3.178 \(\int \frac{1}{x^4 (a+b x)^2} \, dx\)

Optimal. Leaf size=69 \[ -\frac{4 b^3 \log (x)}{a^5}+\frac{4 b^3 \log (a+b x)}{a^5}-\frac{b^3}{a^4 (a+b x)}-\frac{3 b^2}{a^4 x}+\frac{b}{a^3 x^2}-\frac{1}{3 a^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0793353, antiderivative size = 69, 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 \[ -\frac{4 b^3 \log (x)}{a^5}+\frac{4 b^3 \log (a+b x)}{a^5}-\frac{b^3}{a^4 (a+b x)}-\frac{3 b^2}{a^4 x}+\frac{b}{a^3 x^2}-\frac{1}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.725, size = 66, normalized size = 0.96 \[ - \frac{1}{3 a^{2} x^{3}} + \frac{b}{a^{3} x^{2}} - \frac{b^{3}}{a^{4} \left (a + b x\right )} - \frac{3 b^{2}}{a^{4} x} - \frac{4 b^{3} \log{\left (x \right )}}{a^{5}} + \frac{4 b^{3} \log{\left (a + b x \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0893661, size = 66, normalized size = 0.96 \[ -\frac{\frac{a \left (a^3-2 a^2 b x+6 a b^2 x^2+12 b^3 x^3\right )}{x^3 (a+b x)}-12 b^3 \log (a+b x)+12 b^3 \log (x)}{3 a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x)^2),x]

[Out]

-((a*(a^3 - 2*a^2*b*x + 6*a*b^2*x^2 + 12*b^3*x^3))/(x^3*(a + b*x)) + 12*b^3*Log[
x] - 12*b^3*Log[a + b*x])/(3*a^5)

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Maple [A]  time = 0.016, size = 68, normalized size = 1. \[ -{\frac{1}{3\,{a}^{2}{x}^{3}}}+{\frac{b}{{a}^{3}{x}^{2}}}-3\,{\frac{{b}^{2}}{{a}^{4}x}}-{\frac{{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}-4\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{5}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.37126, size = 99, normalized size = 1.43 \[ -\frac{12 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}}{3 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac{4 \, b^{3} \log \left (b x + a\right )}{a^{5}} - \frac{4 \, b^{3} \log \left (x\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^4),x, algorithm="maxima")

[Out]

-1/3*(12*b^3*x^3 + 6*a*b^2*x^2 - 2*a^2*b*x + a^3)/(a^4*b*x^4 + a^5*x^3) + 4*b^3*
log(b*x + a)/a^5 - 4*b^3*log(x)/a^5

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Fricas [A]  time = 0.208677, size = 128, normalized size = 1.86 \[ -\frac{12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4} - 12 \,{\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} x^{4} + a b^{3} x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^4),x, algorithm="fricas")

[Out]

-1/3*(12*a*b^3*x^3 + 6*a^2*b^2*x^2 - 2*a^3*b*x + a^4 - 12*(b^4*x^4 + a*b^3*x^3)*
log(b*x + a) + 12*(b^4*x^4 + a*b^3*x^3)*log(x))/(a^5*b*x^4 + a^6*x^3)

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Sympy [A]  time = 1.84959, size = 66, normalized size = 0.96 \[ - \frac{a^{3} - 2 a^{2} b x + 6 a b^{2} x^{2} + 12 b^{3} x^{3}}{3 a^{5} x^{3} + 3 a^{4} b x^{4}} + \frac{4 b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x+a)**2,x)

[Out]

-(a**3 - 2*a**2*b*x + 6*a*b**2*x**2 + 12*b**3*x**3)/(3*a**5*x**3 + 3*a**4*b*x**4
) + 4*b**3*(-log(x) + log(a/b + x))/a**5

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GIAC/XCAS [A]  time = 0.231857, size = 122, normalized size = 1.77 \[ -\frac{4 \, b^{3}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{5}} - \frac{b^{3}}{{\left (b x + a\right )} a^{4}} - \frac{\frac{30 \, a b^{3}}{b x + a} - \frac{18 \, a^{2} b^{3}}{{\left (b x + a\right )}^{2}} - 13 \, b^{3}}{3 \, a^{5}{\left (\frac{a}{b x + a} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^2*x^4),x, algorithm="giac")

[Out]

-4*b^3*ln(abs(-a/(b*x + a) + 1))/a^5 - b^3/((b*x + a)*a^4) - 1/3*(30*a*b^3/(b*x
+ a) - 18*a^2*b^3/(b*x + a)^2 - 13*b^3)/(a^5*(a/(b*x + a) - 1)^3)