3.1616 \(\int \frac {1}{(a+\frac {b}{x}) x^5} \, dx\)

Optimal. Leaf size=56 \[ -\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (a x+b)}{b^4}-\frac {a^2}{b^3 x}+\frac {a}{2 b^2 x^2}-\frac {1}{3 b x^3} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 44} \[ -\frac {a^2}{b^3 x}-\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (a x+b)}{b^4}+\frac {a}{2 b^2 x^2}-\frac {1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)*x^5),x]

[Out]

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

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])

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 56, normalized size = 1.00 \[ -\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (a x+b)}{b^4}-\frac {a^2}{b^3 x}+\frac {a}{2 b^2 x^2}-\frac {1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)*x^5),x]

[Out]

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

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fricas [A]  time = 0.98, size = 54, normalized size = 0.96 \[ \frac {6 \, a^{3} x^{3} \log \left (a x + b\right ) - 6 \, a^{3} x^{3} \log \relax (x) - 6 \, a^{2} b x^{2} + 3 \, a b^{2} x - 2 \, b^{3}}{6 \, b^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)/x^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 56, normalized size = 1.00 \[ \frac {a^{3} \log \left ({\left | a x + b \right |}\right )}{b^{4}} - \frac {a^{3} \log \left ({\left | x \right |}\right )}{b^{4}} - \frac {6 \, a^{2} b x^{2} - 3 \, a b^{2} x + 2 \, b^{3}}{6 \, b^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 53, normalized size = 0.95 \[ -\frac {a^{3} \ln \relax (x )}{b^{4}}+\frac {a^{3} \ln \left (a x +b \right )}{b^{4}}-\frac {a^{2}}{b^{3} x}+\frac {a}{2 b^{2} x^{2}}-\frac {1}{3 b \,x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 51, normalized size = 0.91 \[ \frac {a^{3} \log \left (a x + b\right )}{b^{4}} - \frac {a^{3} \log \relax (x)}{b^{4}} - \frac {6 \, a^{2} x^{2} - 3 \, a b x + 2 \, b^{2}}{6 \, b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.08, size = 48, normalized size = 0.86 \[ \frac {2\,a^3\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^4}-\frac {a^2\,b\,x^2-\frac {a\,b^2\,x}{2}+\frac {b^3}{3}}{b^4\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b/x)),x)

[Out]

(2*a^3*atanh((2*a*x)/b + 1))/b^4 - (b^3/3 + a^2*b*x^2 - (a*b^2*x)/2)/(b^4*x^3)

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sympy [A]  time = 0.24, size = 44, normalized size = 0.79 \[ \frac {a^{3} \left (- \log {\relax (x )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{4}} + \frac {- 6 a^{2} x^{2} + 3 a b x - 2 b^{2}}{6 b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*(-log(x) + log(x + b/a))/b**4 + (-6*a**2*x**2 + 3*a*b*x - 2*b**2)/(6*b**3*x**3)

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