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

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

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Rubi [A]  time = 0.04, antiderivative size = 76, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.05, size = 68, normalized size = 0.89 \begin {gather*} \frac {\frac {a \left (-a^3+4 a^2 b x+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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 (a+b x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^3*(a + b*x)^3),x]

[Out]

IntegrateAlgebraic[1/(x^3*(a + b*x)^3), x]

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fricas [A]  time = 1.18, size = 130, normalized size = 1.71 \begin {gather*} \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 \relax (x)}{2 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \end {gather*}

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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giac [A]  time = 1.39, size = 73, normalized size = 0.96 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.01, size = 73, normalized size = 0.96 \begin {gather*} \frac {b^{2}}{2 \left (b x +a \right )^{2} a^{3}}+\frac {3 b^{2}}{\left (b x +a \right ) a^{4}}+\frac {6 b^{2} \ln \relax (x )}{a^{5}}-\frac {6 b^{2} \ln \left (b x +a \right )}{a^{5}}+\frac {3 b}{a^{4} x}-\frac {1}{2 a^{3} x^{2}} \end {gather*}

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 = 1.37, size = 86, normalized size = 1.13 \begin {gather*} \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 \relax (x)}{a^{5}} \end {gather*}

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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mupad [B]  time = 0.12, size = 79, normalized size = 1.04 \begin {gather*} \frac {\frac {9\,b^2\,x^2}{a^3}-\frac {1}{2\,a}+\frac {6\,b^3\,x^3}{a^4}+\frac {2\,b\,x}{a^2}}{a^2\,x^2+2\,a\,b\,x^3+b^2\,x^4}-\frac {12\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x)^3),x)

[Out]

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

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sympy [A]  time = 0.41, size = 78, normalized size = 1.03 \begin {gather*} \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 {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{5}} \end {gather*}

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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