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

Optimal. Leaf size=89 \[ -\frac{10 b^3 \log (x)}{a^6}+\frac{10 b^3 \log (a+b x)}{a^6}-\frac{4 b^3}{a^5 (a+b x)}-\frac{6 b^2}{a^5 x}-\frac{b^3}{2 a^4 (a+b x)^2}+\frac{3 b}{2 a^4 x^2}-\frac{1}{3 a^3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.104454, antiderivative size = 89, 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 \[ -\frac{10 b^3 \log (x)}{a^6}+\frac{10 b^3 \log (a+b x)}{a^6}-\frac{4 b^3}{a^5 (a+b x)}-\frac{6 b^2}{a^5 x}-\frac{b^3}{2 a^4 (a+b x)^2}+\frac{3 b}{2 a^4 x^2}-\frac{1}{3 a^3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.791, size = 87, normalized size = 0.98 \[ - \frac{1}{3 a^{3} x^{3}} - \frac{b^{3}}{2 a^{4} \left (a + b x\right )^{2}} + \frac{3 b}{2 a^{4} x^{2}} - \frac{4 b^{3}}{a^{5} \left (a + b x\right )} - \frac{6 b^{2}}{a^{5} x} - \frac{10 b^{3} \log{\left (x \right )}}{a^{6}} + \frac{10 b^{3} \log{\left (a + b x \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x+a)**3,x)

[Out]

-1/(3*a**3*x**3) - b**3/(2*a**4*(a + b*x)**2) + 3*b/(2*a**4*x**2) - 4*b**3/(a**5
*(a + b*x)) - 6*b**2/(a**5*x) - 10*b**3*log(x)/a**6 + 10*b**3*log(a + b*x)/a**6

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Mathematica [A]  time = 0.109444, size = 79, normalized size = 0.89 \[ -\frac{\frac{a \left (2 a^4-5 a^3 b x+20 a^2 b^2 x^2+90 a b^3 x^3+60 b^4 x^4\right )}{x^3 (a+b x)^2}-60 b^3 \log (a+b x)+60 b^3 \log (x)}{6 a^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x)^3),x]

[Out]

-((a*(2*a^4 - 5*a^3*b*x + 20*a^2*b^2*x^2 + 90*a*b^3*x^3 + 60*b^4*x^4))/(x^3*(a +
b*x)^2) + 60*b^3*Log[x] - 60*b^3*Log[a + b*x])/(6*a^6)

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Maple [A]  time = 0.016, size = 84, normalized size = 0.9 \[ -{\frac{1}{3\,{a}^{3}{x}^{3}}}+{\frac{3\,b}{2\,{a}^{4}{x}^{2}}}-6\,{\frac{{b}^{2}}{{a}^{5}x}}-{\frac{{b}^{3}}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}-4\,{\frac{{b}^{3}}{{a}^{5} \left ( bx+a \right ) }}-10\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{6}}}+10\,{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.35138, size = 131, normalized size = 1.47 \[ -\frac{60 \, b^{4} x^{4} + 90 \, a b^{3} x^{3} + 20 \, a^{2} b^{2} x^{2} - 5 \, a^{3} b x + 2 \, a^{4}}{6 \,{\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} + \frac{10 \, b^{3} \log \left (b x + a\right )}{a^{6}} - \frac{10 \, b^{3} \log \left (x\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(60*b^4*x^4 + 90*a*b^3*x^3 + 20*a^2*b^2*x^2 - 5*a^3*b*x + 2*a^4)/(a^5*b^2*x
^5 + 2*a^6*b*x^4 + a^7*x^3) + 10*b^3*log(b*x + a)/a^6 - 10*b^3*log(x)/a^6

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Fricas [A]  time = 0.214948, size = 190, normalized size = 2.13 \[ -\frac{60 \, a b^{4} x^{4} + 90 \, a^{2} b^{3} x^{3} + 20 \, a^{3} b^{2} x^{2} - 5 \, a^{4} b x + 2 \, a^{5} - 60 \,{\left (b^{5} x^{5} + 2 \, a b^{4} x^{4} + a^{2} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{5} x^{5} + 2 \, a b^{4} x^{4} + a^{2} b^{3} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(60*a*b^4*x^4 + 90*a^2*b^3*x^3 + 20*a^3*b^2*x^2 - 5*a^4*b*x + 2*a^5 - 60*(b
^5*x^5 + 2*a*b^4*x^4 + a^2*b^3*x^3)*log(b*x + a) + 60*(b^5*x^5 + 2*a*b^4*x^4 + a
^2*b^3*x^3)*log(x))/(a^6*b^2*x^5 + 2*a^7*b*x^4 + a^8*x^3)

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Sympy [A]  time = 2.21895, size = 92, normalized size = 1.03 \[ - \frac{2 a^{4} - 5 a^{3} b x + 20 a^{2} b^{2} x^{2} + 90 a b^{3} x^{3} + 60 b^{4} x^{4}}{6 a^{7} x^{3} + 12 a^{6} b x^{4} + 6 a^{5} b^{2} x^{5}} + \frac{10 b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x+a)**3,x)

[Out]

-(2*a**4 - 5*a**3*b*x + 20*a**2*b**2*x**2 + 90*a*b**3*x**3 + 60*b**4*x**4)/(6*a*
*7*x**3 + 12*a**6*b*x**4 + 6*a**5*b**2*x**5) + 10*b**3*(-log(x) + log(a/b + x))/
a**6

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GIAC/XCAS [A]  time = 0.205728, size = 116, normalized size = 1.3 \[ \frac{10 \, b^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{6}} - \frac{10 \, b^{3}{\rm ln}\left ({\left | x \right |}\right )}{a^{6}} - \frac{60 \, a b^{4} x^{4} + 90 \, a^{2} b^{3} x^{3} + 20 \, a^{3} b^{2} x^{2} - 5 \, a^{4} b x + 2 \, a^{5}}{6 \,{\left (b x + a\right )}^{2} a^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*b^3*ln(abs(b*x + a))/a^6 - 10*b^3*ln(abs(x))/a^6 - 1/6*(60*a*b^4*x^4 + 90*a^2
*b^3*x^3 + 20*a^3*b^2*x^2 - 5*a^4*b*x + 2*a^5)/((b*x + a)^2*a^6*x^3)