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

Optimal. Leaf size=68 \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3}{a^4 x}+\frac{b^4 \log (x)}{a^5}-\frac{b^4 \log (a+b x)}{a^5}+\frac{b}{3 a^2 x^3}-\frac{1}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.0354385, antiderivative size = 68, normalized size of antiderivative = 1., 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} \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3}{a^4 x}+\frac{b^4 \log (x)}{a^5}-\frac{b^4 \log (a+b x)}{a^5}+\frac{b}{3 a^2 x^3}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x)),x]

[Out]

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

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

Mathematica [A]  time = 0.0075066, size = 68, normalized size = 1. \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3}{a^4 x}+\frac{b^4 \log (x)}{a^5}-\frac{b^4 \log (a+b x)}{a^5}+\frac{b}{3 a^2 x^3}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.006, size = 63, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,a{x}^{4}}}+{\frac{b}{3\,{a}^{2}{x}^{3}}}-{\frac{{b}^{2}}{2\,{a}^{3}{x}^{2}}}+{\frac{{b}^{3}}{{a}^{4}x}}+{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{5}}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01518, size = 84, normalized size = 1.24 \begin{align*} -\frac{b^{4} \log \left (b x + a\right )}{a^{5}} + \frac{b^{4} \log \left (x\right )}{a^{5}} + \frac{12 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 4 \, a^{2} b x - 3 \, a^{3}}{12 \, a^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.5254, size = 154, normalized size = 2.26 \begin{align*} -\frac{12 \, b^{4} x^{4} \log \left (b x + a\right ) - 12 \, b^{4} x^{4} \log \left (x\right ) - 12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 3 \, a^{4}}{12 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.591385, size = 56, normalized size = 0.82 \begin{align*} \frac{- 3 a^{3} + 4 a^{2} b x - 6 a b^{2} x^{2} + 12 b^{3} x^{3}}{12 a^{4} x^{4}} + \frac{b^{4} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18353, size = 90, normalized size = 1.32 \begin{align*} -\frac{b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{5}} + \frac{b^{4} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{12 \, a b^{3} x^{3} - 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x - 3 \, a^{4}}{12 \, a^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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