/b^5/(b*x+a)^3*A+1/3*a^5/b^6/(b*x+a)^3*B

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Maxima [A]  time = 1.02238, size = 193, normalized size = 1.6 \begin{align*} \frac{47 \, B a^{5} - 26 \, A a^{4} b + 12 \,{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 15 \,{\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{6 \,{\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac{B b x^{2} - 2 \,{\left (4 \, B a - A b\right )} x}{2 \, b^{5}} + \frac{2 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/6*(47*B*a^5 - 26*A*a^4*b + 12*(5*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 15*(7*B*a^4*b - 4*A*a^3*b^2)*x)/(b^9*x^3 + 3
*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6) + 1/2*(B*b*x^2 - 2*(4*B*a - A*b)*x)/b^5 + 2*(5*B*a^2 - 2*A*a*b)*log(b*x +
a)/b^6

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Fricas [A]  time = 1.23972, size = 489, normalized size = 4.04 \begin{align*} \frac{3 \, B b^{5} x^{5} + 47 \, B a^{5} - 26 \, A a^{4} b - 3 \,{\left (5 \, B a b^{4} - 2 \, A b^{5}\right )} x^{4} - 9 \,{\left (7 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} - 9 \,{\left (B a^{3} b^{2} + 2 \, A a^{2} b^{3}\right )} x^{2} + 27 \,{\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x + 12 \,{\left (5 \, B a^{5} - 2 \, A a^{4} b +{\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} + 3 \,{\left (5 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 3 \,{\left (5 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/6*(3*B*b^5*x^5 + 47*B*a^5 - 26*A*a^4*b - 3*(5*B*a*b^4 - 2*A*b^5)*x^4 - 9*(7*B*a^2*b^3 - 2*A*a*b^4)*x^3 - 9*(
B*a^3*b^2 + 2*A*a^2*b^3)*x^2 + 27*(3*B*a^4*b - 2*A*a^3*b^2)*x + 12*(5*B*a^5 - 2*A*a^4*b + (5*B*a^2*b^3 - 2*A*a
*b^4)*x^3 + 3*(5*B*a^3*b^2 - 2*A*a^2*b^3)*x^2 + 3*(5*B*a^4*b - 2*A*a^3*b^2)*x)*log(b*x + a))/(b^9*x^3 + 3*a*b^
8*x^2 + 3*a^2*b^7*x + a^3*b^6)

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Sympy [A]  time = 1.55419, size = 143, normalized size = 1.18 \begin{align*} \frac{B x^{2}}{2 b^{4}} + \frac{2 a \left (- 2 A b + 5 B a\right ) \log{\left (a + b x \right )}}{b^{6}} + \frac{- 26 A a^{4} b + 47 B a^{5} + x^{2} \left (- 36 A a^{2} b^{3} + 60 B a^{3} b^{2}\right ) + x \left (- 60 A a^{3} b^{2} + 105 B a^{4} b\right )}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} - \frac{x \left (- A b + 4 B a\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

B*x**2/(2*b**4) + 2*a*(-2*A*b + 5*B*a)*log(a + b*x)/b**6 + (-26*A*a**4*b + 47*B*a**5 + x**2*(-36*A*a**2*b**3 +
60*B*a**3*b**2) + x*(-60*A*a**3*b**2 + 105*B*a**4*b))/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9
*x**3) - x*(-A*b + 4*B*a)/b**5

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Giac [A]  time = 1.15934, size = 167, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (5 \, B a^{2} - 2 \, A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{B b^{4} x^{2} - 8 \, B a b^{3} x + 2 \, A b^{4} x}{2 \, b^{8}} + \frac{47 \, B a^{5} - 26 \, A a^{4} b + 12 \,{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 15 \,{\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{6 \,{\left (b x + a\right )}^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

2*(5*B*a^2 - 2*A*a*b)*log(abs(b*x + a))/b^6 + 1/2*(B*b^4*x^2 - 8*B*a*b^3*x + 2*A*b^4*x)/b^8 + 1/6*(47*B*a^5 -
26*A*a^4*b + 12*(5*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 15*(7*B*a^4*b - 4*A*a^3*b^2)*x)/((b*x + a)^3*b^6)