3.280 \(\int \frac{x^2 (c+d x^3+e x^6+f x^9)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=109 \[ -\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 b^4 \left (a+b x^3\right )^2}-\frac{3 a^2 f-2 a b e+b^2 d}{3 b^4 \left (a+b x^3\right )}+\frac{(b e-3 a f) \log \left (a+b x^3\right )}{3 b^4}+\frac{f x^3}{3 b^3} \]

[Out]

(f*x^3)/(3*b^3) - (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(6*b^4*(a + b*x^3)^2) - (b^2*d - 2*a*b*e + 3*a^2*f)/(3*b
^4*(a + b*x^3)) + ((b*e - 3*a*f)*Log[a + b*x^3])/(3*b^4)

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Rubi [A]  time = 0.151546, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1819, 1850} \[ -\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 b^4 \left (a+b x^3\right )^2}-\frac{3 a^2 f-2 a b e+b^2 d}{3 b^4 \left (a+b x^3\right )}+\frac{(b e-3 a f) \log \left (a+b x^3\right )}{3 b^4}+\frac{f x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(f*x^3)/(3*b^3) - (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(6*b^4*(a + b*x^3)^2) - (b^2*d - 2*a*b*e + 3*a^2*f)/(3*b
^4*(a + b*x^3)) + ((b*e - 3*a*f)*Log[a + b*x^3])/(3*b^4)

Rule 1819

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1)
, Pq, x]*(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && NeQ[m, -1] &&
IGtQ[Simplify[n/(m + 1)], 0] && PolyQ[Pq, x^(m + 1)]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \frac{x^2 \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{f}{b^3}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{b^3 (a+b x)^3}+\frac{b^2 d-2 a b e+3 a^2 f}{b^3 (a+b x)^2}+\frac{b e-3 a f}{b^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{f x^3}{3 b^3}-\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{6 b^4 \left (a+b x^3\right )^2}-\frac{b^2 d-2 a b e+3 a^2 f}{3 b^4 \left (a+b x^3\right )}+\frac{(b e-3 a f) \log \left (a+b x^3\right )}{3 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0541032, size = 105, normalized size = 0.96 \[ \frac{a^2 b \left (3 e-4 f x^3\right )-5 a^3 f+a b^2 \left (-d+4 e x^3+4 f x^6\right )+2 \left (a+b x^3\right )^2 (b e-3 a f) \log \left (a+b x^3\right )-b^3 \left (c+2 d x^3-2 f x^9\right )}{6 b^4 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^3,x]

[Out]

(-5*a^3*f + a^2*b*(3*e - 4*f*x^3) + a*b^2*(-d + 4*e*x^3 + 4*f*x^6) - b^3*(c + 2*d*x^3 - 2*f*x^9) + 2*(b*e - 3*
a*f)*(a + b*x^3)^2*Log[a + b*x^3])/(6*b^4*(a + b*x^3)^2)

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Maple [A]  time = 0.01, size = 156, normalized size = 1.4 \begin{align*}{\frac{f{x}^{3}}{3\,{b}^{3}}}+{\frac{{a}^{3}f}{6\,{b}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{a}^{2}e}{6\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{ad}{6\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{c}{6\,b \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) af}{{b}^{4}}}+{\frac{\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{3}}}-{\frac{{a}^{2}f}{{b}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{2\,ae}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{d}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x)

[Out]

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

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Maxima [A]  time = 0.951215, size = 147, normalized size = 1.35 \begin{align*} \frac{f x^{3}}{3 \, b^{3}} - \frac{b^{3} c + a b^{2} d - 3 \, a^{2} b e + 5 \, a^{3} f + 2 \,{\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{3}}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac{{\left (b e - 3 \, a f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/3*f*x^3/b^3 - 1/6*(b^3*c + a*b^2*d - 3*a^2*b*e + 5*a^3*f + 2*(b^3*d - 2*a*b^2*e + 3*a^2*b*f)*x^3)/(b^6*x^6 +
2*a*b^5*x^3 + a^2*b^4) + 1/3*(b*e - 3*a*f)*log(b*x^3 + a)/b^4

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Fricas [A]  time = 1.165, size = 329, normalized size = 3.02 \begin{align*} \frac{2 \, b^{3} f x^{9} + 4 \, a b^{2} f x^{6} - b^{3} c - a b^{2} d + 3 \, a^{2} b e - 5 \, a^{3} f - 2 \,{\left (b^{3} d - 2 \, a b^{2} e + 2 \, a^{2} b f\right )} x^{3} + 2 \,{\left ({\left (b^{3} e - 3 \, a b^{2} f\right )} x^{6} + a^{2} b e - 3 \, a^{3} f + 2 \,{\left (a b^{2} e - 3 \, a^{2} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \,{\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/6*(2*b^3*f*x^9 + 4*a*b^2*f*x^6 - b^3*c - a*b^2*d + 3*a^2*b*e - 5*a^3*f - 2*(b^3*d - 2*a*b^2*e + 2*a^2*b*f)*x
^3 + 2*((b^3*e - 3*a*b^2*f)*x^6 + a^2*b*e - 3*a^3*f + 2*(a*b^2*e - 3*a^2*b*f)*x^3)*log(b*x^3 + a))/(b^6*x^6 +
2*a*b^5*x^3 + a^2*b^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.07817, size = 135, normalized size = 1.24 \begin{align*} \frac{f x^{3}}{3 \, b^{3}} - \frac{{\left (3 \, a f - b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{4}} - \frac{b^{3} c + a b^{2} d + 5 \, a^{3} f + 2 \,{\left (b^{3} d + 3 \, a^{2} b f - 2 \, a b^{2} e\right )} x^{3} - 3 \, a^{2} b e}{6 \,{\left (b x^{3} + a\right )}^{2} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/3*f*x^3/b^3 - 1/3*(3*a*f - b*e)*log(abs(b*x^3 + a))/b^4 - 1/6*(b^3*c + a*b^2*d + 5*a^3*f + 2*(b^3*d + 3*a^2*
b*f - 2*a*b^2*e)*x^3 - 3*a^2*b*e)/((b*x^3 + a)^2*b^4)