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

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

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Rubi [A]  time = 0.11, antiderivative size = 133, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.09, size = 129, normalized size = 0.97 \begin {gather*} \frac {\frac {a \left (-\left (a^4 (3 A+4 B x)\right )+a^3 b x (5 A+8 B x)-2 a^2 b^2 x^2 (5 A+12 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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{x^5 \left (a^2+2 a b x+b^2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 203, normalized size = 1.53 \begin {gather*} -\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 \relax (x)}{12 \, {\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \end {gather*}

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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giac [A]  time = 0.18, size = 157, normalized size = 1.18 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.06, size = 158, normalized size = 1.19 \begin {gather*} \frac {A \,b^{4}}{\left (b x +a \right ) a^{5}}+\frac {5 A \,b^{4} \ln \relax (x )}{a^{6}}-\frac {5 A \,b^{4} \ln \left (b x +a \right )}{a^{6}}-\frac {B \,b^{3}}{\left (b x +a \right ) a^{4}}-\frac {4 B \,b^{3} \ln \relax (x )}{a^{5}}+\frac {4 B \,b^{3} \ln \left (b x +a \right )}{a^{5}}+\frac {4 A \,b^{3}}{a^{5} x}-\frac {3 B \,b^{2}}{a^{4} x}-\frac {3 A \,b^{2}}{2 a^{4} x^{2}}+\frac {B b}{a^{3} x^{2}}+\frac {2 A b}{3 a^{3} x^{3}}-\frac {B}{3 a^{2} x^{3}}-\frac {A}{4 a^{2} x^{4}} \end {gather*}

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]

-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-1/4*A/a^2/x^4+2/3/a^3/x^3*A*b
-1/3/a^2/x^3*B-3/2*A/a^4*b^2/x^2+B/a^3*b/x^2+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

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maxima [A]  time = 0.50, size = 152, normalized size = 1.14 \begin {gather*} -\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 \relax (x)}{a^{6}} \end {gather*}

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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mupad [B]  time = 1.15, size = 150, normalized size = 1.13 \begin {gather*} \frac {\frac {x\,\left (5\,A\,b-4\,B\,a\right )}{12\,a^2}-\frac {A}{4\,a}+\frac {b^2\,x^3\,\left (5\,A\,b-4\,B\,a\right )}{2\,a^4}+\frac {b^3\,x^4\,\left (5\,A\,b-4\,B\,a\right )}{a^5}-\frac {b\,x^2\,\left (5\,A\,b-4\,B\,a\right )}{6\,a^3}}{b\,x^5+a\,x^4}-\frac {2\,b^3\,\mathrm {atanh}\left (\frac {b^3\,\left (5\,A\,b-4\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (5\,A\,b^4-4\,B\,a\,b^3\right )}\right )\,\left (5\,A\,b-4\,B\,a\right )}{a^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.73, size = 243, normalized size = 1.83 \begin {gather*} \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 {gather*}

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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