3.206 \(\int \frac{1}{x^5 (a+b x)^4} \, dx\)

Optimal. Leaf size=117 \[ -\frac{5 b^2}{a^6 x^2}+\frac{15 b^4}{a^7 (a+b x)}+\frac{5 b^4}{2 a^6 (a+b x)^2}+\frac{b^4}{3 a^5 (a+b x)^3}+\frac{20 b^3}{a^7 x}+\frac{35 b^4 \log (x)}{a^8}-\frac{35 b^4 \log (a+b x)}{a^8}+\frac{4 b}{3 a^5 x^3}-\frac{1}{4 a^4 x^4} \]

[Out]

-1/(4*a^4*x^4) + (4*b)/(3*a^5*x^3) - (5*b^2)/(a^6*x^2) + (20*b^3)/(a^7*x) + b^4/(3*a^5*(a + b*x)^3) + (5*b^4)/
(2*a^6*(a + b*x)^2) + (15*b^4)/(a^7*(a + b*x)) + (35*b^4*Log[x])/a^8 - (35*b^4*Log[a + b*x])/a^8

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Rubi [A]  time = 0.0702539, antiderivative size = 117, 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} \[ -\frac{5 b^2}{a^6 x^2}+\frac{15 b^4}{a^7 (a+b x)}+\frac{5 b^4}{2 a^6 (a+b x)^2}+\frac{b^4}{3 a^5 (a+b x)^3}+\frac{20 b^3}{a^7 x}+\frac{35 b^4 \log (x)}{a^8}-\frac{35 b^4 \log (a+b x)}{a^8}+\frac{4 b}{3 a^5 x^3}-\frac{1}{4 a^4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x)^4),x]

[Out]

-1/(4*a^4*x^4) + (4*b)/(3*a^5*x^3) - (5*b^2)/(a^6*x^2) + (20*b^3)/(a^7*x) + b^4/(3*a^5*(a + b*x)^3) + (5*b^4)/
(2*a^6*(a + b*x)^2) + (15*b^4)/(a^7*(a + b*x)) + (35*b^4*Log[x])/a^8 - (35*b^4*Log[a + b*x])/a^8

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^5 (a+b x)^4} \, dx &=\int \left (\frac{1}{a^4 x^5}-\frac{4 b}{a^5 x^4}+\frac{10 b^2}{a^6 x^3}-\frac{20 b^3}{a^7 x^2}+\frac{35 b^4}{a^8 x}-\frac{b^5}{a^5 (a+b x)^4}-\frac{5 b^5}{a^6 (a+b x)^3}-\frac{15 b^5}{a^7 (a+b x)^2}-\frac{35 b^5}{a^8 (a+b x)}\right ) \, dx\\ &=-\frac{1}{4 a^4 x^4}+\frac{4 b}{3 a^5 x^3}-\frac{5 b^2}{a^6 x^2}+\frac{20 b^3}{a^7 x}+\frac{b^4}{3 a^5 (a+b x)^3}+\frac{5 b^4}{2 a^6 (a+b x)^2}+\frac{15 b^4}{a^7 (a+b x)}+\frac{35 b^4 \log (x)}{a^8}-\frac{35 b^4 \log (a+b x)}{a^8}\\ \end{align*}

Mathematica [A]  time = 0.10412, size = 101, normalized size = 0.86 \[ \frac{\frac{a \left (-21 a^4 b^2 x^2+105 a^3 b^3 x^3+770 a^2 b^4 x^4+7 a^5 b x-3 a^6+1050 a b^5 x^5+420 b^6 x^6\right )}{x^4 (a+b x)^3}-420 b^4 \log (a+b x)+420 b^4 \log (x)}{12 a^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x)^4),x]

[Out]

((a*(-3*a^6 + 7*a^5*b*x - 21*a^4*b^2*x^2 + 105*a^3*b^3*x^3 + 770*a^2*b^4*x^4 + 1050*a*b^5*x^5 + 420*b^6*x^6))/
(x^4*(a + b*x)^3) + 420*b^4*Log[x] - 420*b^4*Log[a + b*x])/(12*a^8)

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Maple [A]  time = 0.012, size = 110, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{a}^{4}{x}^{4}}}+{\frac{4\,b}{3\,{a}^{5}{x}^{3}}}-5\,{\frac{{b}^{2}}{{a}^{6}{x}^{2}}}+20\,{\frac{{b}^{3}}{{a}^{7}x}}+{\frac{{b}^{4}}{3\,{a}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{5\,{b}^{4}}{2\,{a}^{6} \left ( bx+a \right ) ^{2}}}+15\,{\frac{{b}^{4}}{{a}^{7} \left ( bx+a \right ) }}+35\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{8}}}-35\,{\frac{{b}^{4}\ln \left ( bx+a \right ) }{{a}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x+a)^4,x)

[Out]

-1/4/a^4/x^4+4/3*b/a^5/x^3-5*b^2/a^6/x^2+20*b^3/a^7/x+1/3*b^4/a^5/(b*x+a)^3+5/2*b^4/a^6/(b*x+a)^2+15*b^4/a^7/(
b*x+a)+35*b^4*ln(x)/a^8-35*b^4*ln(b*x+a)/a^8

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Maxima [A]  time = 1.00662, size = 176, normalized size = 1.5 \begin{align*} \frac{420 \, b^{6} x^{6} + 1050 \, a b^{5} x^{5} + 770 \, a^{2} b^{4} x^{4} + 105 \, a^{3} b^{3} x^{3} - 21 \, a^{4} b^{2} x^{2} + 7 \, a^{5} b x - 3 \, a^{6}}{12 \,{\left (a^{7} b^{3} x^{7} + 3 \, a^{8} b^{2} x^{6} + 3 \, a^{9} b x^{5} + a^{10} x^{4}\right )}} - \frac{35 \, b^{4} \log \left (b x + a\right )}{a^{8}} + \frac{35 \, b^{4} \log \left (x\right )}{a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^4,x, algorithm="maxima")

[Out]

1/12*(420*b^6*x^6 + 1050*a*b^5*x^5 + 770*a^2*b^4*x^4 + 105*a^3*b^3*x^3 - 21*a^4*b^2*x^2 + 7*a^5*b*x - 3*a^6)/(
a^7*b^3*x^7 + 3*a^8*b^2*x^6 + 3*a^9*b*x^5 + a^10*x^4) - 35*b^4*log(b*x + a)/a^8 + 35*b^4*log(x)/a^8

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Fricas [A]  time = 1.60078, size = 419, normalized size = 3.58 \begin{align*} \frac{420 \, a b^{6} x^{6} + 1050 \, a^{2} b^{5} x^{5} + 770 \, a^{3} b^{4} x^{4} + 105 \, a^{4} b^{3} x^{3} - 21 \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x - 3 \, a^{7} - 420 \,{\left (b^{7} x^{7} + 3 \, a b^{6} x^{6} + 3 \, a^{2} b^{5} x^{5} + a^{3} b^{4} x^{4}\right )} \log \left (b x + a\right ) + 420 \,{\left (b^{7} x^{7} + 3 \, a b^{6} x^{6} + 3 \, a^{2} b^{5} x^{5} + a^{3} b^{4} x^{4}\right )} \log \left (x\right )}{12 \,{\left (a^{8} b^{3} x^{7} + 3 \, a^{9} b^{2} x^{6} + 3 \, a^{10} b x^{5} + a^{11} x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/12*(420*a*b^6*x^6 + 1050*a^2*b^5*x^5 + 770*a^3*b^4*x^4 + 105*a^4*b^3*x^3 - 21*a^5*b^2*x^2 + 7*a^6*b*x - 3*a^
7 - 420*(b^7*x^7 + 3*a*b^6*x^6 + 3*a^2*b^5*x^5 + a^3*b^4*x^4)*log(b*x + a) + 420*(b^7*x^7 + 3*a*b^6*x^6 + 3*a^
2*b^5*x^5 + a^3*b^4*x^4)*log(x))/(a^8*b^3*x^7 + 3*a^9*b^2*x^6 + 3*a^10*b*x^5 + a^11*x^4)

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Sympy [A]  time = 1.02134, size = 128, normalized size = 1.09 \begin{align*} \frac{- 3 a^{6} + 7 a^{5} b x - 21 a^{4} b^{2} x^{2} + 105 a^{3} b^{3} x^{3} + 770 a^{2} b^{4} x^{4} + 1050 a b^{5} x^{5} + 420 b^{6} x^{6}}{12 a^{10} x^{4} + 36 a^{9} b x^{5} + 36 a^{8} b^{2} x^{6} + 12 a^{7} b^{3} x^{7}} + \frac{35 b^{4} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x+a)**4,x)

[Out]

(-3*a**6 + 7*a**5*b*x - 21*a**4*b**2*x**2 + 105*a**3*b**3*x**3 + 770*a**2*b**4*x**4 + 1050*a*b**5*x**5 + 420*b
**6*x**6)/(12*a**10*x**4 + 36*a**9*b*x**5 + 36*a**8*b**2*x**6 + 12*a**7*b**3*x**7) + 35*b**4*(log(x) - log(a/b
+ x))/a**8

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Giac [A]  time = 1.16894, size = 146, normalized size = 1.25 \begin{align*} -\frac{35 \, b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{8}} + \frac{35 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac{420 \, a b^{6} x^{6} + 1050 \, a^{2} b^{5} x^{5} + 770 \, a^{3} b^{4} x^{4} + 105 \, a^{4} b^{3} x^{3} - 21 \, a^{5} b^{2} x^{2} + 7 \, a^{6} b x - 3 \, a^{7}}{12 \,{\left (b x + a\right )}^{3} a^{8} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x+a)^4,x, algorithm="giac")

[Out]

-35*b^4*log(abs(b*x + a))/a^8 + 35*b^4*log(abs(x))/a^8 + 1/12*(420*a*b^6*x^6 + 1050*a^2*b^5*x^5 + 770*a^3*b^4*
x^4 + 105*a^4*b^3*x^3 - 21*a^5*b^2*x^2 + 7*a^6*b*x - 3*a^7)/((b*x + a)^3*a^8*x^4)