3.188 \(\int \frac{1}{x (a+b x)^3} \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{a^2 (a+b x)}-\frac{\log (a+b x)}{a^3}+\frac{\log (x)}{a^3}+\frac{1}{2 a (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0194961, antiderivative size = 43, 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} \[ \frac{1}{a^2 (a+b x)}-\frac{\log (a+b x)}{a^3}+\frac{\log (x)}{a^3}+\frac{1}{2 a (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0405402, size = 37, normalized size = 0.86 \[ \frac{\frac{a (3 a+2 b x)}{(a+b x)^2}-2 \log (a+b x)+2 \log (x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x)^3),x]

[Out]

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

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Maple [A]  time = 0.008, size = 42, normalized size = 1. \begin{align*}{\frac{1}{2\,a \left ( bx+a \right ) ^{2}}}+{\frac{1}{{a}^{2} \left ( bx+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{3}}}-{\frac{\ln \left ( bx+a \right ) }{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.10759, size = 69, normalized size = 1.6 \begin{align*} \frac{2 \, b x + 3 \, a}{2 \,{\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} - \frac{\log \left (b x + a\right )}{a^{3}} + \frac{\log \left (x\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.44843, size = 182, normalized size = 4.23 \begin{align*} \frac{2 \, a b x + 3 \, a^{2} - 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.670261, size = 46, normalized size = 1.07 \begin{align*} \frac{3 a + 2 b x}{2 a^{4} + 4 a^{3} b x + 2 a^{2} b^{2} x^{2}} + \frac{\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)**3,x)

[Out]

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

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Giac [A]  time = 1.16206, size = 58, normalized size = 1.35 \begin{align*} -\frac{\log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac{\log \left ({\left | x \right |}\right )}{a^{3}} + \frac{2 \, a b x + 3 \, a^{2}}{2 \,{\left (b x + a\right )}^{2} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x+a)^3,x, algorithm="giac")

[Out]

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