\(\int \frac {1}{x^4 (a+b x)^4} \, dx\) [205]

Optimal result

Integrand size = 11, antiderivative size = 102 \[ \int \frac {1}{x^4 (a+b x)^4} \, dx=-\frac {1}{3 a^4 x^3}+\frac {2 b}{a^5 x^2}-\frac {10 b^2}{a^6 x}-\frac {b^3}{3 a^4 (a+b x)^3}-\frac {2 b^3}{a^5 (a+b x)^2}-\frac {10 b^3}{a^6 (a+b x)}-\frac {20 b^3 \log (x)}{a^7}+\frac {20 b^3 \log (a+b x)}{a^7} \]

[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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^4 (a+b x)^4} \, dx=-\frac {20 b^3 \log (x)}{a^7}+\frac {20 b^3 \log (a+b x)}{a^7}-\frac {10 b^3}{a^6 (a+b x)}-\frac {10 b^2}{a^6 x}-\frac {2 b^3}{a^5 (a+b x)^2}+\frac {2 b}{a^5 x^2}-\frac {b^3}{3 a^4 (a+b x)^3}-\frac {1}{3 a^4 x^3} \]

[In]

Int[1/(x^4*(a + b*x)^4),x]

[Out]

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^4 (a+b x)^4} \, dx=-\frac {\frac {a \left (a^5-3 a^4 b x+15 a^3 b^2 x^2+110 a^2 b^3 x^3+150 a b^4 x^4+60 b^5 x^5\right )}{x^3 (a+b x)^3}+60 b^3 \log (x)-60 b^3 \log (a+b x)}{3 a^7} \]

[In]

Integrate[1/(x^4*(a + b*x)^4),x]

[Out]

-1/3*((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])/a^7

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.91

method	result	size
norman	\(\frac {\frac {b x}{a^{2}}-\frac {1}{3 a}-\frac {5 b^{2} x^{2}}{a^{3}}+\frac {60 b^{4} x^{4}}{a^{5}}+\frac {90 b^{5} x^{5}}{a^{6}}+\frac {110 b^{6} x^{6}}{3 a^{7}}}{x^{3} \left (b x +a \right )^{3}}-\frac {20 b^{3} \ln \left (x \right )}{a^{7}}+\frac {20 b^{3} \ln \left (b x +a \right )}{a^{7}}\)	\(93\)
risch	\(\frac {-\frac {20 b^{5} x^{5}}{a^{6}}-\frac {50 b^{4} x^{4}}{a^{5}}-\frac {110 b^{3} x^{3}}{3 a^{4}}-\frac {5 b^{2} x^{2}}{a^{3}}+\frac {b x}{a^{2}}-\frac {1}{3 a}}{x^{3} \left (b x +a \right )^{3}}+\frac {20 b^{3} \ln \left (-b x -a \right )}{a^{7}}-\frac {20 b^{3} \ln \left (x \right )}{a^{7}}\)	\(96\)
default	\(-\frac {1}{3 a^{4} x^{3}}+\frac {2 b}{a^{5} x^{2}}-\frac {10 b^{2}}{a^{6} x}-\frac {b^{3}}{3 a^{4} \left (b x +a \right )^{3}}-\frac {2 b^{3}}{a^{5} \left (b x +a \right )^{2}}-\frac {10 b^{3}}{a^{6} \left (b x +a \right )}-\frac {20 b^{3} \ln \left (x \right )}{a^{7}}+\frac {20 b^{3} \ln \left (b x +a \right )}{a^{7}}\)	\(99\)
parallelrisch	\(-\frac {60 \ln \left (x \right ) x^{6} b^{6}-60 \ln \left (b x +a \right ) x^{6} b^{6}+180 \ln \left (x \right ) x^{5} a \,b^{5}-180 \ln \left (b x +a \right ) x^{5} a \,b^{5}-110 b^{6} x^{6}+180 \ln \left (x \right ) x^{4} a^{2} b^{4}-180 \ln \left (b x +a \right ) x^{4} a^{2} b^{4}-270 a \,x^{5} b^{5}+60 \ln \left (x \right ) x^{3} a^{3} b^{3}-60 \ln \left (b x +a \right ) x^{3} a^{3} b^{3}-180 a^{2} x^{4} b^{4}+15 a^{4} x^{2} b^{2}-3 a^{5} x b +a^{6}}{3 a^{7} x^{3} \left (b x +a \right )^{3}}\)	\(176\)

[In]

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

[Out]

(b/a^2*x-1/3/a-5*b^2/a^3*x^2+60*b^4/a^5*x^4+90*b^5/a^6*x^5+110/3*b^6/a^7*x^6)/x^3/(b*x+a)^3-20*b^3*ln(x)/a^7+2
0*b^3*ln(b*x+a)/a^7

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x^4 (a+b x)^4} \, dx=-\frac {60 \, a b^{5} x^{5} + 150 \, a^{2} b^{4} x^{4} + 110 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} - 3 \, a^{5} b x + a^{6} - 60 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{5} + 3 \, a^{2} b^{4} x^{4} + a^{3} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{5} + 3 \, a^{2} b^{4} x^{4} + a^{3} b^{3} x^{3}\right )} \log \left (x\right )}{3 \, {\left (a^{7} b^{3} x^{6} + 3 \, a^{8} b^{2} x^{5} + 3 \, a^{9} b x^{4} + a^{10} x^{3}\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^4 (a+b x)^4} \, dx=\frac {- a^{5} + 3 a^{4} b x - 15 a^{3} b^{2} x^{2} - 110 a^{2} b^{3} x^{3} - 150 a b^{4} x^{4} - 60 b^{5} x^{5}}{3 a^{9} x^{3} + 9 a^{8} b x^{4} + 9 a^{7} b^{2} x^{5} + 3 a^{6} b^{3} x^{6}} + \frac {20 b^{3} \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{7}} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^4 (a+b x)^4} \, dx=-\frac {60 \, b^{5} x^{5} + 150 \, a b^{4} x^{4} + 110 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} - 3 \, a^{4} b x + a^{5}}{3 \, {\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{5} + 3 \, a^{8} b x^{4} + a^{9} x^{3}\right )}} + \frac {20 \, b^{3} \log \left (b x + a\right )}{a^{7}} - \frac {20 \, b^{3} \log \left (x\right )}{a^{7}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^4 (a+b x)^4} \, dx=\frac {20 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{7}} - \frac {20 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{7}} - \frac {60 \, b^{5} x^{5} + 150 \, a b^{4} x^{4} + 110 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} - 3 \, a^{4} b x + a^{5}}{3 \, {\left (b x^{2} + a x\right )}^{3} a^{6}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^4 (a+b x)^4} \, dx=\frac {40\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^7}-\frac {\frac {1}{3\,a}+\frac {5\,b^2\,x^2}{a^3}+\frac {110\,b^3\,x^3}{3\,a^4}+\frac {50\,b^4\,x^4}{a^5}+\frac {20\,b^5\,x^5}{a^6}-\frac {b\,x}{a^2}}{a^3\,x^3+3\,a^2\,b\,x^4+3\,a\,b^2\,x^5+b^3\,x^6} \]

[In]

int(1/(x^4*(a + b*x)^4),x)

[Out]

(40*b^3*atanh((2*b*x)/a + 1))/a^7 - (1/(3*a) + (5*b^2*x^2)/a^3 + (110*b^3*x^3)/(3*a^4) + (50*b^4*x^4)/a^5 + (2
0*b^5*x^5)/a^6 - (b*x)/a^2)/(a^3*x^3 + b^3*x^6 + 3*a^2*b*x^4 + 3*a*b^2*x^5)