3.224 \(\int \frac{x^8 (c+d x^3+e x^6+f x^9)}{a+b x^3} \, dx\)

Optimal. Leaf size=170 \[ \frac{x^6 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^4}-\frac{a x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5}+\frac{a^2 \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^6}+\frac{x^9 \left (a^2 f-a b e+b^2 d\right )}{9 b^3}+\frac{x^{12} (b e-a f)}{12 b^2}+\frac{f x^{15}}{15 b} \]

[Out]

-(a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/(3*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^6)/(6*b^4) + ((b
^2*d - a*b*e + a^2*f)*x^9)/(9*b^3) + ((b*e - a*f)*x^12)/(12*b^2) + (f*x^15)/(15*b) + (a^2*(b^3*c - a*b^2*d + a
^2*b*e - a^3*f)*Log[a + b*x^3])/(3*b^6)

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Rubi [A]  time = 0.242332, antiderivative size = 170, 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} \[ \frac{x^6 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^4}-\frac{a x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5}+\frac{a^2 \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^6}+\frac{x^9 \left (a^2 f-a b e+b^2 d\right )}{9 b^3}+\frac{x^{12} (b e-a f)}{12 b^2}+\frac{f x^{15}}{15 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

-(a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/(3*b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^6)/(6*b^4) + ((b
^2*d - a*b*e + a^2*f)*x^9)/(9*b^3) + ((b*e - a*f)*x^12)/(12*b^2) + (f*x^15)/(15*b) + (a^2*(b^3*c - a*b^2*d + a
^2*b*e - a^3*f)*Log[a + b*x^3])/(3*b^6)

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^8 \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^2 \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{a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^2}{b^3}+\frac{(b e-a f) x^3}{b^2}+\frac{f x^4}{b}-\frac{a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^5}+\frac{\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{6 b^4}+\frac{\left (b^2 d-a b e+a^2 f\right ) x^9}{9 b^3}+\frac{(b e-a f) x^{12}}{12 b^2}+\frac{f x^{15}}{15 b}+\frac{a^2 \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^6}\\ \end{align*}

Mathematica [A]  time = 0.0659395, size = 154, normalized size = 0.91 \[ \frac{b x^3 \left (10 a^2 b^2 \left (6 d+3 e x^3+2 f x^6\right )-30 a^3 b \left (2 e+f x^3\right )+60 a^4 f-5 a b^3 \left (12 c+6 d x^3+4 e x^6+3 f x^9\right )+b^4 x^3 \left (30 c+20 d x^3+15 e x^6+12 f x^9\right )\right )-60 a^2 \log \left (a+b x^3\right ) \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{180 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^8*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

(b*x^3*(60*a^4*f - 30*a^3*b*(2*e + f*x^3) + 10*a^2*b^2*(6*d + 3*e*x^3 + 2*f*x^6) - 5*a*b^3*(12*c + 6*d*x^3 + 4
*e*x^6 + 3*f*x^9) + b^4*x^3*(30*c + 20*d*x^3 + 15*e*x^6 + 12*f*x^9)) - 60*a^2*(-(b^3*c) + a*b^2*d - a^2*b*e +
a^3*f)*Log[a + b*x^3])/(180*b^6)

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Maple [A]  time = 0.004, size = 218, normalized size = 1.3 \begin{align*}{\frac{f{x}^{15}}{15\,b}}-{\frac{{x}^{12}af}{12\,{b}^{2}}}+{\frac{{x}^{12}e}{12\,b}}+{\frac{{x}^{9}{a}^{2}f}{9\,{b}^{3}}}-{\frac{{x}^{9}ae}{9\,{b}^{2}}}+{\frac{{x}^{9}d}{9\,b}}-{\frac{{a}^{3}f{x}^{6}}{6\,{b}^{4}}}+{\frac{{a}^{2}e{x}^{6}}{6\,{b}^{3}}}-{\frac{ad{x}^{6}}{6\,{b}^{2}}}+{\frac{{x}^{6}c}{6\,b}}+{\frac{{a}^{4}f{x}^{3}}{3\,{b}^{5}}}-{\frac{{a}^{3}e{x}^{3}}{3\,{b}^{4}}}+{\frac{{a}^{2}d{x}^{3}}{3\,{b}^{3}}}-{\frac{ac{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{5}\ln \left ( b{x}^{3}+a \right ) f}{3\,{b}^{6}}}+{\frac{{a}^{4}\ln \left ( b{x}^{3}+a \right ) e}{3\,{b}^{5}}}-{\frac{{a}^{3}\ln \left ( b{x}^{3}+a \right ) d}{3\,{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) c}{3\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)

[Out]

1/15*f*x^15/b-1/12/b^2*x^12*a*f+1/12/b*x^12*e+1/9/b^3*x^9*a^2*f-1/9/b^2*x^9*a*e+1/9/b*x^9*d-1/6/b^4*x^6*a^3*f+
1/6/b^3*x^6*a^2*e-1/6/b^2*x^6*a*d+1/6/b*x^6*c+1/3/b^5*a^4*f*x^3-1/3/b^4*a^3*e*x^3+1/3/b^3*a^2*d*x^3-1/3/b^2*a*
c*x^3-1/3*a^5/b^6*ln(b*x^3+a)*f+1/3*a^4/b^5*ln(b*x^3+a)*e-1/3*a^3/b^4*ln(b*x^3+a)*d+1/3*a^2/b^3*ln(b*x^3+a)*c

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Maxima [A]  time = 0.95531, size = 228, normalized size = 1.34 \begin{align*} \frac{12 \, b^{4} f x^{15} + 15 \,{\left (b^{4} e - a b^{3} f\right )} x^{12} + 20 \,{\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{9} + 30 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{6} - 60 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{3}}{180 \, b^{5}} + \frac{{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="maxima")

[Out]

1/180*(12*b^4*f*x^15 + 15*(b^4*e - a*b^3*f)*x^12 + 20*(b^4*d - a*b^3*e + a^2*b^2*f)*x^9 + 30*(b^4*c - a*b^3*d
+ a^2*b^2*e - a^3*b*f)*x^6 - 60*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^3)/b^5 + 1/3*(a^2*b^3*c - a^3*b^2*d
+ a^4*b*e - a^5*f)*log(b*x^3 + a)/b^6

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Fricas [A]  time = 1.234, size = 351, normalized size = 2.06 \begin{align*} \frac{12 \, b^{5} f x^{15} + 15 \,{\left (b^{5} e - a b^{4} f\right )} x^{12} + 20 \,{\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{9} + 30 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{6} - 60 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} + 60 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \log \left (b x^{3} + a\right )}{180 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="fricas")

[Out]

1/180*(12*b^5*f*x^15 + 15*(b^5*e - a*b^4*f)*x^12 + 20*(b^5*d - a*b^4*e + a^2*b^3*f)*x^9 + 30*(b^5*c - a*b^4*d
+ a^2*b^3*e - a^3*b^2*f)*x^6 - 60*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^3 + 60*(a^2*b^3*c - a^3*b^2*d
+ a^4*b*e - a^5*f)*log(b*x^3 + a))/b^6

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Sympy [A]  time = 1.13667, size = 155, normalized size = 0.91 \begin{align*} - \frac{a^{2} \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^{6}} + \frac{f x^{15}}{15 b} - \frac{x^{12} \left (a f - b e\right )}{12 b^{2}} + \frac{x^{9} \left (a^{2} f - a b e + b^{2} d\right )}{9 b^{3}} - \frac{x^{6} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{6 b^{4}} + \frac{x^{3} \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{3 b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

-a**2*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a + b*x**3)/(3*b**6) + f*x**15/(15*b) - x**12*(a*f - b*e)/(1
2*b**2) + x**9*(a**2*f - a*b*e + b**2*d)/(9*b**3) - x**6*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(6*b**4) + x*
*3*(a**4*f - a**3*b*e + a**2*b**2*d - a*b**3*c)/(3*b**5)

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Giac [A]  time = 1.06393, size = 266, normalized size = 1.56 \begin{align*} \frac{12 \, b^{4} f x^{15} - 15 \, a b^{3} f x^{12} + 15 \, b^{4} x^{12} e + 20 \, b^{4} d x^{9} + 20 \, a^{2} b^{2} f x^{9} - 20 \, a b^{3} x^{9} e + 30 \, b^{4} c x^{6} - 30 \, a b^{3} d x^{6} - 30 \, a^{3} b f x^{6} + 30 \, a^{2} b^{2} x^{6} e - 60 \, a b^{3} c x^{3} + 60 \, a^{2} b^{2} d x^{3} + 60 \, a^{4} f x^{3} - 60 \, a^{3} b x^{3} e}{180 \, b^{5}} + \frac{{\left (a^{2} b^{3} c - a^{3} b^{2} d - a^{5} f + a^{4} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="giac")

[Out]

1/180*(12*b^4*f*x^15 - 15*a*b^3*f*x^12 + 15*b^4*x^12*e + 20*b^4*d*x^9 + 20*a^2*b^2*f*x^9 - 20*a*b^3*x^9*e + 30
*b^4*c*x^6 - 30*a*b^3*d*x^6 - 30*a^3*b*f*x^6 + 30*a^2*b^2*x^6*e - 60*a*b^3*c*x^3 + 60*a^2*b^2*d*x^3 + 60*a^4*f
*x^3 - 60*a^3*b*x^3*e)/b^5 + 1/3*(a^2*b^3*c - a^3*b^2*d - a^5*f + a^4*b*e)*log(abs(b*x^3 + a))/b^6