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

Optimal. Leaf size=96 \[ \frac{\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4}+\frac{x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}+\frac{x^6 (b e-a f)}{6 b^2}+\frac{f x^9}{9 b} \]

[Out]

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

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Rubi [A]  time = 0.139912, antiderivative size = 96, 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} \[ \frac{\log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4}+\frac{x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}+\frac{x^6 (b e-a f)}{6 b^2}+\frac{f x^9}{9 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0430123, size = 88, normalized size = 0.92 \[ \frac{6 \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+b x^3 \left (6 a^2 f-3 a b \left (2 e+f x^3\right )+b^2 \left (6 d+3 e x^3+2 f x^6\right )\right )}{18 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^3*(6*a^2*f - 3*a*b*(2*e + f*x^3) + b^2*(6*d + 3*e*x^3 + 2*f*x^6)) + 6*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)
*Log[a + b*x^3])/(18*b^4)

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Maple [A]  time = 0.003, size = 124, normalized size = 1.3 \begin{align*}{\frac{f{x}^{9}}{9\,b}}-{\frac{{x}^{6}af}{6\,{b}^{2}}}+{\frac{e{x}^{6}}{6\,b}}+{\frac{{a}^{2}f{x}^{3}}{3\,{b}^{3}}}-{\frac{ae{x}^{3}}{3\,{b}^{2}}}+{\frac{d{x}^{3}}{3\,b}}-{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{3}f}{3\,{b}^{4}}}+{\frac{\ln \left ( b{x}^{3}+a \right ){a}^{2}e}{3\,{b}^{3}}}-{\frac{\ln \left ( b{x}^{3}+a \right ) ad}{3\,{b}^{2}}}+{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \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),x)

[Out]

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

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Maxima [A]  time = 0.948623, size = 123, normalized size = 1.28 \begin{align*} \frac{2 \, b^{2} f x^{9} + 3 \,{\left (b^{2} e - a b f\right )} x^{6} + 6 \,{\left (b^{2} d - a b e + a^{2} f\right )} x^{3}}{18 \, b^{3}} + \frac{{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} 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),x, algorithm="maxima")

[Out]

1/18*(2*b^2*f*x^9 + 3*(b^2*e - a*b*f)*x^6 + 6*(b^2*d - a*b*e + a^2*f)*x^3)/b^3 + 1/3*(b^3*c - a*b^2*d + a^2*b*
e - a^3*f)*log(b*x^3 + a)/b^4

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Fricas [A]  time = 1.19505, size = 190, normalized size = 1.98 \begin{align*} \frac{2 \, b^{3} f x^{9} + 3 \,{\left (b^{3} e - a b^{2} f\right )} x^{6} + 6 \,{\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{3} + 6 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (b x^{3} + a\right )}{18 \, 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),x, algorithm="fricas")

[Out]

1/18*(2*b^3*f*x^9 + 3*(b^3*e - a*b^2*f)*x^6 + 6*(b^3*d - a*b^2*e + a^2*b*f)*x^3 + 6*(b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*log(b*x^3 + a))/b^4

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Sympy [A]  time = 1.00665, size = 83, normalized size = 0.86 \begin{align*} \frac{f x^{9}}{9 b} - \frac{x^{6} \left (a f - b e\right )}{6 b^{2}} + \frac{x^{3} \left (a^{2} f - a b e + b^{2} d\right )}{3 b^{3}} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a + b x^{3} \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),x)

[Out]

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

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Giac [A]  time = 1.06947, size = 136, normalized size = 1.42 \begin{align*} \frac{2 \, b^{2} f x^{9} - 3 \, a b f x^{6} + 3 \, b^{2} x^{6} e + 6 \, b^{2} d x^{3} + 6 \, a^{2} f x^{3} - 6 \, a b x^{3} e}{18 \, b^{3}} + \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\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),x, algorithm="giac")

[Out]

1/18*(2*b^2*f*x^9 - 3*a*b*f*x^6 + 3*b^2*x^6*e + 6*b^2*d*x^3 + 6*a^2*f*x^3 - 6*a*b*x^3*e)/b^3 + 1/3*(b^3*c - a*
b^2*d - a^3*f + a^2*b*e)*log(abs(b*x^3 + a))/b^4