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

Optimal. Leaf size=70 \[ -\frac{3 b}{a^4 (a+b x)}-\frac{b}{a^3 (a+b x)^2}-\frac{b}{3 a^2 (a+b x)^3}-\frac{4 b \log (x)}{a^5}+\frac{4 b \log (a+b x)}{a^5}-\frac{1}{a^4 x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0388295, antiderivative size = 70, 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} \[ -\frac{3 b}{a^4 (a+b x)}-\frac{b}{a^3 (a+b x)^2}-\frac{b}{3 a^2 (a+b x)^3}-\frac{4 b \log (x)}{a^5}+\frac{4 b \log (a+b x)}{a^5}-\frac{1}{a^4 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x)^4),x]

[Out]

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

Mathematica [A]  time = 0.102326, size = 64, normalized size = 0.91 \[ -\frac{\frac{a \left (22 a^2 b x+3 a^3+30 a b^2 x^2+12 b^3 x^3\right )}{x (a+b x)^3}-12 b \log (a+b x)+12 b \log (x)}{3 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x)^4),x]

[Out]

-((a*(3*a^3 + 22*a^2*b*x + 30*a*b^2*x^2 + 12*b^3*x^3))/(x*(a + b*x)^3) + 12*b*Log[x] - 12*b*Log[a + b*x])/(3*a
^5)

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 69, normalized size = 1. \begin{align*} -{\frac{1}{{a}^{4}x}}-{\frac{b}{3\,{a}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{b}{{a}^{3} \left ( bx+a \right ) ^{2}}}-3\,{\frac{b}{{a}^{4} \left ( bx+a \right ) }}-4\,{\frac{b\ln \left ( x \right ) }{{a}^{5}}}+4\,{\frac{b\ln \left ( bx+a \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.04235, size = 123, normalized size = 1.76 \begin{align*} -\frac{12 \, b^{3} x^{3} + 30 \, a b^{2} x^{2} + 22 \, a^{2} b x + 3 \, a^{3}}{3 \,{\left (a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{3} + 3 \, a^{6} b x^{2} + a^{7} x\right )}} + \frac{4 \, b \log \left (b x + a\right )}{a^{5}} - \frac{4 \, b \log \left (x\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(12*b^3*x^3 + 30*a*b^2*x^2 + 22*a^2*b*x + 3*a^3)/(a^4*b^3*x^4 + 3*a^5*b^2*x^3 + 3*a^6*b*x^2 + a^7*x) + 4*
b*log(b*x + a)/a^5 - 4*b*log(x)/a^5

________________________________________________________________________________________

Fricas [B]  time = 1.39027, size = 325, normalized size = 4.64 \begin{align*} -\frac{12 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 22 \, a^{3} b x + 3 \, a^{4} - 12 \,{\left (b^{4} x^{4} + 3 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + a^{3} b x\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} x^{4} + 3 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + a^{3} b x\right )} \log \left (x\right )}{3 \,{\left (a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{3} + 3 \, a^{7} b x^{2} + a^{8} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(12*a*b^3*x^3 + 30*a^2*b^2*x^2 + 22*a^3*b*x + 3*a^4 - 12*(b^4*x^4 + 3*a*b^3*x^3 + 3*a^2*b^2*x^2 + a^3*b*x
)*log(b*x + a) + 12*(b^4*x^4 + 3*a*b^3*x^3 + 3*a^2*b^2*x^2 + a^3*b*x)*log(x))/(a^5*b^3*x^4 + 3*a^6*b^2*x^3 + 3
*a^7*b*x^2 + a^8*x)

________________________________________________________________________________________

Sympy [A]  time = 0.793333, size = 88, normalized size = 1.26 \begin{align*} - \frac{3 a^{3} + 22 a^{2} b x + 30 a b^{2} x^{2} + 12 b^{3} x^{3}}{3 a^{7} x + 9 a^{6} b x^{2} + 9 a^{5} b^{2} x^{3} + 3 a^{4} b^{3} x^{4}} + \frac{4 b \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**2/(b*x+a)**4,x)

[Out]

-(3*a**3 + 22*a**2*b*x + 30*a*b**2*x**2 + 12*b**3*x**3)/(3*a**7*x + 9*a**6*b*x**2 + 9*a**5*b**2*x**3 + 3*a**4*
b**3*x**4) + 4*b*(-log(x) + log(a/b + x))/a**5

________________________________________________________________________________________

Giac [A]  time = 1.1801, size = 96, normalized size = 1.37 \begin{align*} \frac{4 \, b \log \left ({\left | b x + a \right |}\right )}{a^{5}} - \frac{4 \, b \log \left ({\left | x \right |}\right )}{a^{5}} - \frac{12 \, a b^{3} x^{3} + 30 \, a^{2} b^{2} x^{2} + 22 \, a^{3} b x + 3 \, a^{4}}{3 \,{\left (b x + a\right )}^{3} a^{5} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b*log(abs(b*x + a))/a^5 - 4*b*log(abs(x))/a^5 - 1/3*(12*a*b^3*x^3 + 30*a^2*b^2*x^2 + 22*a^3*b*x + 3*a^4)/((b
*x + a)^3*a^5*x)