3.192 \(\int \frac{1}{x^5 (a+b x)^3} \, dx\)

Optimal. Leaf size=97 \[ \frac{15 b^4 \log (x)}{a^7}-\frac{15 b^4 \log (a+b x)}{a^7}+\frac{5 b^4}{a^6 (a+b x)}+\frac{10 b^3}{a^6 x}+\frac{b^4}{2 a^5 (a+b x)^2}-\frac{3 b^2}{a^5 x^2}+\frac{b}{a^4 x^3}-\frac{1}{4 a^3 x^4} \]

[Out]

-1/(4*a^3*x^4) + b/(a^4*x^3) - (3*b^2)/(a^5*x^2) + (10*b^3)/(a^6*x) + b^4/(2*a^5
*(a + b*x)^2) + (5*b^4)/(a^6*(a + b*x)) + (15*b^4*Log[x])/a^7 - (15*b^4*Log[a +
b*x])/a^7

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Rubi [A]  time = 0.124366, antiderivative size = 97, 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 \[ \frac{15 b^4 \log (x)}{a^7}-\frac{15 b^4 \log (a+b x)}{a^7}+\frac{5 b^4}{a^6 (a+b x)}+\frac{10 b^3}{a^6 x}+\frac{b^4}{2 a^5 (a+b x)^2}-\frac{3 b^2}{a^5 x^2}+\frac{b}{a^4 x^3}-\frac{1}{4 a^3 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x)^3),x]

[Out]

-1/(4*a^3*x^4) + b/(a^4*x^3) - (3*b^2)/(a^5*x^2) + (10*b^3)/(a^6*x) + b^4/(2*a^5
*(a + b*x)^2) + (5*b^4)/(a^6*(a + b*x)) + (15*b^4*Log[x])/a^7 - (15*b^4*Log[a +
b*x])/a^7

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Rubi in Sympy [A]  time = 20.5086, size = 95, normalized size = 0.98 \[ - \frac{1}{4 a^{3} x^{4}} + \frac{b}{a^{4} x^{3}} + \frac{b^{4}}{2 a^{5} \left (a + b x\right )^{2}} - \frac{3 b^{2}}{a^{5} x^{2}} + \frac{5 b^{4}}{a^{6} \left (a + b x\right )} + \frac{10 b^{3}}{a^{6} x} + \frac{15 b^{4} \log{\left (x \right )}}{a^{7}} - \frac{15 b^{4} \log{\left (a + b x \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x+a)**3,x)

[Out]

-1/(4*a**3*x**4) + b/(a**4*x**3) + b**4/(2*a**5*(a + b*x)**2) - 3*b**2/(a**5*x**
2) + 5*b**4/(a**6*(a + b*x)) + 10*b**3/(a**6*x) + 15*b**4*log(x)/a**7 - 15*b**4*
log(a + b*x)/a**7

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Mathematica [A]  time = 0.105073, size = 90, normalized size = 0.93 \[ \frac{\frac{a \left (-a^5+2 a^4 b x-5 a^3 b^2 x^2+20 a^2 b^3 x^3+90 a b^4 x^4+60 b^5 x^5\right )}{x^4 (a+b x)^2}-60 b^4 \log (a+b x)+60 b^4 \log (x)}{4 a^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x)^3),x]

[Out]

((a*(-a^5 + 2*a^4*b*x - 5*a^3*b^2*x^2 + 20*a^2*b^3*x^3 + 90*a*b^4*x^4 + 60*b^5*x
^5))/(x^4*(a + b*x)^2) + 60*b^4*Log[x] - 60*b^4*Log[a + b*x])/(4*a^7)

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Maple [A]  time = 0.015, size = 94, normalized size = 1. \[ -{\frac{1}{4\,{a}^{3}{x}^{4}}}+{\frac{b}{{a}^{4}{x}^{3}}}-3\,{\frac{{b}^{2}}{{a}^{5}{x}^{2}}}+10\,{\frac{{b}^{3}}{{a}^{6}x}}+{\frac{{b}^{4}}{2\,{a}^{5} \left ( bx+a \right ) ^{2}}}+5\,{\frac{{b}^{4}}{{a}^{6} \left ( bx+a \right ) }}+15\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{7}}}-15\,{\frac{{b}^{4}\ln \left ( bx+a \right ) }{{a}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x+a)^3,x)

[Out]

-1/4/a^3/x^4+b/a^4/x^3-3*b^2/a^5/x^2+10*b^3/a^6/x+1/2*b^4/a^5/(b*x+a)^2+5*b^4/a^
6/(b*x+a)+15*b^4*ln(x)/a^7-15*b^4*ln(b*x+a)/a^7

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Maxima [A]  time = 1.35077, size = 146, normalized size = 1.51 \[ \frac{60 \, b^{5} x^{5} + 90 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} - 5 \, a^{3} b^{2} x^{2} + 2 \, a^{4} b x - a^{5}}{4 \,{\left (a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{5} + a^{8} x^{4}\right )}} - \frac{15 \, b^{4} \log \left (b x + a\right )}{a^{7}} + \frac{15 \, b^{4} \log \left (x\right )}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*x^5),x, algorithm="maxima")

[Out]

1/4*(60*b^5*x^5 + 90*a*b^4*x^4 + 20*a^2*b^3*x^3 - 5*a^3*b^2*x^2 + 2*a^4*b*x - a^
5)/(a^6*b^2*x^6 + 2*a^7*b*x^5 + a^8*x^4) - 15*b^4*log(b*x + a)/a^7 + 15*b^4*log(
x)/a^7

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Fricas [A]  time = 0.217177, size = 205, normalized size = 2.11 \[ \frac{60 \, a b^{5} x^{5} + 90 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{4} b^{2} x^{2} + 2 \, a^{5} b x - a^{6} - 60 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} + a^{2} b^{4} x^{4}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} + a^{2} b^{4} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{7} b^{2} x^{6} + 2 \, a^{8} b x^{5} + a^{9} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*x^5),x, algorithm="fricas")

[Out]

1/4*(60*a*b^5*x^5 + 90*a^2*b^4*x^4 + 20*a^3*b^3*x^3 - 5*a^4*b^2*x^2 + 2*a^5*b*x
- a^6 - 60*(b^6*x^6 + 2*a*b^5*x^5 + a^2*b^4*x^4)*log(b*x + a) + 60*(b^6*x^6 + 2*
a*b^5*x^5 + a^2*b^4*x^4)*log(x))/(a^7*b^2*x^6 + 2*a^8*b*x^5 + a^9*x^4)

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Sympy [A]  time = 2.5107, size = 102, normalized size = 1.05 \[ \frac{- a^{5} + 2 a^{4} b x - 5 a^{3} b^{2} x^{2} + 20 a^{2} b^{3} x^{3} + 90 a b^{4} x^{4} + 60 b^{5} x^{5}}{4 a^{8} x^{4} + 8 a^{7} b x^{5} + 4 a^{6} b^{2} x^{6}} + \frac{15 b^{4} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x+a)**3,x)

[Out]

(-a**5 + 2*a**4*b*x - 5*a**3*b**2*x**2 + 20*a**2*b**3*x**3 + 90*a*b**4*x**4 + 60
*b**5*x**5)/(4*a**8*x**4 + 8*a**7*b*x**5 + 4*a**6*b**2*x**6) + 15*b**4*(log(x) -
log(a/b + x))/a**7

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GIAC/XCAS [A]  time = 0.21023, size = 131, normalized size = 1.35 \[ -\frac{15 \, b^{4}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{7}} + \frac{15 \, b^{4}{\rm ln}\left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{5} + 90 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{4} b^{2} x^{2} + 2 \, a^{5} b x - a^{6}}{4 \,{\left (b x + a\right )}^{2} a^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^3*x^5),x, algorithm="giac")

[Out]

-15*b^4*ln(abs(b*x + a))/a^7 + 15*b^4*ln(abs(x))/a^7 + 1/4*(60*a*b^5*x^5 + 90*a^
2*b^4*x^4 + 20*a^3*b^3*x^3 - 5*a^4*b^2*x^2 + 2*a^5*b*x - a^6)/((b*x + a)^2*a^7*x
^4)