3.352 \(\int \frac{x^8}{a-b x^3} \, dx\)

Optimal. Leaf size=41 \[ -\frac{a^2 \log \left (a-b x^3\right )}{3 b^3}-\frac{a x^3}{3 b^2}-\frac{x^6}{6 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0293576, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {266, 43} \[ -\frac{a^2 \log \left (a-b x^3\right )}{3 b^3}-\frac{a x^3}{3 b^2}-\frac{x^6}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a - b*x^3),x]

[Out]

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

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^8}{a-b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{a-b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a}{b^2}-\frac{x}{b}-\frac{a^2}{b^2 (-a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{a x^3}{3 b^2}-\frac{x^6}{6 b}-\frac{a^2 \log \left (a-b x^3\right )}{3 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0068252, size = 41, normalized size = 1. \[ -\frac{a^2 \log \left (a-b x^3\right )}{3 b^3}-\frac{a x^3}{3 b^2}-\frac{x^6}{6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a - b*x^3),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 37, normalized size = 0.9 \begin{align*} -{\frac{{x}^{6}}{6\,b}}-{\frac{a{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{2}\ln \left ( b{x}^{3}-a \right ) }{3\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(-b*x^3+a),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02048, size = 49, normalized size = 1.2 \begin{align*} -\frac{a^{2} \log \left (b x^{3} - a\right )}{3 \, b^{3}} - \frac{b x^{6} + 2 \, a x^{3}}{6 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-b*x^3+a),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.48899, size = 74, normalized size = 1.8 \begin{align*} -\frac{b^{2} x^{6} + 2 \, a b x^{3} + 2 \, a^{2} \log \left (b x^{3} - a\right )}{6 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-b*x^3+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.496991, size = 34, normalized size = 0.83 \begin{align*} - \frac{a^{2} \log{\left (- a + b x^{3} \right )}}{3 b^{3}} - \frac{a x^{3}}{3 b^{2}} - \frac{x^{6}}{6 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(-b*x**3+a),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.10224, size = 50, normalized size = 1.22 \begin{align*} -\frac{a^{2} \log \left ({\left | b x^{3} - a \right |}\right )}{3 \, b^{3}} - \frac{b x^{6} + 2 \, a x^{3}}{6 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(-b*x^3+a),x, algorithm="giac")

[Out]

-1/3*a^2*log(abs(b*x^3 - a))/b^3 - 1/6*(b*x^6 + 2*a*x^3)/b^2