3.225 \(\int \frac{x^5 (c+d x^3+e x^6+f x^9)}{a+b x^3} \, dx\)

Optimal. Leaf size=132 \[ \frac{x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4}-\frac{a \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^6 \left (a^2 f-a b e+b^2 d\right )}{6 b^3}+\frac{x^9 (b e-a f)}{9 b^2}+\frac{f x^{12}}{12 b} \]

[Out]

((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/(3*b^4) + ((b^2*d - a*b*e + a^2*f)*x^6)/(6*b^3) + ((b*e - a*f)*x^9)/
(9*b^2) + (f*x^12)/(12*b) - (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a + b*x^3])/(3*b^5)

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Rubi [A]  time = 0.183319, antiderivative size = 132, 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 = {1821, 1620} \[ \frac{x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4}-\frac{a \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5}+\frac{x^6 \left (a^2 f-a b e+b^2 d\right )}{6 b^3}+\frac{x^9 (b e-a f)}{9 b^2}+\frac{f x^{12}}{12 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/(3*b^4) + ((b^2*d - a*b*e + a^2*f)*x^6)/(6*b^3) + ((b*e - a*f)*x^9)/
(9*b^2) + (f*x^12)/(12*b) - (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a + b*x^3])/(3*b^5)

Rule 1821

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

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \frac{x^5 \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{x \left (c+d x+e x^2+f x^3\right )}{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x}{b^3}+\frac{(b e-a f) x^2}{b^2}+\frac{f x^3}{b}+\frac{a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^6}{6 b^3}+\frac{(b e-a f) x^9}{9 b^2}+\frac{f x^{12}}{12 b}-\frac{a \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^5}\\ \end{align*}

Mathematica [A]  time = 0.052875, size = 119, normalized size = 0.9 \[ \frac{b x^3 \left (6 a^2 b \left (2 e+f x^3\right )-12 a^3 f-2 a b^2 \left (6 d+3 e x^3+2 f x^6\right )+b^3 \left (12 c+6 d x^3+4 e x^6+3 f x^9\right )\right )+12 a \log \left (a+b x^3\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{36 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

(b*x^3*(-12*a^3*f + 6*a^2*b*(2*e + f*x^3) - 2*a*b^2*(6*d + 3*e*x^3 + 2*f*x^6) + b^3*(12*c + 6*d*x^3 + 4*e*x^6
+ 3*f*x^9)) + 12*a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a + b*x^3])/(36*b^5)

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Maple [A]  time = 0.003, size = 170, normalized size = 1.3 \begin{align*}{\frac{f{x}^{12}}{12\,b}}-{\frac{{x}^{9}af}{9\,{b}^{2}}}+{\frac{{x}^{9}e}{9\,b}}+{\frac{{a}^{2}f{x}^{6}}{6\,{b}^{3}}}-{\frac{ae{x}^{6}}{6\,{b}^{2}}}+{\frac{d{x}^{6}}{6\,b}}-{\frac{{a}^{3}f{x}^{3}}{3\,{b}^{4}}}+{\frac{{a}^{2}e{x}^{3}}{3\,{b}^{3}}}-{\frac{ad{x}^{3}}{3\,{b}^{2}}}+{\frac{c{x}^{3}}{3\,b}}+{\frac{{a}^{4}\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{5}}}-{\frac{{a}^{3}\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{3}}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)

[Out]

1/12*f*x^12/b-1/9/b^2*x^9*a*f+1/9/b*x^9*e+1/6/b^3*x^6*a^2*f-1/6/b^2*x^6*a*e+1/6/b*x^6*d-1/3/b^4*a^3*f*x^3+1/3/
b^3*a^2*e*x^3-1/3/b^2*a*d*x^3+1/3/b*c*x^3+1/3*a^4/b^5*ln(b*x^3+a)*f-1/3*a^3/b^4*ln(b*x^3+a)*e+1/3*a^2/b^3*ln(b
*x^3+a)*d-1/3*a/b^2*ln(b*x^3+a)*c

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Maxima [A]  time = 0.953117, size = 174, normalized size = 1.32 \begin{align*} \frac{3 \, b^{3} f x^{12} + 4 \,{\left (b^{3} e - a b^{2} f\right )} x^{9} + 6 \,{\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{6} + 12 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3}}{36 \, b^{4}} - \frac{{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="maxima")

[Out]

1/36*(3*b^3*f*x^12 + 4*(b^3*e - a*b^2*f)*x^9 + 6*(b^3*d - a*b^2*e + a^2*b*f)*x^6 + 12*(b^3*c - a*b^2*d + a^2*b
*e - a^3*f)*x^3)/b^4 - 1/3*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(b*x^3 + a)/b^5

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Fricas [A]  time = 1.27328, size = 267, normalized size = 2.02 \begin{align*} \frac{3 \, b^{4} f x^{12} + 4 \,{\left (b^{4} e - a b^{3} f\right )} x^{9} + 6 \,{\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{6} + 12 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{3} - 12 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (b x^{3} + a\right )}{36 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="fricas")

[Out]

1/36*(3*b^4*f*x^12 + 4*(b^4*e - a*b^3*f)*x^9 + 6*(b^4*d - a*b^3*e + a^2*b^2*f)*x^6 + 12*(b^4*c - a*b^3*d + a^2
*b^2*e - a^3*b*f)*x^3 - 12*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(b*x^3 + a))/b^5

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Sympy [A]  time = 1.08249, size = 117, normalized size = 0.89 \begin{align*} \frac{a \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^{5}} + \frac{f x^{12}}{12 b} - \frac{x^{9} \left (a f - b e\right )}{9 b^{2}} + \frac{x^{6} \left (a^{2} f - a b e + b^{2} d\right )}{6 b^{3}} - \frac{x^{3} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

a*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a + b*x**3)/(3*b**5) + f*x**12/(12*b) - x**9*(a*f - b*e)/(9*b**2
) + x**6*(a**2*f - a*b*e + b**2*d)/(6*b**3) - x**3*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(3*b**4)

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Giac [A]  time = 1.06881, size = 200, normalized size = 1.52 \begin{align*} \frac{3 \, b^{3} f x^{12} - 4 \, a b^{2} f x^{9} + 4 \, b^{3} x^{9} e + 6 \, b^{3} d x^{6} + 6 \, a^{2} b f x^{6} - 6 \, a b^{2} x^{6} e + 12 \, b^{3} c x^{3} - 12 \, a b^{2} d x^{3} - 12 \, a^{3} f x^{3} + 12 \, a^{2} b x^{3} e}{36 \, b^{4}} - \frac{{\left (a b^{3} c - a^{2} b^{2} d - a^{4} f + a^{3} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="giac")

[Out]

1/36*(3*b^3*f*x^12 - 4*a*b^2*f*x^9 + 4*b^3*x^9*e + 6*b^3*d*x^6 + 6*a^2*b*f*x^6 - 6*a*b^2*x^6*e + 12*b^3*c*x^3
- 12*a*b^2*d*x^3 - 12*a^3*f*x^3 + 12*a^2*b*x^3*e)/b^4 - 1/3*(a*b^3*c - a^2*b^2*d - a^4*f + a^3*b*e)*log(abs(b*
x^3 + a))/b^5