3.232 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{16} (a+b x^3)} \, dx\)

Optimal. Leaf size=205 \[ -\frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 a^4 x^6}+\frac{b^2 \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^6}-\frac{b^2 \log (x) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^6}-\frac{a^2 e-a b d+b^2 c}{9 a^3 x^9}+\frac{b c-a d}{12 a^2 x^{12}}-\frac{c}{15 a x^{15}} \]

[Out]

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

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Rubi [A]  time = 0.208872, antiderivative size = 205, 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} \[ -\frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 x^3}+\frac{a^2 b e+a^3 (-f)-a b^2 d+b^3 c}{6 a^4 x^6}+\frac{b^2 \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^6}-\frac{b^2 \log (x) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{a^6}-\frac{a^2 e-a b d+b^2 c}{9 a^3 x^9}+\frac{b c-a d}{12 a^2 x^{12}}-\frac{c}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^16*(a + b*x^3)),x]

[Out]

-c/(15*a*x^15) + (b*c - a*d)/(12*a^2*x^12) - (b^2*c - a*b*d + a^2*e)/(9*a^3*x^9) + (b^3*c - a*b^2*d + a^2*b*e
- a^3*f)/(6*a^4*x^6) - (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(3*a^5*x^3) - (b^2*(b^3*c - a*b^2*d + a^2*b*e -
a^3*f)*Log[x])/a^6 + (b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a + b*x^3])/(3*a^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{c+d x^3+e x^6+f x^9}{x^{16} \left (a+b x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{x^6 (a+b x)} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{c}{a x^6}+\frac{-b c+a d}{a^2 x^5}+\frac{b^2 c-a b d+a^2 e}{a^3 x^4}+\frac{-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^4 x^3}-\frac{b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 x^2}+\frac{b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^6 x}-\frac{b^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^6 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{c}{15 a x^{15}}+\frac{b c-a d}{12 a^2 x^{12}}-\frac{b^2 c-a b d+a^2 e}{9 a^3 x^9}+\frac{b^3 c-a b^2 d+a^2 b e-a^3 f}{6 a^4 x^6}-\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 x^3}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \log (x)}{a^6}+\frac{b^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 a^6}\\ \end{align*}

Mathematica [A]  time = 0.143637, size = 194, normalized size = 0.95 \[ -\frac{\frac{a \left (10 a^2 b^2 x^6 \left (2 c+3 d x^3+6 e x^6\right )-5 a^3 b x^3 \left (3 c+4 d x^3+6 e x^6+12 f x^9\right )+a^4 \left (12 c+15 d x^3+20 e x^6+30 f x^9\right )-30 a b^3 x^9 \left (c+2 d x^3\right )+60 b^4 c x^{12}\right )}{x^{15}}-60 b^2 \log \left (a+b x^3\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+180 b^2 \log (x) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{180 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^16*(a + b*x^3)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 260, normalized size = 1.3 \begin{align*} -{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) f}{3\,{a}^{3}}}+{\frac{{b}^{3}\ln \left ( b{x}^{3}+a \right ) e}{3\,{a}^{4}}}-{\frac{{b}^{4}\ln \left ( b{x}^{3}+a \right ) d}{3\,{a}^{5}}}+{\frac{{b}^{5}\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{6}}}-{\frac{c}{15\,a{x}^{15}}}-{\frac{d}{12\,a{x}^{12}}}+{\frac{bc}{12\,{a}^{2}{x}^{12}}}-{\frac{e}{9\,a{x}^{9}}}+{\frac{bd}{9\,{a}^{2}{x}^{9}}}-{\frac{{b}^{2}c}{9\,{a}^{3}{x}^{9}}}-{\frac{f}{6\,a{x}^{6}}}+{\frac{be}{6\,{a}^{2}{x}^{6}}}-{\frac{{b}^{2}d}{6\,{a}^{3}{x}^{6}}}+{\frac{{b}^{3}c}{6\,{a}^{4}{x}^{6}}}+{\frac{{b}^{2}\ln \left ( x \right ) f}{{a}^{3}}}-{\frac{{b}^{3}\ln \left ( x \right ) e}{{a}^{4}}}+{\frac{{b}^{4}\ln \left ( x \right ) d}{{a}^{5}}}-{\frac{{b}^{5}\ln \left ( x \right ) c}{{a}^{6}}}+{\frac{bf}{3\,{x}^{3}{a}^{2}}}-{\frac{{b}^{2}e}{3\,{a}^{3}{x}^{3}}}+{\frac{{b}^{3}d}{3\,{a}^{4}{x}^{3}}}-{\frac{{b}^{4}c}{3\,{a}^{5}{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^16/(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 0.958911, size = 281, normalized size = 1.37 \begin{align*} \frac{{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{6}} - \frac{{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} \log \left (x^{3}\right )}{3 \, a^{6}} - \frac{60 \,{\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{12} - 30 \,{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{9} + 20 \,{\left (a^{2} b^{2} c - a^{3} b d + a^{4} e\right )} x^{6} + 12 \, a^{4} c - 15 \,{\left (a^{3} b c - a^{4} d\right )} x^{3}}{180 \, a^{5} x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^16/(b*x^3+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.6669, size = 441, normalized size = 2.15 \begin{align*} \frac{60 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} \log \left (b x^{3} + a\right ) - 180 \,{\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{15} \log \left (x\right ) - 60 \,{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{12} + 30 \,{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x^{9} - 20 \,{\left (a^{3} b^{2} c - a^{4} b d + a^{5} e\right )} x^{6} - 12 \, a^{5} c + 15 \,{\left (a^{4} b c - a^{5} d\right )} x^{3}}{180 \, a^{6} x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^16/(b*x^3+a),x, algorithm="fricas")

[Out]

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

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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((f*x**9+e*x**6+d*x**3+c)/x**16/(b*x**3+a),x)

[Out]

Timed out

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Giac [A]  time = 1.06632, size = 387, normalized size = 1.89 \begin{align*} -\frac{{\left (b^{5} c - a b^{4} d - a^{3} b^{2} f + a^{2} b^{3} e\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{{\left (b^{6} c - a b^{5} d - a^{3} b^{3} f + a^{2} b^{4} e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{6} b} + \frac{137 \, b^{5} c x^{15} - 137 \, a b^{4} d x^{15} - 137 \, a^{3} b^{2} f x^{15} + 137 \, a^{2} b^{3} x^{15} e - 60 \, a b^{4} c x^{12} + 60 \, a^{2} b^{3} d x^{12} + 60 \, a^{4} b f x^{12} - 60 \, a^{3} b^{2} x^{12} e + 30 \, a^{2} b^{3} c x^{9} - 30 \, a^{3} b^{2} d x^{9} - 30 \, a^{5} f x^{9} + 30 \, a^{4} b x^{9} e - 20 \, a^{3} b^{2} c x^{6} + 20 \, a^{4} b d x^{6} - 20 \, a^{5} x^{6} e + 15 \, a^{4} b c x^{3} - 15 \, a^{5} d x^{3} - 12 \, a^{5} c}{180 \, a^{6} x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^16/(b*x^3+a),x, algorithm="giac")

[Out]

-(b^5*c - a*b^4*d - a^3*b^2*f + a^2*b^3*e)*log(abs(x))/a^6 + 1/3*(b^6*c - a*b^5*d - a^3*b^3*f + a^2*b^4*e)*log
(abs(b*x^3 + a))/(a^6*b) + 1/180*(137*b^5*c*x^15 - 137*a*b^4*d*x^15 - 137*a^3*b^2*f*x^15 + 137*a^2*b^3*x^15*e
- 60*a*b^4*c*x^12 + 60*a^2*b^3*d*x^12 + 60*a^4*b*f*x^12 - 60*a^3*b^2*x^12*e + 30*a^2*b^3*c*x^9 - 30*a^3*b^2*d*
x^9 - 30*a^5*f*x^9 + 30*a^4*b*x^9*e - 20*a^3*b^2*c*x^6 + 20*a^4*b*d*x^6 - 20*a^5*x^6*e + 15*a^4*b*c*x^3 - 15*a
^5*d*x^3 - 12*a^5*c)/(a^6*x^15)