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

Optimal. Leaf size=113 \[ -\frac {b^2 \log (x) (4 A b-3 a B)}{a^5}+\frac {b^2 (4 A b-3 a B) \log (a+b x)}{a^5}-\frac {b^2 (A b-a B)}{a^4 (a+b x)}-\frac {b (3 A b-2 a B)}{a^4 x}+\frac {2 A b-a B}{2 a^3 x^2}-\frac {A}{3 a^2 x^3} \]

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Rubi [A]  time = 0.09, antiderivative size = 113, 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^2 (A b-a B)}{a^4 (a+b x)}-\frac {b^2 \log (x) (4 A b-3 a B)}{a^5}+\frac {b^2 (4 A b-3 a B) \log (a+b x)}{a^5}+\frac {2 A b-a B}{2 a^3 x^2}-\frac {b (3 A b-2 a B)}{a^4 x}-\frac {A}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.08, size = 106, normalized size = 0.94 \begin {gather*} \frac {-\frac {2 a^3 A}{x^3}-\frac {3 a^2 (a B-2 A b)}{x^2}+\frac {6 a b^2 (a B-A b)}{a+b x}+6 b^2 \log (x) (3 a B-4 A b)+6 b^2 (4 A b-3 a B) \log (a+b x)+\frac {6 a b (2 a B-3 A b)}{x}}{6 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{x^4 \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^4*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

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

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fricas [A]  time = 0.41, size = 179, normalized size = 1.58 \begin {gather*} -\frac {2 \, A a^{4} - 6 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 3 \, {\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} + {\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x + 6 \, {\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} + {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} + {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3}\right )} \log \relax (x)}{6 \, {\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 133, normalized size = 1.18 \begin {gather*} \frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {{\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac {2 \, A a^{4} - 6 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 3 \, {\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} + {\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x}{6 \, {\left (b x + a\right )} a^{5} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 134, normalized size = 1.19 \begin {gather*} -\frac {A \,b^{3}}{\left (b x +a \right ) a^{4}}-\frac {4 A \,b^{3} \ln \relax (x )}{a^{5}}+\frac {4 A \,b^{3} \ln \left (b x +a \right )}{a^{5}}+\frac {B \,b^{2}}{\left (b x +a \right ) a^{3}}+\frac {3 B \,b^{2} \ln \relax (x )}{a^{4}}-\frac {3 B \,b^{2} \ln \left (b x +a \right )}{a^{4}}-\frac {3 A \,b^{2}}{a^{4} x}+\frac {2 B b}{a^{3} x}+\frac {A b}{a^{3} x^{2}}-\frac {B}{2 a^{2} x^{2}}-\frac {A}{3 a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.50, size = 128, normalized size = 1.13 \begin {gather*} -\frac {2 \, A a^{3} - 6 \, {\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \, {\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} + {\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} x}{6 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} - \frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{5}} + \frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \relax (x)}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.11, size = 131, normalized size = 1.16 \begin {gather*} \frac {2\,b^2\,\mathrm {atanh}\left (\frac {b^2\,\left (4\,A\,b-3\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (4\,A\,b^3-3\,B\,a\,b^2\right )}\right )\,\left (4\,A\,b-3\,B\,a\right )}{a^5}-\frac {\frac {A}{3\,a}-\frac {x\,\left (4\,A\,b-3\,B\,a\right )}{6\,a^2}+\frac {b^2\,x^3\,\left (4\,A\,b-3\,B\,a\right )}{a^4}+\frac {b\,x^2\,\left (4\,A\,b-3\,B\,a\right )}{2\,a^3}}{b\,x^4+a\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.65, size = 219, normalized size = 1.94 \begin {gather*} \frac {- 2 A a^{3} + x^{3} \left (- 24 A b^{3} + 18 B a b^{2}\right ) + x^{2} \left (- 12 A a b^{2} + 9 B a^{2} b\right ) + x \left (4 A a^{2} b - 3 B a^{3}\right )}{6 a^{5} x^{3} + 6 a^{4} b x^{4}} + \frac {b^{2} \left (- 4 A b + 3 B a\right ) \log {\left (x + \frac {- 4 A a b^{3} + 3 B a^{2} b^{2} - a b^{2} \left (- 4 A b + 3 B a\right )}{- 8 A b^{4} + 6 B a b^{3}} \right )}}{a^{5}} - \frac {b^{2} \left (- 4 A b + 3 B a\right ) \log {\left (x + \frac {- 4 A a b^{3} + 3 B a^{2} b^{2} + a b^{2} \left (- 4 A b + 3 B a\right )}{- 8 A b^{4} + 6 B a b^{3}} \right )}}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*A*a**3 + x**3*(-24*A*b**3 + 18*B*a*b**2) + x**2*(-12*A*a*b**2 + 9*B*a**2*b) + x*(4*A*a**2*b - 3*B*a**3))/(
6*a**5*x**3 + 6*a**4*b*x**4) + b**2*(-4*A*b + 3*B*a)*log(x + (-4*A*a*b**3 + 3*B*a**2*b**2 - a*b**2*(-4*A*b + 3
*B*a))/(-8*A*b**4 + 6*B*a*b**3))/a**5 - b**2*(-4*A*b + 3*B*a)*log(x + (-4*A*a*b**3 + 3*B*a**2*b**2 + a*b**2*(-
4*A*b + 3*B*a))/(-8*A*b**4 + 6*B*a*b**3))/a**5

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