\(\int \frac {1}{x^3 (a+b x)^4} \, dx\) [204]

Optimal result

Integrand size = 11, antiderivative size = 93 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=-\frac {1}{2 a^4 x^2}+\frac {4 b}{a^5 x}+\frac {b^2}{3 a^3 (a+b x)^3}+\frac {3 b^2}{2 a^4 (a+b x)^2}+\frac {6 b^2}{a^5 (a+b x)}+\frac {10 b^2 \log (x)}{a^6}-\frac {10 b^2 \log (a+b x)}{a^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 93, 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} \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {10 b^2 \log (x)}{a^6}-\frac {10 b^2 \log (a+b x)}{a^6}+\frac {6 b^2}{a^5 (a+b x)}+\frac {4 b}{a^5 x}+\frac {3 b^2}{2 a^4 (a+b x)^2}-\frac {1}{2 a^4 x^2}+\frac {b^2}{3 a^3 (a+b x)^3} \]

[In]

Int[1/(x^3*(a + b*x)^4),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {\frac {a \left (-3 a^4+15 a^3 b x+110 a^2 b^2 x^2+150 a b^3 x^3+60 b^4 x^4\right )}{x^2 (a+b x)^3}+60 b^2 \log (x)-60 b^2 \log (a+b x)}{6 a^6} \]

[In]

Integrate[1/(x^3*(a + b*x)^4),x]

[Out]

((a*(-3*a^4 + 15*a^3*b*x + 110*a^2*b^2*x^2 + 150*a*b^3*x^3 + 60*b^4*x^4))/(x^2*(a + b*x)^3) + 60*b^2*Log[x] -
60*b^2*Log[a + b*x])/(6*a^6)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.89

method	result	size
norman	\(\frac {-\frac {1}{2 a}+\frac {5 b x}{2 a^{2}}-\frac {30 b^{3} x^{3}}{a^{4}}-\frac {45 b^{4} x^{4}}{a^{5}}-\frac {55 b^{5} x^{5}}{3 a^{6}}}{x^{2} \left (b x +a \right )^{3}}+\frac {10 b^{2} \ln \left (x \right )}{a^{6}}-\frac {10 b^{2} \ln \left (b x +a \right )}{a^{6}}\)	\(83\)
risch	\(\frac {\frac {10 b^{4} x^{4}}{a^{5}}+\frac {25 b^{3} x^{3}}{a^{4}}+\frac {55 b^{2} x^{2}}{3 a^{3}}+\frac {5 b x}{2 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b x +a \right )^{3}}-\frac {10 b^{2} \ln \left (b x +a \right )}{a^{6}}+\frac {10 b^{2} \ln \left (-x \right )}{a^{6}}\)	\(85\)
default	\(-\frac {1}{2 a^{4} x^{2}}+\frac {4 b}{a^{5} x}+\frac {b^{2}}{3 a^{3} \left (b x +a \right )^{3}}+\frac {3 b^{2}}{2 a^{4} \left (b x +a \right )^{2}}+\frac {6 b^{2}}{a^{5} \left (b x +a \right )}+\frac {10 b^{2} \ln \left (x \right )}{a^{6}}-\frac {10 b^{2} \ln \left (b x +a \right )}{a^{6}}\)	\(88\)
parallelrisch	\(\frac {60 b^{5} \ln \left (x \right ) x^{5}-60 \ln \left (b x +a \right ) x^{5} b^{5}+180 a \,b^{4} \ln \left (x \right ) x^{4}-180 \ln \left (b x +a \right ) x^{4} a \,b^{4}-110 b^{5} x^{5}+180 a^{2} b^{3} \ln \left (x \right ) x^{3}-180 \ln \left (b x +a \right ) x^{3} a^{2} b^{3}-270 a \,b^{4} x^{4}+60 a^{3} b^{2} \ln \left (x \right ) x^{2}-60 \ln \left (b x +a \right ) x^{2} a^{3} b^{2}-180 a^{2} b^{3} x^{3}+15 a^{4} b x -3 a^{5}}{6 a^{6} x^{2} \left (b x +a \right )^{3}}\)	\(167\)

[In]

int(1/x^3/(b*x+a)^4,x,method=_RETURNVERBOSE)

[Out]

(-1/2/a+5/2*b/a^2*x-30*b^3/a^4*x^3-45*b^4/a^5*x^4-55/3*b^5/a^6*x^5)/x^2/(b*x+a)^3+10*b^2*ln(x)/a^6-10*b^2*ln(b
*x+a)/a^6

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.87 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {60 \, a b^{4} x^{4} + 150 \, a^{2} b^{3} x^{3} + 110 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x - 3 \, a^{5} - 60 \, {\left (b^{5} x^{5} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + a^{3} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} x^{5} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + a^{3} b^{2} x^{2}\right )} \log \left (x\right )}{6 \, {\left (a^{6} b^{3} x^{5} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{3} + a^{9} x^{2}\right )}} \]

[In]

integrate(1/x^3/(b*x+a)^4,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {- 3 a^{4} + 15 a^{3} b x + 110 a^{2} b^{2} x^{2} + 150 a b^{3} x^{3} + 60 b^{4} x^{4}}{6 a^{8} x^{2} + 18 a^{7} b x^{3} + 18 a^{6} b^{2} x^{4} + 6 a^{5} b^{3} x^{5}} + \frac {10 b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{6}} \]

[In]

integrate(1/x**3/(b*x+a)**4,x)

[Out]

(-3*a**4 + 15*a**3*b*x + 110*a**2*b**2*x**2 + 150*a*b**3*x**3 + 60*b**4*x**4)/(6*a**8*x**2 + 18*a**7*b*x**3 +
18*a**6*b**2*x**4 + 6*a**5*b**3*x**5) + 10*b**2*(log(x) - log(a/b + x))/a**6

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {60 \, b^{4} x^{4} + 150 \, a b^{3} x^{3} + 110 \, a^{2} b^{2} x^{2} + 15 \, a^{3} b x - 3 \, a^{4}}{6 \, {\left (a^{5} b^{3} x^{5} + 3 \, a^{6} b^{2} x^{4} + 3 \, a^{7} b x^{3} + a^{8} x^{2}\right )}} - \frac {10 \, b^{2} \log \left (b x + a\right )}{a^{6}} + \frac {10 \, b^{2} \log \left (x\right )}{a^{6}} \]

[In]

integrate(1/x^3/(b*x+a)^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=-\frac {10 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{6}} + \frac {10 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {60 \, a b^{4} x^{4} + 150 \, a^{2} b^{3} x^{3} + 110 \, a^{3} b^{2} x^{2} + 15 \, a^{4} b x - 3 \, a^{5}}{6 \, {\left (b x + a\right )}^{3} a^{6} x^{2}} \]

[In]

integrate(1/x^3/(b*x+a)^4,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 (a+b x)^4} \, dx=\frac {\frac {55\,b^2\,x^2}{3\,a^3}-\frac {1}{2\,a}+\frac {25\,b^3\,x^3}{a^4}+\frac {10\,b^4\,x^4}{a^5}+\frac {5\,b\,x}{2\,a^2}}{a^3\,x^2+3\,a^2\,b\,x^3+3\,a\,b^2\,x^4+b^3\,x^5}-\frac {20\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^6} \]

[In]

int(1/(x^3*(a + b*x)^4),x)

[Out]

((55*b^2*x^2)/(3*a^3) - 1/(2*a) + (25*b^3*x^3)/a^4 + (10*b^4*x^4)/a^5 + (5*b*x)/(2*a^2))/(a^3*x^2 + b^3*x^5 +
3*a^2*b*x^3 + 3*a*b^2*x^4) - (20*b^2*atanh((2*b*x)/a + 1))/a^6