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

Optimal. Leaf size=76 \[ \frac{3 b^2}{a^4 (a+b x)}+\frac{b^2}{2 a^3 (a+b x)^2}+\frac{6 b^2 \log (x)}{a^5}-\frac{6 b^2 \log (a+b x)}{a^5}+\frac{3 b}{a^4 x}-\frac{1}{2 a^3 x^2} \]

[Out]

-1/(2*a^3*x^2) + (3*b)/(a^4*x) + b^2/(2*a^3*(a + b*x)^2) + (3*b^2)/(a^4*(a + b*x)) + (6*b^2*Log[x])/a^5 - (6*b
^2*Log[a + b*x])/a^5

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Rubi [A]  time = 0.0359509, antiderivative size = 76, 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} \[ \frac{3 b^2}{a^4 (a+b x)}+\frac{b^2}{2 a^3 (a+b x)^2}+\frac{6 b^2 \log (x)}{a^5}-\frac{6 b^2 \log (a+b x)}{a^5}+\frac{3 b}{a^4 x}-\frac{1}{2 a^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x)^3),x]

[Out]

-1/(2*a^3*x^2) + (3*b)/(a^4*x) + b^2/(2*a^3*(a + b*x)^2) + (3*b^2)/(a^4*(a + b*x)) + (6*b^2*Log[x])/a^5 - (6*b
^2*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^3 (a+b x)^3} \, dx &=\int \left (\frac{1}{a^3 x^3}-\frac{3 b}{a^4 x^2}+\frac{6 b^2}{a^5 x}-\frac{b^3}{a^3 (a+b x)^3}-\frac{3 b^3}{a^4 (a+b x)^2}-\frac{6 b^3}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{1}{2 a^3 x^2}+\frac{3 b}{a^4 x}+\frac{b^2}{2 a^3 (a+b x)^2}+\frac{3 b^2}{a^4 (a+b x)}+\frac{6 b^2 \log (x)}{a^5}-\frac{6 b^2 \log (a+b x)}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.0622125, size = 68, normalized size = 0.89 \[ \frac{\frac{a \left (4 a^2 b x-a^3+18 a b^2 x^2+12 b^3 x^3\right )}{x^2 (a+b x)^2}-12 b^2 \log (a+b x)+12 b^2 \log (x)}{2 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x)^3),x]

[Out]

((a*(-a^3 + 4*a^2*b*x + 18*a*b^2*x^2 + 12*b^3*x^3))/(x^2*(a + b*x)^2) + 12*b^2*Log[x] - 12*b^2*Log[a + b*x])/(
2*a^5)

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Maple [A]  time = 0.01, size = 73, normalized size = 1. \begin{align*} -{\frac{1}{2\,{a}^{3}{x}^{2}}}+3\,{\frac{b}{{a}^{4}x}}+{\frac{{b}^{2}}{2\,{a}^{3} \left ( bx+a \right ) ^{2}}}+3\,{\frac{{b}^{2}}{{a}^{4} \left ( bx+a \right ) }}+6\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{5}}}-6\,{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03154, size = 116, normalized size = 1.53 \begin{align*} \frac{12 \, b^{3} x^{3} + 18 \, a b^{2} x^{2} + 4 \, a^{2} b x - a^{3}}{2 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} - \frac{6 \, b^{2} \log \left (b x + a\right )}{a^{5}} + \frac{6 \, b^{2} \log \left (x\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(12*b^3*x^3 + 18*a*b^2*x^2 + 4*a^2*b*x - a^3)/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2) - 6*b^2*log(b*x + a)/a
^5 + 6*b^2*log(x)/a^5

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Fricas [A]  time = 1.50816, size = 269, normalized size = 3.54 \begin{align*} \frac{12 \, a b^{3} x^{3} + 18 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x - a^{4} - 12 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(12*a*b^3*x^3 + 18*a^2*b^2*x^2 + 4*a^3*b*x - a^4 - 12*(b^4*x^4 + 2*a*b^3*x^3 + a^2*b^2*x^2)*log(b*x + a) +
12*(b^4*x^4 + 2*a*b^3*x^3 + a^2*b^2*x^2)*log(x))/(a^5*b^2*x^4 + 2*a^6*b*x^3 + a^7*x^2)

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Sympy [A]  time = 0.761018, size = 78, normalized size = 1.03 \begin{align*} \frac{- a^{3} + 4 a^{2} b x + 18 a b^{2} x^{2} + 12 b^{3} x^{3}}{2 a^{6} x^{2} + 4 a^{5} b x^{3} + 2 a^{4} b^{2} x^{4}} + \frac{6 b^{2} \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**3/(b*x+a)**3,x)

[Out]

(-a**3 + 4*a**2*b*x + 18*a*b**2*x**2 + 12*b**3*x**3)/(2*a**6*x**2 + 4*a**5*b*x**3 + 2*a**4*b**2*x**4) + 6*b**2
*(log(x) - log(a/b + x))/a**5

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Giac [A]  time = 1.23645, size = 99, normalized size = 1.3 \begin{align*} -\frac{6 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{5}} + \frac{6 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{12 \, b^{3} x^{3} + 18 \, a b^{2} x^{2} + 4 \, a^{2} b x - a^{3}}{2 \,{\left (b x^{2} + a x\right )}^{2} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*b^2*log(abs(b*x + a))/a^5 + 6*b^2*log(abs(x))/a^5 + 1/2*(12*b^3*x^3 + 18*a*b^2*x^2 + 4*a^2*b*x - a^3)/((b*x
^2 + a*x)^2*a^4)