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

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

[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 = 0.0487344, 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, Rules used = {266, 44} \[ -\frac{2 b}{3 a^3 \left (a+b x^3\right )}-\frac{b}{6 a^2 \left (a+b x^3\right )^2}+\frac{b \log \left (a+b x^3\right )}{a^4}-\frac{3 b \log (x)}{a^4}-\frac{1}{3 a^3 x^3} \]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^2}-\frac{3 b}{a^4 x}+\frac{b^2}{a^2 (a+b x)^3}+\frac{2 b^2}{a^3 (a+b x)^2}+\frac{3 b^2}{a^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{1}{3 a^3 x^3}-\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{3 b \log (x)}{a^4}+\frac{b \log \left (a+b x^3\right )}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.050518, 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*Log[a + b*x^3])/(6*a^4)

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Maple [A]  time = 0.015, size = 61, normalized size = 0.9 \begin{align*} -{\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}}} \end{align*}

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 = 0.993694, size = 103, normalized size = 1.56 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.43089, size = 247, normalized size = 3.74 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^3+a)^3,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 = 3.07543, size = 75, normalized size = 1.14 \begin{align*} - \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}} \end{align*}

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*l
og(a/b + x**3)/a**4

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Giac [A]  time = 1.09526, size = 108, normalized size = 1.64 \begin{align*} \frac{b \log \left ({\left | b x^{3} + a \right |}\right )}{a^{4}} - \frac{3 \, b \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(abs(b*x^3 + a))/a^4 - 3*b*log(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)