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

Optimal. Leaf size=66 \[ \frac{b \log \left (a+b x^3\right )}{a^4}-\frac{3 b \log (x)}{a^4}-\frac{2 b}{3 a^3 \left (a+b x^3\right )}-\frac{1}{3 a^3 x^3}-\frac{b}{6 a^2 \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.103139, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{b \log \left (a+b x^3\right )}{a^4}-\frac{3 b \log (x)}{a^4}-\frac{2 b}{3 a^3 \left (a+b x^3\right )}-\frac{1}{3 a^3 x^3}-\frac{b}{6 a^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^3)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.545, size = 63, normalized size = 0.95 \[ - \frac{b}{6 a^{2} \left (a + b x^{3}\right )^{2}} - \frac{2 b}{3 a^{3} \left (a + b x^{3}\right )} - \frac{1}{3 a^{3} x^{3}} - \frac{b \log{\left (x^{3} \right )}}{a^{4}} + \frac{b \log{\left (a + b x^{3} \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b/(6*a**2*(a + b*x**3)**2) - 2*b/(3*a**3*(a + b*x**3)) - 1/(3*a**3*x**3) - b*lo
g(x**3)/a**4 + b*log(a + b*x**3)/a**4

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Mathematica [A]  time = 0.0985324, size = 59, normalized size = 0.89 \[ -\frac{\frac{a \left (2 a^2+9 a b x^3+6 b^2 x^6\right )}{x^3 \left (a+b x^3\right )^2}-6 b \log \left (a+b x^3\right )+18 b \log (x)}{6 a^4} \]

Antiderivative was successfully verified.

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

[Out]

-((a*(2*a^2 + 9*a*b*x^3 + 6*b^2*x^6))/(x^3*(a + b*x^3)^2) + 18*b*Log[x] - 6*b*Lo
g[a + b*x^3])/(6*a^4)

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Maple [A]  time = 0.014, size = 61, normalized size = 0.9 \[ -{\frac{1}{3\,{a}^{3}{x}^{3}}}-{\frac{b}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{2\,b}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-3\,{\frac{b\ln \left ( x \right ) }{{a}^{4}}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) }{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^3+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.45083, size = 103, normalized size = 1.56 \[ -\frac{6 \, b^{2} x^{6} + 9 \, a b x^{3} + 2 \, a^{2}}{6 \,{\left (a^{3} b^{2} x^{9} + 2 \, a^{4} b x^{6} + a^{5} x^{3}\right )}} + \frac{b \log \left (b x^{3} + a\right )}{a^{4}} - \frac{b \log \left (x^{3}\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(6*b^2*x^6 + 9*a*b*x^3 + 2*a^2)/(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3) + b*l
og(b*x^3 + a)/a^4 - b*log(x^3)/a^4

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Fricas [A]  time = 0.220969, size = 161, normalized size = 2.44 \[ -\frac{6 \, a b^{2} x^{6} + 9 \, a^{2} b x^{3} + 2 \, a^{3} - 6 \,{\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (b x^{3} + a\right ) + 18 \,{\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(6*a*b^2*x^6 + 9*a^2*b*x^3 + 2*a^3 - 6*(b^3*x^9 + 2*a*b^2*x^6 + a^2*b*x^3)*
log(b*x^3 + a) + 18*(b^3*x^9 + 2*a*b^2*x^6 + a^2*b*x^3)*log(x))/(a^4*b^2*x^9 + 2
*a^5*b*x^6 + a^6*x^3)

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Sympy [A]  time = 10.7012, size = 75, normalized size = 1.14 \[ - \frac{2 a^{2} + 9 a b x^{3} + 6 b^{2} x^{6}}{6 a^{5} x^{3} + 12 a^{4} b x^{6} + 6 a^{3} b^{2} x^{9}} - \frac{3 b \log{\left (x \right )}}{a^{4}} + \frac{b \log{\left (\frac{a}{b} + x^{3} \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a**2 + 9*a*b*x**3 + 6*b**2*x**6)/(6*a**5*x**3 + 12*a**4*b*x**6 + 6*a**3*b**2
*x**9) - 3*b*log(x)/a**4 + b*log(a/b + x**3)/a**4

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GIAC/XCAS [A]  time = 0.217764, size = 108, normalized size = 1.64 \[ \frac{b{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{a^{4}} - \frac{3 \, b{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{9 \, b^{3} x^{6} + 22 \, a b^{2} x^{3} + 14 \, a^{2} b}{6 \,{\left (b x^{3} + a\right )}^{2} a^{4}} + \frac{3 \, b x^{3} - a}{3 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*ln(abs(b*x^3 + a))/a^4 - 3*b*ln(abs(x))/a^4 - 1/6*(9*b^3*x^6 + 22*a*b^2*x^3 +
14*a^2*b)/((b*x^3 + a)^2*a^4) + 1/3*(3*b*x^3 - a)/(a^4*x^3)