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

Optimal. Leaf size=133 \[ \frac{b^3 (A b-a B)}{a^5 (a+b x)}+\frac{b^2 (4 A b-3 a B)}{a^5 x}+\frac{b^3 \log (x) (5 A b-4 a B)}{a^6}-\frac{b^3 (5 A b-4 a B) \log (a+b x)}{a^6}-\frac{b (3 A b-2 a B)}{2 a^4 x^2}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{4 a^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.114113, antiderivative size = 133, 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} \[ \frac{b^3 (A b-a B)}{a^5 (a+b x)}+\frac{b^2 (4 A b-3 a B)}{a^5 x}+\frac{b^3 \log (x) (5 A b-4 a B)}{a^6}-\frac{b^3 (5 A b-4 a B) \log (a+b x)}{a^6}-\frac{b (3 A b-2 a B)}{2 a^4 x^2}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{A}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0939845, size = 129, normalized size = 0.97 \[ \frac{\frac{a \left (-2 a^2 b^2 x^2 (5 A+12 B x)+a^3 b x (5 A+8 B x)+a^4 (-(3 A+4 B x))+6 a b^3 x^3 (5 A-8 B x)+60 A b^4 x^4\right )}{x^4 (a+b x)}+12 b^3 \log (x) (5 A b-4 a B)+12 b^3 (4 a B-5 A b) \log (a+b x)}{12 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(60*A*b^4*x^4 + 6*a*b^3*x^3*(5*A - 8*B*x) - a^4*(3*A + 4*B*x) + a^3*b*x*(5*A + 8*B*x) - 2*a^2*b^2*x^2*(5*A
+ 12*B*x)))/(x^4*(a + b*x)) + 12*b^3*(5*A*b - 4*a*B)*Log[x] + 12*b^3*(-5*A*b + 4*a*B)*Log[a + b*x])/(12*a^6)

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Maple [A]  time = 0.011, size = 158, normalized size = 1.2 \begin{align*} -{\frac{A}{4\,{a}^{2}{x}^{4}}}+{\frac{2\,Ab}{3\,{a}^{3}{x}^{3}}}-{\frac{B}{3\,{a}^{2}{x}^{3}}}-{\frac{3\,A{b}^{2}}{2\,{a}^{4}{x}^{2}}}+{\frac{bB}{{a}^{3}{x}^{2}}}+5\,{\frac{A\ln \left ( x \right ){b}^{4}}{{a}^{6}}}-4\,{\frac{B\ln \left ( x \right ){b}^{3}}{{a}^{5}}}+4\,{\frac{A{b}^{3}}{{a}^{5}x}}-3\,{\frac{{b}^{2}B}{{a}^{4}x}}-5\,{\frac{{b}^{4}\ln \left ( bx+a \right ) A}{{a}^{6}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) B}{{a}^{5}}}+{\frac{A{b}^{4}}{{a}^{5} \left ( bx+a \right ) }}-{\frac{{b}^{3}B}{{a}^{4} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01029, size = 205, normalized size = 1.54 \begin{align*} -\frac{3 \, A a^{4} + 12 \,{\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} + 6 \,{\left (4 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} - 2 \,{\left (4 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} +{\left (4 \, B a^{4} - 5 \, A a^{3} b\right )} x}{12 \,{\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} + \frac{{\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left (b x + a\right )}{a^{6}} - \frac{{\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left (x\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*A*a^4 + 12*(4*B*a*b^3 - 5*A*b^4)*x^4 + 6*(4*B*a^2*b^2 - 5*A*a*b^3)*x^3 - 2*(4*B*a^3*b - 5*A*a^2*b^2)*
x^2 + (4*B*a^4 - 5*A*a^3*b)*x)/(a^5*b*x^5 + a^6*x^4) + (4*B*a*b^3 - 5*A*b^4)*log(b*x + a)/a^6 - (4*B*a*b^3 - 5
*A*b^4)*log(x)/a^6

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Fricas [A]  time = 1.23806, size = 432, normalized size = 3.25 \begin{align*} -\frac{3 \, A a^{5} + 12 \,{\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 6 \,{\left (4 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 2 \,{\left (4 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} +{\left (4 \, B a^{5} - 5 \, A a^{4} b\right )} x - 12 \,{\left ({\left (4 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} +{\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4}\right )} \log \left (b x + a\right ) + 12 \,{\left ({\left (4 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} +{\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4}\right )} \log \left (x\right )}{12 \,{\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.15655, size = 243, normalized size = 1.83 \begin{align*} - \frac{3 A a^{4} + x^{4} \left (- 60 A b^{4} + 48 B a b^{3}\right ) + x^{3} \left (- 30 A a b^{3} + 24 B a^{2} b^{2}\right ) + x^{2} \left (10 A a^{2} b^{2} - 8 B a^{3} b\right ) + x \left (- 5 A a^{3} b + 4 B a^{4}\right )}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} - \frac{b^{3} \left (- 5 A b + 4 B a\right ) \log{\left (x + \frac{- 5 A a b^{4} + 4 B a^{2} b^{3} - a b^{3} \left (- 5 A b + 4 B a\right )}{- 10 A b^{5} + 8 B a b^{4}} \right )}}{a^{6}} + \frac{b^{3} \left (- 5 A b + 4 B a\right ) \log{\left (x + \frac{- 5 A a b^{4} + 4 B a^{2} b^{3} + a b^{3} \left (- 5 A b + 4 B a\right )}{- 10 A b^{5} + 8 B a b^{4}} \right )}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*A*a**4 + x**4*(-60*A*b**4 + 48*B*a*b**3) + x**3*(-30*A*a*b**3 + 24*B*a**2*b**2) + x**2*(10*A*a**2*b**2 - 8
*B*a**3*b) + x*(-5*A*a**3*b + 4*B*a**4))/(12*a**6*x**4 + 12*a**5*b*x**5) - b**3*(-5*A*b + 4*B*a)*log(x + (-5*A
*a*b**4 + 4*B*a**2*b**3 - a*b**3*(-5*A*b + 4*B*a))/(-10*A*b**5 + 8*B*a*b**4))/a**6 + b**3*(-5*A*b + 4*B*a)*log
(x + (-5*A*a*b**4 + 4*B*a**2*b**3 + a*b**3*(-5*A*b + 4*B*a))/(-10*A*b**5 + 8*B*a*b**4))/a**6

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Giac [A]  time = 1.16947, size = 212, normalized size = 1.59 \begin{align*} -\frac{{\left (4 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{{\left (4 \, B a b^{4} - 5 \, A b^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac{3 \, A a^{5} + 12 \,{\left (4 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + 6 \,{\left (4 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 2 \,{\left (4 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} +{\left (4 \, B a^{5} - 5 \, A a^{4} b\right )} x}{12 \,{\left (b x + a\right )} a^{6} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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