3.2.79 \(\int \frac {1}{x^5 (a+b x)^2} \, dx\)

Optimal. Leaf size=84 \[ \frac {5 b^4 \log (x)}{a^6}-\frac {5 b^4 \log (a+b x)}{a^6}+\frac {b^4}{a^5 (a+b x)}+\frac {4 b^3}{a^5 x}-\frac {3 b^2}{2 a^4 x^2}+\frac {2 b}{3 a^3 x^3}-\frac {1}{4 a^2 x^4} \]

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Rubi [A]  time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {44} \begin {gather*} -\frac {3 b^2}{2 a^4 x^2}+\frac {b^4}{a^5 (a+b x)}+\frac {4 b^3}{a^5 x}+\frac {5 b^4 \log (x)}{a^6}-\frac {5 b^4 \log (a+b x)}{a^6}+\frac {2 b}{3 a^3 x^3}-\frac {1}{4 a^2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x)^2),x]

[Out]

-1/(4*a^2*x^4) + (2*b)/(3*a^3*x^3) - (3*b^2)/(2*a^4*x^2) + (4*b^3)/(a^5*x) + b^4/(a^5*(a + b*x)) + (5*b^4*Log[
x])/a^6 - (5*b^4*Log[a + b*x])/a^6

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 79, normalized size = 0.94 \begin {gather*} \frac {\frac {a \left (-3 a^4+5 a^3 b x-10 a^2 b^2 x^2+30 a b^3 x^3+60 b^4 x^4\right )}{x^4 (a+b x)}-60 b^4 \log (a+b x)+60 b^4 \log (x)}{12 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x)^2),x]

[Out]

((a*(-3*a^4 + 5*a^3*b*x - 10*a^2*b^2*x^2 + 30*a*b^3*x^3 + 60*b^4*x^4))/(x^4*(a + b*x)) + 60*b^4*Log[x] - 60*b^
4*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 {1}{x^5 (a+b x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^5*(a + b*x)^2),x]

[Out]

IntegrateAlgebraic[1/(x^5*(a + b*x)^2), x]

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fricas [A]  time = 1.18, size = 108, normalized size = 1.29 \begin {gather*} \frac {60 \, a b^{4} x^{4} + 30 \, a^{2} b^{3} x^{3} - 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x - 3 \, a^{5} - 60 \, {\left (b^{5} x^{5} + a b^{4} x^{4}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} x^{5} + a b^{4} 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(1/x^5/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/12*(60*a*b^4*x^4 + 30*a^2*b^3*x^3 - 10*a^3*b^2*x^2 + 5*a^4*b*x - 3*a^5 - 60*(b^5*x^5 + a*b^4*x^4)*log(b*x +
a) + 60*(b^5*x^5 + a*b^4*x^4)*log(x))/(a^6*b*x^5 + a^7*x^4)

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giac [A]  time = 0.99, size = 104, normalized size = 1.24 \begin {gather*} \frac {5 \, b^{4} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{6}} + \frac {b^{4}}{{\left (b x + a\right )} a^{5}} - \frac {\frac {260 \, a b^{4}}{b x + a} - \frac {300 \, a^{2} b^{4}}{{\left (b x + a\right )}^{2}} + \frac {120 \, a^{3} b^{4}}{{\left (b x + a\right )}^{3}} - 77 \, b^{4}}{12 \, a^{6} {\left (\frac {a}{b x + a} - 1\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^2,x, algorithm="giac")

[Out]

5*b^4*log(abs(-a/(b*x + a) + 1))/a^6 + b^4/((b*x + a)*a^5) - 1/12*(260*a*b^4/(b*x + a) - 300*a^2*b^4/(b*x + a)
^2 + 120*a^3*b^4/(b*x + a)^3 - 77*b^4)/(a^6*(a/(b*x + a) - 1)^4)

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maple [A]  time = 0.01, size = 79, normalized size = 0.94 \begin {gather*} \frac {b^{4}}{\left (b x +a \right ) a^{5}}+\frac {5 b^{4} \ln \relax (x )}{a^{6}}-\frac {5 b^{4} \ln \left (b x +a \right )}{a^{6}}+\frac {4 b^{3}}{a^{5} x}-\frac {3 b^{2}}{2 a^{4} x^{2}}+\frac {2 b}{3 a^{3} x^{3}}-\frac {1}{4 a^{2} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x+a)^2,x)

[Out]

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

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maxima [A]  time = 1.34, size = 86, normalized size = 1.02 \begin {gather*} \frac {60 \, b^{4} x^{4} + 30 \, a b^{3} x^{3} - 10 \, a^{2} b^{2} x^{2} + 5 \, a^{3} b x - 3 \, a^{4}}{12 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} - \frac {5 \, b^{4} \log \left (b x + a\right )}{a^{6}} + \frac {5 \, b^{4} \log \relax (x)}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^2,x, algorithm="maxima")

[Out]

1/12*(60*b^4*x^4 + 30*a*b^3*x^3 - 10*a^2*b^2*x^2 + 5*a^3*b*x - 3*a^4)/(a^5*b*x^5 + a^6*x^4) - 5*b^4*log(b*x +
a)/a^6 + 5*b^4*log(x)/a^6

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mupad [B]  time = 0.12, size = 79, normalized size = 0.94 \begin {gather*} \frac {\frac {5\,b^3\,x^3}{2\,a^4}-\frac {5\,b^2\,x^2}{6\,a^3}-\frac {1}{4\,a}+\frac {5\,b^4\,x^4}{a^5}+\frac {5\,b\,x}{12\,a^2}}{b\,x^5+a\,x^4}-\frac {10\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b*x)^2),x)

[Out]

((5*b^3*x^3)/(2*a^4) - (5*b^2*x^2)/(6*a^3) - 1/(4*a) + (5*b^4*x^4)/a^5 + (5*b*x)/(12*a^2))/(a*x^4 + b*x^5) - (
10*b^4*atanh((2*b*x)/a + 1))/a^6

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sympy [A]  time = 0.39, size = 80, normalized size = 0.95 \begin {gather*} \frac {- 3 a^{4} + 5 a^{3} b x - 10 a^{2} b^{2} x^{2} + 30 a b^{3} x^{3} + 60 b^{4} x^{4}}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} + \frac {5 b^{4} \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x+a)**2,x)

[Out]

(-3*a**4 + 5*a**3*b*x - 10*a**2*b**2*x**2 + 30*a*b**3*x**3 + 60*b**4*x**4)/(12*a**6*x**4 + 12*a**5*b*x**5) + 5
*b**4*(log(x) - log(a/b + x))/a**6

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