3.259 \(\int \frac{c+d x^3+e x^6+f x^9}{x^{13} (a+b x^3)^2} \, dx\)

Optimal. Leaf size=214 \[ \frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{3 a^5 x^3}-\frac{b \log \left (a+b x^3\right ) \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{3 a^6}+\frac{b \log (x) \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{a^6}-\frac{a^2 e-2 a b d+3 b^2 c}{6 a^4 x^6}+\frac{2 b c-a d}{9 a^3 x^9}-\frac{c}{12 a^2 x^{12}} \]

[Out]

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

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Rubi [A]  time = 0.233997, antiderivative size = 214, 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} \[ \frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{3 a^5 x^3}-\frac{b \log \left (a+b x^3\right ) \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{3 a^6}+\frac{b \log (x) \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{a^6}-\frac{a^2 e-2 a b d+3 b^2 c}{6 a^4 x^6}+\frac{2 b c-a d}{9 a^3 x^9}-\frac{c}{12 a^2 x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)^2),x]

[Out]

-c/(12*a^2*x^12) + (2*b*c - a*d)/(9*a^3*x^9) - (3*b^2*c - 2*a*b*d + a^2*e)/(6*a^4*x^6) + (4*b^3*c - 3*a*b^2*d
+ 2*a^2*b*e - a^3*f)/(3*a^5*x^3) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(3*a^5*(a + b*x^3)) + (b*(5*b^3*c -
4*a*b^2*d + 3*a^2*b*e - 2*a^3*f)*Log[x])/a^6 - (b*(5*b^3*c - 4*a*b^2*d + 3*a^2*b*e - 2*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^{13} \left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{x^5 (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{c}{a^2 x^5}+\frac{-2 b c+a d}{a^3 x^4}+\frac{3 b^2 c-2 a b d+a^2 e}{a^4 x^3}+\frac{-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f}{a^5 x^2}-\frac{b \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^6 x}+\frac{b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 (a+b x)^2}+\frac{b^2 \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^6 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{c}{12 a^2 x^{12}}+\frac{2 b c-a d}{9 a^3 x^9}-\frac{3 b^2 c-2 a b d+a^2 e}{6 a^4 x^6}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 x^3}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 \left (a+b x^3\right )}+\frac{b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) \log (x)}{a^6}-\frac{b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) \log \left (a+b x^3\right )}{3 a^6}\\ \end{align*}

Mathematica [A]  time = 0.202961, size = 198, normalized size = 0.93 \[ -\frac{\frac{12 a b \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{a+b x^3}+\frac{12 a \left (-2 a^2 b e+a^3 f+3 a b^2 d-4 b^3 c\right )}{x^3}+12 b \log \left (a+b x^3\right ) \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )-36 b \log (x) \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )+\frac{6 a^2 \left (a^2 e-2 a b d+3 b^2 c\right )}{x^6}+\frac{4 a^3 (a d-2 b c)}{x^9}+\frac{3 a^4 c}{x^{12}}}{36 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^13*(a + b*x^3)^2),x]

[Out]

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

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Maple [A]  time = 0.02, size = 282, normalized size = 1.3 \begin{align*}{\frac{2\,b\ln \left ( b{x}^{3}+a \right ) f}{3\,{a}^{3}}}-{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) e}{{a}^{4}}}+{\frac{4\,{b}^{3}\ln \left ( b{x}^{3}+a \right ) d}{3\,{a}^{5}}}-{\frac{5\,{b}^{4}\ln \left ( b{x}^{3}+a \right ) c}{3\,{a}^{6}}}-{\frac{fb}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{e{b}^{2}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{{b}^{3}d}{3\,{a}^{4} \left ( b{x}^{3}+a \right ) }}+{\frac{c{b}^{4}}{3\,{a}^{5} \left ( b{x}^{3}+a \right ) }}-{\frac{c}{12\,{a}^{2}{x}^{12}}}-{\frac{d}{9\,{a}^{2}{x}^{9}}}+{\frac{2\,bc}{9\,{a}^{3}{x}^{9}}}-{\frac{e}{6\,{a}^{2}{x}^{6}}}+{\frac{bd}{3\,{a}^{3}{x}^{6}}}-{\frac{{b}^{2}c}{2\,{a}^{4}{x}^{6}}}-{\frac{f}{3\,{x}^{3}{a}^{2}}}+{\frac{2\,be}{3\,{a}^{3}{x}^{3}}}-{\frac{{b}^{2}d}{{a}^{4}{x}^{3}}}+{\frac{4\,{b}^{3}c}{3\,{a}^{5}{x}^{3}}}-2\,{\frac{b\ln \left ( x \right ) f}{{a}^{3}}}+3\,{\frac{{b}^{2}\ln \left ( x \right ) e}{{a}^{4}}}-4\,{\frac{{b}^{3}\ln \left ( x \right ) d}{{a}^{5}}}+5\,{\frac{{b}^{4}\ln \left ( x \right ) c}{{a}^{6}}} \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^13/(b*x^3+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.01741, size = 305, normalized size = 1.43 \begin{align*} \frac{12 \,{\left (5 \, b^{4} c - 4 \, a b^{3} d + 3 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} x^{12} + 6 \,{\left (5 \, a b^{3} c - 4 \, a^{2} b^{2} d + 3 \, a^{3} b e - 2 \, a^{4} f\right )} x^{9} - 2 \,{\left (5 \, a^{2} b^{2} c - 4 \, a^{3} b d + 3 \, a^{4} e\right )} x^{6} - 3 \, a^{4} c +{\left (5 \, a^{3} b c - 4 \, a^{4} d\right )} x^{3}}{36 \,{\left (a^{5} b x^{15} + a^{6} x^{12}\right )}} - \frac{{\left (5 \, b^{4} c - 4 \, a b^{3} d + 3 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{6}} + \frac{{\left (5 \, b^{4} c - 4 \, a b^{3} d + 3 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} \log \left (x^{3}\right )}{3 \, a^{6}} \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^13/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/36*(12*(5*b^4*c - 4*a*b^3*d + 3*a^2*b^2*e - 2*a^3*b*f)*x^12 + 6*(5*a*b^3*c - 4*a^2*b^2*d + 3*a^3*b*e - 2*a^4
*f)*x^9 - 2*(5*a^2*b^2*c - 4*a^3*b*d + 3*a^4*e)*x^6 - 3*a^4*c + (5*a^3*b*c - 4*a^4*d)*x^3)/(a^5*b*x^15 + a^6*x
^12) - 1/3*(5*b^4*c - 4*a*b^3*d + 3*a^2*b^2*e - 2*a^3*b*f)*log(b*x^3 + a)/a^6 + 1/3*(5*b^4*c - 4*a*b^3*d + 3*a
^2*b^2*e - 2*a^3*b*f)*log(x^3)/a^6

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Fricas [A]  time = 1.60778, size = 670, normalized size = 3.13 \begin{align*} \frac{12 \,{\left (5 \, a b^{4} c - 4 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{12} + 6 \,{\left (5 \, a^{2} b^{3} c - 4 \, a^{3} b^{2} d + 3 \, a^{4} b e - 2 \, a^{5} f\right )} x^{9} - 2 \,{\left (5 \, a^{3} b^{2} c - 4 \, a^{4} b d + 3 \, a^{5} e\right )} x^{6} - 3 \, a^{5} c +{\left (5 \, a^{4} b c - 4 \, a^{5} d\right )} x^{3} - 12 \,{\left ({\left (5 \, b^{5} c - 4 \, a b^{4} d + 3 \, a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} x^{15} +{\left (5 \, a b^{4} c - 4 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{12}\right )} \log \left (b x^{3} + a\right ) + 36 \,{\left ({\left (5 \, b^{5} c - 4 \, a b^{4} d + 3 \, a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} x^{15} +{\left (5 \, a b^{4} c - 4 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{12}\right )} \log \left (x\right )}{36 \,{\left (a^{6} b x^{15} + a^{7} x^{12}\right )}} \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^13/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/36*(12*(5*a*b^4*c - 4*a^2*b^3*d + 3*a^3*b^2*e - 2*a^4*b*f)*x^12 + 6*(5*a^2*b^3*c - 4*a^3*b^2*d + 3*a^4*b*e -
2*a^5*f)*x^9 - 2*(5*a^3*b^2*c - 4*a^4*b*d + 3*a^5*e)*x^6 - 3*a^5*c + (5*a^4*b*c - 4*a^5*d)*x^3 - 12*((5*b^5*c
- 4*a*b^4*d + 3*a^2*b^3*e - 2*a^3*b^2*f)*x^15 + (5*a*b^4*c - 4*a^2*b^3*d + 3*a^3*b^2*e - 2*a^4*b*f)*x^12)*log
(b*x^3 + a) + 36*((5*b^5*c - 4*a*b^4*d + 3*a^2*b^3*e - 2*a^3*b^2*f)*x^15 + (5*a*b^4*c - 4*a^2*b^3*d + 3*a^3*b^
2*e - 2*a^4*b*f)*x^12)*log(x))/(a^6*b*x^15 + a^7*x^12)

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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**13/(b*x**3+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.07398, size = 447, normalized size = 2.09 \begin{align*} \frac{{\left (5 \, b^{4} c - 4 \, a b^{3} d - 2 \, a^{3} b f + 3 \, a^{2} b^{2} e\right )} \log \left ({\left | x \right |}\right )}{a^{6}} - \frac{{\left (5 \, b^{5} c - 4 \, a b^{4} d - 2 \, a^{3} b^{2} f + 3 \, a^{2} b^{3} e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{6} b} + \frac{5 \, b^{5} c x^{3} - 4 \, a b^{4} d x^{3} - 2 \, a^{3} b^{2} f x^{3} + 3 \, a^{2} b^{3} x^{3} e + 6 \, a b^{4} c - 5 \, a^{2} b^{3} d - 3 \, a^{4} b f + 4 \, a^{3} b^{2} e}{3 \,{\left (b x^{3} + a\right )} a^{6}} - \frac{125 \, b^{4} c x^{12} - 100 \, a b^{3} d x^{12} - 50 \, a^{3} b f x^{12} + 75 \, a^{2} b^{2} x^{12} e - 48 \, a b^{3} c x^{9} + 36 \, a^{2} b^{2} d x^{9} + 12 \, a^{4} f x^{9} - 24 \, a^{3} b x^{9} e + 18 \, a^{2} b^{2} c x^{6} - 12 \, a^{3} b d x^{6} + 6 \, a^{4} x^{6} e - 8 \, a^{3} b c x^{3} + 4 \, a^{4} d x^{3} + 3 \, a^{4} c}{36 \, a^{6} x^{12}} \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^13/(b*x^3+a)^2,x, algorithm="giac")

[Out]

(5*b^4*c - 4*a*b^3*d - 2*a^3*b*f + 3*a^2*b^2*e)*log(abs(x))/a^6 - 1/3*(5*b^5*c - 4*a*b^4*d - 2*a^3*b^2*f + 3*a
^2*b^3*e)*log(abs(b*x^3 + a))/(a^6*b) + 1/3*(5*b^5*c*x^3 - 4*a*b^4*d*x^3 - 2*a^3*b^2*f*x^3 + 3*a^2*b^3*x^3*e +
6*a*b^4*c - 5*a^2*b^3*d - 3*a^4*b*f + 4*a^3*b^2*e)/((b*x^3 + a)*a^6) - 1/36*(125*b^4*c*x^12 - 100*a*b^3*d*x^1
2 - 50*a^3*b*f*x^12 + 75*a^2*b^2*x^12*e - 48*a*b^3*c*x^9 + 36*a^2*b^2*d*x^9 + 12*a^4*f*x^9 - 24*a^3*b*x^9*e +
18*a^2*b^2*c*x^6 - 12*a^3*b*d*x^6 + 6*a^4*x^6*e - 8*a^3*b*c*x^3 + 4*a^4*d*x^3 + 3*a^4*c)/(a^6*x^12)