3.645 \(\int \frac{x^4 (A+B x)}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=106 \[ -\frac{3 a^2 B}{b^6 (a+b x)^2}+\frac{4 a^3 B}{3 b^6 (a+b x)^3}-\frac{a^4 B}{4 b^6 (a+b x)^4}+\frac{x^5 (A b-a B)}{5 a b (a+b x)^5}+\frac{4 a B}{b^6 (a+b x)}+\frac{B \log (a+b x)}{b^6} \]

[Out]

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

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Rubi [A]  time = 0.0617757, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {27, 78, 43} \[ -\frac{3 a^2 B}{b^6 (a+b x)^2}+\frac{4 a^3 B}{3 b^6 (a+b x)^3}-\frac{a^4 B}{4 b^6 (a+b x)^4}+\frac{x^5 (A b-a B)}{5 a b (a+b x)^5}+\frac{4 a B}{b^6 (a+b x)}+\frac{B \log (a+b x)}{b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*x^5)/(5*a*b*(a + b*x)^5) - (a^4*B)/(4*b^6*(a + b*x)^4) + (4*a^3*B)/(3*b^6*(a + b*x)^3) - (3*a^2*B
)/(b^6*(a + b*x)^2) + (4*a*B)/(b^6*(a + b*x)) + (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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^4 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac{x^4 (A+B x)}{(a+b x)^6} \, dx\\ &=\frac{(A b-a B) x^5}{5 a b (a+b x)^5}+\frac{B \int \frac{x^4}{(a+b x)^5} \, dx}{b}\\ &=\frac{(A b-a B) x^5}{5 a b (a+b x)^5}+\frac{B \int \left (\frac{a^4}{b^4 (a+b x)^5}-\frac{4 a^3}{b^4 (a+b x)^4}+\frac{6 a^2}{b^4 (a+b x)^3}-\frac{4 a}{b^4 (a+b x)^2}+\frac{1}{b^4 (a+b x)}\right ) \, dx}{b}\\ &=\frac{(A b-a B) x^5}{5 a b (a+b x)^5}-\frac{a^4 B}{4 b^6 (a+b x)^4}+\frac{4 a^3 B}{3 b^6 (a+b x)^3}-\frac{3 a^2 B}{b^6 (a+b x)^2}+\frac{4 a B}{b^6 (a+b x)}+\frac{B \log (a+b x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0426029, size = 113, normalized size = 1.07 \[ \frac{60 a^2 b^3 x^2 (15 B x-2 A)+20 a^3 b^2 x (55 B x-3 A)+a^4 (625 b B x-12 A b)+137 a^5 B+60 a b^4 x^3 (5 B x-2 A)+60 B (a+b x)^5 \log (a+b x)-60 A b^5 x^4}{60 b^6 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(137*a^5*B - 60*A*b^5*x^4 + 60*a*b^4*x^3*(-2*A + 5*B*x) + 60*a^2*b^3*x^2*(-2*A + 15*B*x) + 20*a^3*b^2*x*(-3*A
+ 55*B*x) + a^4*(-12*A*b + 625*b*B*x) + 60*B*(a + b*x)^5*Log[a + b*x])/(60*b^6*(a + b*x)^5)

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Maple [A]  time = 0.006, size = 165, normalized size = 1.6 \begin{align*} -{\frac{A}{{b}^{5} \left ( bx+a \right ) }}+5\,{\frac{aB}{{b}^{6} \left ( bx+a \right ) }}+{\frac{B\ln \left ( bx+a \right ) }{{b}^{6}}}-{\frac{A{a}^{4}}{5\,{b}^{5} \left ( bx+a \right ) ^{5}}}+{\frac{B{a}^{5}}{5\,{b}^{6} \left ( bx+a \right ) ^{5}}}+{\frac{A{a}^{3}}{{b}^{5} \left ( bx+a \right ) ^{4}}}-{\frac{5\,B{a}^{4}}{4\,{b}^{6} \left ( bx+a \right ) ^{4}}}+2\,{\frac{aA}{{b}^{5} \left ( bx+a \right ) ^{2}}}-5\,{\frac{B{a}^{2}}{{b}^{6} \left ( bx+a \right ) ^{2}}}-2\,{\frac{A{a}^{2}}{{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{10\,B{a}^{3}}{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)^3,x)

[Out]

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

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Maxima [A]  time = 1.07605, size = 230, normalized size = 2.17 \begin{align*} \frac{137 \, B a^{5} - 12 \, A a^{4} b + 60 \,{\left (5 \, B a b^{4} - A b^{5}\right )} x^{4} + 60 \,{\left (15 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} + 20 \,{\left (55 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{2} + 5 \,{\left (125 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x}{60 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} + \frac{B \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)^3,x, algorithm="maxima")

[Out]

1/60*(137*B*a^5 - 12*A*a^4*b + 60*(5*B*a*b^4 - A*b^5)*x^4 + 60*(15*B*a^2*b^3 - 2*A*a*b^4)*x^3 + 20*(55*B*a^3*b
^2 - 6*A*a^2*b^3)*x^2 + 5*(125*B*a^4*b - 12*A*a^3*b^2)*x)/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b
^8*x^2 + 5*a^4*b^7*x + a^5*b^6) + B*log(b*x + a)/b^6

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Fricas [B]  time = 1.34417, size = 485, normalized size = 4.58 \begin{align*} \frac{137 \, B a^{5} - 12 \, A a^{4} b + 60 \,{\left (5 \, B a b^{4} - A b^{5}\right )} x^{4} + 60 \,{\left (15 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} + 20 \,{\left (55 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{2} + 5 \,{\left (125 \, B a^{4} b - 12 \, A a^{3} b^{2}\right )} x + 60 \,{\left (B b^{5} x^{5} + 5 \, B a b^{4} x^{4} + 10 \, B a^{2} b^{3} x^{3} + 10 \, B a^{3} b^{2} x^{2} + 5 \, B a^{4} b x + B a^{5}\right )} \log \left (b x + a\right )}{60 \,{\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} 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)^3,x, algorithm="fricas")

[Out]

1/60*(137*B*a^5 - 12*A*a^4*b + 60*(5*B*a*b^4 - A*b^5)*x^4 + 60*(15*B*a^2*b^3 - 2*A*a*b^4)*x^3 + 20*(55*B*a^3*b
^2 - 6*A*a^2*b^3)*x^2 + 5*(125*B*a^4*b - 12*A*a^3*b^2)*x + 60*(B*b^5*x^5 + 5*B*a*b^4*x^4 + 10*B*a^2*b^3*x^3 +
10*B*a^3*b^2*x^2 + 5*B*a^4*b*x + B*a^5)*log(b*x + a))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x
^2 + 5*a^4*b^7*x + a^5*b^6)

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Sympy [A]  time = 2.68125, size = 172, normalized size = 1.62 \begin{align*} \frac{B \log{\left (a + b x \right )}}{b^{6}} + \frac{- 12 A a^{4} b + 137 B a^{5} + x^{4} \left (- 60 A b^{5} + 300 B a b^{4}\right ) + x^{3} \left (- 120 A a b^{4} + 900 B a^{2} b^{3}\right ) + x^{2} \left (- 120 A a^{2} b^{3} + 1100 B a^{3} b^{2}\right ) + x \left (- 60 A a^{3} b^{2} + 625 B a^{4} b\right )}{60 a^{5} b^{6} + 300 a^{4} b^{7} x + 600 a^{3} b^{8} x^{2} + 600 a^{2} b^{9} x^{3} + 300 a b^{10} x^{4} + 60 b^{11} x^{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)**3,x)

[Out]

B*log(a + b*x)/b**6 + (-12*A*a**4*b + 137*B*a**5 + x**4*(-60*A*b**5 + 300*B*a*b**4) + x**3*(-120*A*a*b**4 + 90
0*B*a**2*b**3) + x**2*(-120*A*a**2*b**3 + 1100*B*a**3*b**2) + x*(-60*A*a**3*b**2 + 625*B*a**4*b))/(60*a**5*b**
6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5)

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Giac [A]  time = 1.14564, size = 167, normalized size = 1.58 \begin{align*} \frac{B \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{60 \,{\left (5 \, B a b^{3} - A b^{4}\right )} x^{4} + 60 \,{\left (15 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 20 \,{\left (55 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} x^{2} + 5 \,{\left (125 \, B a^{4} - 12 \, A a^{3} b\right )} x + \frac{137 \, B a^{5} - 12 \, A a^{4} b}{b}}{60 \,{\left (b x + a\right )}^{5} 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)^3,x, algorithm="giac")

[Out]

B*log(abs(b*x + a))/b^6 + 1/60*(60*(5*B*a*b^3 - A*b^4)*x^4 + 60*(15*B*a^2*b^2 - 2*A*a*b^3)*x^3 + 20*(55*B*a^3*
b - 6*A*a^2*b^2)*x^2 + 5*(125*B*a^4 - 12*A*a^3*b)*x + (137*B*a^5 - 12*A*a^4*b)/b)/((b*x + a)^5*b^5)