3.178 \(\int \frac{1}{x^4 (a+b x)^2} \, dx\)

Optimal. Leaf size=69 \[ -\frac{b^3}{a^4 (a+b x)}-\frac{3 b^2}{a^4 x}-\frac{4 b^3 \log (x)}{a^5}+\frac{4 b^3 \log (a+b x)}{a^5}+\frac{b}{a^3 x^2}-\frac{1}{3 a^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0347626, antiderivative size = 69, 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} \[ -\frac{b^3}{a^4 (a+b x)}-\frac{3 b^2}{a^4 x}-\frac{4 b^3 \log (x)}{a^5}+\frac{4 b^3 \log (a+b x)}{a^5}+\frac{b}{a^3 x^2}-\frac{1}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x)^2),x]

[Out]

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

Mathematica [A]  time = 0.0924556, size = 66, normalized size = 0.96 \[ -\frac{\frac{a \left (-2 a^2 b x+a^3+6 a b^2 x^2+12 b^3 x^3\right )}{x^3 (a+b x)}-12 b^3 \log (a+b x)+12 b^3 \log (x)}{3 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x)^2),x]

[Out]

-((a*(a^3 - 2*a^2*b*x + 6*a*b^2*x^2 + 12*b^3*x^3))/(x^3*(a + b*x)) + 12*b^3*Log[x] - 12*b^3*Log[a + b*x])/(3*a
^5)

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Maple [A]  time = 0.01, size = 68, normalized size = 1. \begin{align*} -{\frac{1}{3\,{a}^{2}{x}^{3}}}+{\frac{b}{{a}^{3}{x}^{2}}}-3\,{\frac{{b}^{2}}{{a}^{4}x}}-{\frac{{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}-4\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{5}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.05616, size = 99, normalized size = 1.43 \begin{align*} -\frac{12 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}}{3 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac{4 \, b^{3} \log \left (b x + a\right )}{a^{5}} - \frac{4 \, b^{3} \log \left (x\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.70433, size = 204, normalized size = 2.96 \begin{align*} -\frac{12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4} - 12 \,{\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.660023, size = 66, normalized size = 0.96 \begin{align*} - \frac{a^{3} - 2 a^{2} b x + 6 a b^{2} x^{2} + 12 b^{3} x^{3}}{3 a^{5} x^{3} + 3 a^{4} b x^{4}} + \frac{4 b^{3} \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**4/(b*x+a)**2,x)

[Out]

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

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Giac [A]  time = 1.23453, size = 122, normalized size = 1.77 \begin{align*} -\frac{4 \, b^{3} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{5}} - \frac{b^{3}}{{\left (b x + a\right )} a^{4}} - \frac{\frac{30 \, a b^{3}}{b x + a} - \frac{18 \, a^{2} b^{3}}{{\left (b x + a\right )}^{2}} - 13 \, b^{3}}{3 \, a^{5}{\left (\frac{a}{b x + a} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x+a)^2,x, algorithm="giac")

[Out]

-4*b^3*log(abs(-a/(b*x + a) + 1))/a^5 - b^3/((b*x + a)*a^4) - 1/3*(30*a*b^3/(b*x + a) - 18*a^2*b^3/(b*x + a)^2
- 13*b^3)/(a^5*(a/(b*x + a) - 1)^3)