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

Optimal. Leaf size=35 \[ \frac{b \log \left (a+b x^3\right )}{3 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0212956, antiderivative size = 35, 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} \[ \frac{b \log \left (a+b x^3\right )}{3 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^3)),x]

[Out]

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

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

Mathematica [A]  time = 0.006412, size = 35, normalized size = 1. \[ \frac{b \log \left (a+b x^3\right )}{3 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^3)),x]

[Out]

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

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Maple [A]  time = 0.005, size = 32, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,a{x}^{3}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 0.95816, size = 45, normalized size = 1.29 \begin{align*} \frac{b \log \left (b x^{3} + a\right )}{3 \, a^{2}} - \frac{b \log \left (x^{3}\right )}{3 \, a^{2}} - \frac{1}{3 \, a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43564, size = 80, normalized size = 2.29 \begin{align*} \frac{b x^{3} \log \left (b x^{3} + a\right ) - 3 \, b x^{3} \log \left (x\right ) - a}{3 \, a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.460066, size = 31, normalized size = 0.89 \begin{align*} - \frac{1}{3 a x^{3}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13638, size = 57, normalized size = 1.63 \begin{align*} \frac{b \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} - \frac{b \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{b x^{3} - a}{3 \, a^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*log(abs(b*x^3 + a))/a^2 - b*log(abs(x))/a^2 + 1/3*(b*x^3 - a)/(a^2*x^3)