3.644 \(\int \frac{x^5 (A+B x)}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=146 \[ \frac{a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac{a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac{5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac{5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac{5 a (A b-3 a B)}{b^7 (a+b x)}+\frac{(A b-6 a B) \log (a+b x)}{b^7}+\frac{B x}{b^6} \]

[Out]

(B*x)/b^6 + (a^5*(A*b - a*B))/(5*b^7*(a + b*x)^5) - (a^4*(5*A*b - 6*a*B))/(4*b^7*(a + b*x)^4) + (5*a^3*(2*A*b
- 3*a*B))/(3*b^7*(a + b*x)^3) - (5*a^2*(A*b - 2*a*B))/(b^7*(a + b*x)^2) + (5*a*(A*b - 3*a*B))/(b^7*(a + b*x))
+ ((A*b - 6*a*B)*Log[a + b*x])/b^7

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Rubi [A]  time = 0.163201, antiderivative size = 146, 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} \[ \frac{a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac{a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac{5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac{5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac{5 a (A b-3 a B)}{b^7 (a+b x)}+\frac{(A b-6 a B) \log (a+b x)}{b^7}+\frac{B x}{b^6} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(B*x)/b^6 + (a^5*(A*b - a*B))/(5*b^7*(a + b*x)^5) - (a^4*(5*A*b - 6*a*B))/(4*b^7*(a + b*x)^4) + (5*a^3*(2*A*b
- 3*a*B))/(3*b^7*(a + b*x)^3) - (5*a^2*(A*b - 2*a*B))/(b^7*(a + b*x)^2) + (5*a*(A*b - 3*a*B))/(b^7*(a + b*x))
+ ((A*b - 6*a*B)*Log[a + b*x])/b^7

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^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{x^5 (A+B x)}{(a+b x)^6} \, dx\\ &=\int \left (\frac{B}{b^6}+\frac{a^5 (-A b+a B)}{b^6 (a+b x)^6}-\frac{a^4 (-5 A b+6 a B)}{b^6 (a+b x)^5}+\frac{5 a^3 (-2 A b+3 a B)}{b^6 (a+b x)^4}-\frac{10 a^2 (-A b+2 a B)}{b^6 (a+b x)^3}+\frac{5 a (-A b+3 a B)}{b^6 (a+b x)^2}+\frac{A b-6 a B}{b^6 (a+b x)}\right ) \, dx\\ &=\frac{B x}{b^6}+\frac{a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac{a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac{5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac{5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac{5 a (A b-3 a B)}{b^7 (a+b x)}+\frac{(A b-6 a B) \log (a+b x)}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.0838994, size = 130, normalized size = 0.89 \[ \frac{\frac{12 a^5 (A b-a B)}{(a+b x)^5}+\frac{15 a^4 (6 a B-5 A b)}{(a+b x)^4}+\frac{100 a^3 (2 A b-3 a B)}{(a+b x)^3}+\frac{300 a^2 (2 a B-A b)}{(a+b x)^2}+\frac{300 a (A b-3 a B)}{a+b x}+60 (A b-6 a B) \log (a+b x)+60 b B x}{60 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(60*b*B*x + (12*a^5*(A*b - a*B))/(a + b*x)^5 + (15*a^4*(-5*A*b + 6*a*B))/(a + b*x)^4 + (100*a^3*(2*A*b - 3*a*B
))/(a + b*x)^3 + (300*a^2*(-(A*b) + 2*a*B))/(a + b*x)^2 + (300*a*(A*b - 3*a*B))/(a + b*x) + 60*(A*b - 6*a*B)*L
og[a + b*x])/(60*b^7)

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Maple [A]  time = 0.008, size = 190, normalized size = 1.3 \begin{align*}{\frac{Bx}{{b}^{6}}}+5\,{\frac{aA}{{b}^{6} \left ( bx+a \right ) }}-15\,{\frac{B{a}^{2}}{{b}^{7} \left ( bx+a \right ) }}-5\,{\frac{A{a}^{2}}{{b}^{6} \left ( bx+a \right ) ^{2}}}+10\,{\frac{B{a}^{3}}{{b}^{7} \left ( bx+a \right ) ^{2}}}+{\frac{\ln \left ( bx+a \right ) A}{{b}^{6}}}-6\,{\frac{\ln \left ( bx+a \right ) aB}{{b}^{7}}}+{\frac{{a}^{5}A}{5\,{b}^{6} \left ( bx+a \right ) ^{5}}}-{\frac{B{a}^{6}}{5\,{b}^{7} \left ( bx+a \right ) ^{5}}}-{\frac{5\,{a}^{4}A}{4\,{b}^{6} \left ( bx+a \right ) ^{4}}}+{\frac{3\,B{a}^{5}}{2\,{b}^{7} \left ( bx+a \right ) ^{4}}}+{\frac{10\,A{a}^{3}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}}-5\,{\frac{B{a}^{4}}{{b}^{7} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

B*x/b^6+5*a/b^6/(b*x+a)*A-15*a^2/b^7/(b*x+a)*B-5*a^2/b^6/(b*x+a)^2*A+10*a^3/b^7/(b*x+a)^2*B+1/b^6*ln(b*x+a)*A-
6/b^7*ln(b*x+a)*a*B+1/5*a^5/b^6/(b*x+a)^5*A-1/5*a^6/b^7/(b*x+a)^5*B-5/4*a^4/b^6/(b*x+a)^4*A+3/2*a^5/b^7/(b*x+a
)^4*B+10/3*a^3/b^6/(b*x+a)^3*A-5*a^4/b^7/(b*x+a)^3*B

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Maxima [A]  time = 1.04954, size = 257, normalized size = 1.76 \begin{align*} -\frac{522 \, B a^{6} - 137 \, A a^{5} b + 300 \,{\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 300 \,{\left (10 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 100 \,{\left (39 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (462 \, B a^{5} b - 125 \, A a^{4} b^{2}\right )} x}{60 \,{\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} + \frac{B x}{b^{6}} - \frac{{\left (6 \, B a - A b\right )} \log \left (b x + a\right )}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/60*(522*B*a^6 - 137*A*a^5*b + 300*(3*B*a^2*b^4 - A*a*b^5)*x^4 + 300*(10*B*a^3*b^3 - 3*A*a^2*b^4)*x^3 + 100*
(39*B*a^4*b^2 - 11*A*a^3*b^3)*x^2 + 5*(462*B*a^5*b - 125*A*a^4*b^2)*x)/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*
x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7) + B*x/b^6 - (6*B*a - A*b)*log(b*x + a)/b^7

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Fricas [B]  time = 1.31446, size = 663, normalized size = 4.54 \begin{align*} \frac{60 \, B b^{6} x^{6} + 300 \, B a b^{5} x^{5} - 522 \, B a^{6} + 137 \, A a^{5} b - 300 \,{\left (B a^{2} b^{4} - A a b^{5}\right )} x^{4} - 300 \,{\left (8 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} - 100 \,{\left (36 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} - 125 \,{\left (18 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x - 60 \,{\left (6 \, B a^{6} - A a^{5} b +{\left (6 \, B a b^{5} - A b^{6}\right )} x^{5} + 5 \,{\left (6 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 10 \,{\left (6 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 10 \,{\left (6 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (6 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \log \left (b x + a\right )}{60 \,{\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/60*(60*B*b^6*x^6 + 300*B*a*b^5*x^5 - 522*B*a^6 + 137*A*a^5*b - 300*(B*a^2*b^4 - A*a*b^5)*x^4 - 300*(8*B*a^3*
b^3 - 3*A*a^2*b^4)*x^3 - 100*(36*B*a^4*b^2 - 11*A*a^3*b^3)*x^2 - 125*(18*B*a^5*b - 5*A*a^4*b^2)*x - 60*(6*B*a^
6 - A*a^5*b + (6*B*a*b^5 - A*b^6)*x^5 + 5*(6*B*a^2*b^4 - A*a*b^5)*x^4 + 10*(6*B*a^3*b^3 - A*a^2*b^4)*x^3 + 10*
(6*B*a^4*b^2 - A*a^3*b^3)*x^2 + 5*(6*B*a^5*b - A*a^4*b^2)*x)*log(b*x + a))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b
^10*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7)

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Sympy [A]  time = 3.45064, size = 190, normalized size = 1.3 \begin{align*} \frac{B x}{b^{6}} - \frac{- 137 A a^{5} b + 522 B a^{6} + x^{4} \left (- 300 A a b^{5} + 900 B a^{2} b^{4}\right ) + x^{3} \left (- 900 A a^{2} b^{4} + 3000 B a^{3} b^{3}\right ) + x^{2} \left (- 1100 A a^{3} b^{3} + 3900 B a^{4} b^{2}\right ) + x \left (- 625 A a^{4} b^{2} + 2310 B a^{5} b\right )}{60 a^{5} b^{7} + 300 a^{4} b^{8} x + 600 a^{3} b^{9} x^{2} + 600 a^{2} b^{10} x^{3} + 300 a b^{11} x^{4} + 60 b^{12} x^{5}} - \frac{\left (- A b + 6 B a\right ) \log{\left (a + b x \right )}}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

B*x/b**6 - (-137*A*a**5*b + 522*B*a**6 + x**4*(-300*A*a*b**5 + 900*B*a**2*b**4) + x**3*(-900*A*a**2*b**4 + 300
0*B*a**3*b**3) + x**2*(-1100*A*a**3*b**3 + 3900*B*a**4*b**2) + x*(-625*A*a**4*b**2 + 2310*B*a**5*b))/(60*a**5*
b**7 + 300*a**4*b**8*x + 600*a**3*b**9*x**2 + 600*a**2*b**10*x**3 + 300*a*b**11*x**4 + 60*b**12*x**5) - (-A*b
+ 6*B*a)*log(a + b*x)/b**7

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Giac [A]  time = 1.17584, size = 194, normalized size = 1.33 \begin{align*} \frac{B x}{b^{6}} - \frac{{\left (6 \, B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{522 \, B a^{6} - 137 \, A a^{5} b + 300 \,{\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 300 \,{\left (10 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 100 \,{\left (39 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} + 5 \,{\left (462 \, B a^{5} b - 125 \, A a^{4} b^{2}\right )} x}{60 \,{\left (b x + a\right )}^{5} b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

B*x/b^6 - (6*B*a - A*b)*log(abs(b*x + a))/b^7 - 1/60*(522*B*a^6 - 137*A*a^5*b + 300*(3*B*a^2*b^4 - A*a*b^5)*x^
4 + 300*(10*B*a^3*b^3 - 3*A*a^2*b^4)*x^3 + 100*(39*B*a^4*b^2 - 11*A*a^3*b^3)*x^2 + 5*(462*B*a^5*b - 125*A*a^4*
b^2)*x)/((b*x + a)^5*b^7)