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

Optimal. Leaf size=102 \[ -\frac{10 b^3}{a^6 (a+b x)}-\frac{2 b^3}{a^5 (a+b x)^2}-\frac{b^3}{3 a^4 (a+b x)^3}-\frac{10 b^2}{a^6 x}-\frac{20 b^3 \log (x)}{a^7}+\frac{20 b^3 \log (a+b x)}{a^7}+\frac{2 b}{a^5 x^2}-\frac{1}{3 a^4 x^3} \]

[Out]

-1/(3*a^4*x^3) + (2*b)/(a^5*x^2) - (10*b^2)/(a^6*x) - b^3/(3*a^4*(a + b*x)^3) - (2*b^3)/(a^5*(a + b*x)^2) - (1
0*b^3)/(a^6*(a + b*x)) - (20*b^3*Log[x])/a^7 + (20*b^3*Log[a + b*x])/a^7

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Rubi [A]  time = 0.0529369, antiderivative size = 102, 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} \[ -\frac{10 b^3}{a^6 (a+b x)}-\frac{2 b^3}{a^5 (a+b x)^2}-\frac{b^3}{3 a^4 (a+b x)^3}-\frac{10 b^2}{a^6 x}-\frac{20 b^3 \log (x)}{a^7}+\frac{20 b^3 \log (a+b x)}{a^7}+\frac{2 b}{a^5 x^2}-\frac{1}{3 a^4 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x)^4),x]

[Out]

-1/(3*a^4*x^3) + (2*b)/(a^5*x^2) - (10*b^2)/(a^6*x) - b^3/(3*a^4*(a + b*x)^3) - (2*b^3)/(a^5*(a + b*x)^2) - (1
0*b^3)/(a^6*(a + b*x)) - (20*b^3*Log[x])/a^7 + (20*b^3*Log[a + b*x])/a^7

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)^4} \, dx &=\int \left (\frac{1}{a^4 x^4}-\frac{4 b}{a^5 x^3}+\frac{10 b^2}{a^6 x^2}-\frac{20 b^3}{a^7 x}+\frac{b^4}{a^4 (a+b x)^4}+\frac{4 b^4}{a^5 (a+b x)^3}+\frac{10 b^4}{a^6 (a+b x)^2}+\frac{20 b^4}{a^7 (a+b x)}\right ) \, dx\\ &=-\frac{1}{3 a^4 x^3}+\frac{2 b}{a^5 x^2}-\frac{10 b^2}{a^6 x}-\frac{b^3}{3 a^4 (a+b x)^3}-\frac{2 b^3}{a^5 (a+b x)^2}-\frac{10 b^3}{a^6 (a+b x)}-\frac{20 b^3 \log (x)}{a^7}+\frac{20 b^3 \log (a+b x)}{a^7}\\ \end{align*}

Mathematica [A]  time = 0.0965544, size = 88, normalized size = 0.86 \[ -\frac{\frac{a \left (15 a^3 b^2 x^2+110 a^2 b^3 x^3-3 a^4 b x+a^5+150 a b^4 x^4+60 b^5 x^5\right )}{x^3 (a+b x)^3}-60 b^3 \log (a+b x)+60 b^3 \log (x)}{3 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x)^4),x]

[Out]

-((a*(a^5 - 3*a^4*b*x + 15*a^3*b^2*x^2 + 110*a^2*b^3*x^3 + 150*a*b^4*x^4 + 60*b^5*x^5))/(x^3*(a + b*x)^3) + 60
*b^3*Log[x] - 60*b^3*Log[a + b*x])/(3*a^7)

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Maple [A]  time = 0.012, size = 99, normalized size = 1. \begin{align*} -{\frac{1}{3\,{a}^{4}{x}^{3}}}+2\,{\frac{b}{{a}^{5}{x}^{2}}}-10\,{\frac{{b}^{2}}{{a}^{6}x}}-{\frac{{b}^{3}}{3\,{a}^{4} \left ( bx+a \right ) ^{3}}}-2\,{\frac{{b}^{3}}{{a}^{5} \left ( bx+a \right ) ^{2}}}-10\,{\frac{{b}^{3}}{{a}^{6} \left ( bx+a \right ) }}-20\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{7}}}+20\,{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/a^4/x^3+2*b/a^5/x^2-10*b^2/a^6/x-1/3*b^3/a^4/(b*x+a)^3-2*b^3/a^5/(b*x+a)^2-10*b^3/a^6/(b*x+a)-20*b^3*ln(x
)/a^7+20*b^3*ln(b*x+a)/a^7

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Maxima [A]  time = 1.06512, size = 158, normalized size = 1.55 \begin{align*} -\frac{60 \, b^{5} x^{5} + 150 \, a b^{4} x^{4} + 110 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} - 3 \, a^{4} b x + a^{5}}{3 \,{\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{5} + 3 \, a^{8} b x^{4} + a^{9} x^{3}\right )}} + \frac{20 \, b^{3} \log \left (b x + a\right )}{a^{7}} - \frac{20 \, b^{3} \log \left (x\right )}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(60*b^5*x^5 + 150*a*b^4*x^4 + 110*a^2*b^3*x^3 + 15*a^3*b^2*x^2 - 3*a^4*b*x + a^5)/(a^6*b^3*x^6 + 3*a^7*b^
2*x^5 + 3*a^8*b*x^4 + a^9*x^3) + 20*b^3*log(b*x + a)/a^7 - 20*b^3*log(x)/a^7

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Fricas [A]  time = 1.61725, size = 385, normalized size = 3.77 \begin{align*} -\frac{60 \, a b^{5} x^{5} + 150 \, a^{2} b^{4} x^{4} + 110 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} - 3 \, a^{5} b x + a^{6} - 60 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{5} + 3 \, a^{2} b^{4} x^{4} + a^{3} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{5} + 3 \, a^{2} b^{4} x^{4} + a^{3} b^{3} x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{7} b^{3} x^{6} + 3 \, a^{8} b^{2} x^{5} + 3 \, a^{9} b x^{4} + a^{10} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(60*a*b^5*x^5 + 150*a^2*b^4*x^4 + 110*a^3*b^3*x^3 + 15*a^4*b^2*x^2 - 3*a^5*b*x + a^6 - 60*(b^6*x^6 + 3*a*
b^5*x^5 + 3*a^2*b^4*x^4 + a^3*b^3*x^3)*log(b*x + a) + 60*(b^6*x^6 + 3*a*b^5*x^5 + 3*a^2*b^4*x^4 + a^3*b^3*x^3)
*log(x))/(a^7*b^3*x^6 + 3*a^8*b^2*x^5 + 3*a^9*b*x^4 + a^10*x^3)

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Sympy [A]  time = 1.00234, size = 114, normalized size = 1.12 \begin{align*} - \frac{a^{5} - 3 a^{4} b x + 15 a^{3} b^{2} x^{2} + 110 a^{2} b^{3} x^{3} + 150 a b^{4} x^{4} + 60 b^{5} x^{5}}{3 a^{9} x^{3} + 9 a^{8} b x^{4} + 9 a^{7} b^{2} x^{5} + 3 a^{6} b^{3} x^{6}} + \frac{20 b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x+a)**4,x)

[Out]

-(a**5 - 3*a**4*b*x + 15*a**3*b**2*x**2 + 110*a**2*b**3*x**3 + 150*a*b**4*x**4 + 60*b**5*x**5)/(3*a**9*x**3 +
9*a**8*b*x**4 + 9*a**7*b**2*x**5 + 3*a**6*b**3*x**6) + 20*b**3*(-log(x) + log(a/b + x))/a**7

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Giac [A]  time = 1.22387, size = 126, normalized size = 1.24 \begin{align*} \frac{20 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{7}} - \frac{20 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{7}} - \frac{60 \, b^{5} x^{5} + 150 \, a b^{4} x^{4} + 110 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} - 3 \, a^{4} b x + a^{5}}{3 \,{\left (b x^{2} + a x\right )}^{3} a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20*b^3*log(abs(b*x + a))/a^7 - 20*b^3*log(abs(x))/a^7 - 1/3*(60*b^5*x^5 + 150*a*b^4*x^4 + 110*a^2*b^3*x^3 + 15
*a^3*b^2*x^2 - 3*a^4*b*x + a^5)/((b*x^2 + a*x)^3*a^6)