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

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

[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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Rubi [A]  time = 0.02, antiderivative size = 43, 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} \[ \frac {1}{a^2 (a+b x)}-\frac {\log (a+b x)}{a^3}+\frac {\log (x)}{a^3}+\frac {1}{2 a (a+b x)^2} \]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.45, size = 80, normalized size = 1.86 \[ \frac {2 \, a b x + 3 \, a^{2} - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \relax (x)}{2 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}} \]

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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giac [A]  time = 1.03, size = 43, normalized size = 1.00 \[ -\frac {\log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b x + 3 \, a^{2}}{2 \, {\left (b x + a\right )}^{2} a^{3}} \]

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)

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maple [A]  time = 0.01, size = 42, normalized size = 0.98 \[ \frac {1}{2 \left (b x +a \right )^{2} a}+\frac {1}{\left (b x +a \right ) a^{2}}+\frac {\ln \relax (x )}{a^{3}}-\frac {\ln \left (b x +a \right )}{a^{3}} \]

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 = 1.35, size = 51, normalized size = 1.19 \[ \frac {2 \, b x + 3 \, a}{2 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} - \frac {\log \left (b x + a\right )}{a^{3}} + \frac {\log \relax (x)}{a^{3}} \]

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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mupad [B]  time = 0.10, size = 43, normalized size = 1.00 \[ \frac {\frac {1}{a^2+b\,x\,a}-\frac {\ln \left (\frac {a+b\,x}{x}\right )}{a^2}}{a}+\frac {1}{2\,a\,{\left (a+b\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x)^3),x)

[Out]

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

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sympy [A]  time = 0.35, size = 46, normalized size = 1.07 \[ \frac {3 a + 2 b x}{2 a^{4} + 4 a^{3} b x + 2 a^{2} b^{2} x^{2}} + \frac {\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}}{a^{3}} \]

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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