3.189 \(\int \frac{1}{x^2 (a+b x)^3} \, dx\)

Optimal. Leaf size=57 \[ -\frac{2 b}{a^3 (a+b x)}-\frac{b}{2 a^2 (a+b x)^2}-\frac{3 b \log (x)}{a^4}+\frac{3 b \log (a+b x)}{a^4}-\frac{1}{a^3 x} \]

[Out]

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

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Rubi [A]  time = 0.0292758, antiderivative size = 57, 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} \[ -\frac{2 b}{a^3 (a+b x)}-\frac{b}{2 a^2 (a+b x)^2}-\frac{3 b \log (x)}{a^4}+\frac{3 b \log (a+b x)}{a^4}-\frac{1}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^3),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0767011, size = 53, normalized size = 0.93 \[ -\frac{\frac{a \left (2 a^2+9 a b x+6 b^2 x^2\right )}{x (a+b x)^2}-6 b \log (a+b x)+6 b \log (x)}{2 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^3),x]

[Out]

-((a*(2*a^2 + 9*a*b*x + 6*b^2*x^2))/(x*(a + b*x)^2) + 6*b*Log[x] - 6*b*Log[a + b*x])/(2*a^4)

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Maple [A]  time = 0.01, size = 56, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{3}x}}-{\frac{b}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}}-2\,{\frac{b}{{a}^{3} \left ( bx+a \right ) }}-3\,{\frac{b\ln \left ( x \right ) }{{a}^{4}}}+3\,{\frac{b\ln \left ( bx+a \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.08097, size = 93, normalized size = 1.63 \begin{align*} -\frac{6 \, b^{2} x^{2} + 9 \, a b x + 2 \, a^{2}}{2 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} + \frac{3 \, b \log \left (b x + a\right )}{a^{4}} - \frac{3 \, b \log \left (x\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*(6*b^2*x^2 + 9*a*b*x + 2*a^2)/(a^3*b^2*x^3 + 2*a^4*b*x^2 + a^5*x) + 3*b*log(b*x + a)/a^4 - 3*b*log(x)/a^4

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Fricas [A]  time = 1.67475, size = 232, normalized size = 4.07 \begin{align*} -\frac{6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3} - 6 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \log \left (x\right )}{2 \,{\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/2*(6*a*b^2*x^2 + 9*a^2*b*x + 2*a^3 - 6*(b^3*x^3 + 2*a*b^2*x^2 + a^2*b*x)*log(b*x + a) + 6*(b^3*x^3 + 2*a*b^
2*x^2 + a^2*b*x)*log(x))/(a^4*b^2*x^3 + 2*a^5*b*x^2 + a^6*x)

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Sympy [A]  time = 0.643188, size = 65, normalized size = 1.14 \begin{align*} - \frac{2 a^{2} + 9 a b x + 6 b^{2} x^{2}}{2 a^{5} x + 4 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac{3 b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x+a)**3,x)

[Out]

-(2*a**2 + 9*a*b*x + 6*b**2*x**2)/(2*a**5*x + 4*a**4*b*x**2 + 2*a**3*b**2*x**3) + 3*b*(-log(x) + log(a/b + x))
/a**4

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Giac [A]  time = 1.23607, size = 81, normalized size = 1.42 \begin{align*} \frac{3 \, b \log \left ({\left | b x + a \right |}\right )}{a^{4}} - \frac{3 \, b \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3}}{2 \,{\left (b x + a\right )}^{2} a^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(b*x+a)^3,x, algorithm="giac")

[Out]

3*b*log(abs(b*x + a))/a^4 - 3*b*log(abs(x))/a^4 - 1/2*(6*a*b^2*x^2 + 9*a^2*b*x + 2*a^3)/((b*x + a)^2*a^4*x)