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

Optimal. Leaf size=258 \[ \frac{b \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{3 a^6 x^3}-\frac{b \log \left (a+b x^3\right ) \left (6 a^2 b e-3 a^3 f-10 a b^2 d+15 b^3 c\right )}{3 a^7}+\frac{b \log (x) \left (6 a^2 b e-3 a^3 f-10 a b^2 d+15 b^3 c\right )}{a^7}-\frac{a^2 e-3 a b d+6 b^2 c}{6 a^5 x^6}+\frac{3 b c-a d}{9 a^4 x^9}-\frac{c}{12 a^3 x^{12}} \]

[Out]

-c/(12*a^3*x^12) + (3*b*c - a*d)/(9*a^4*x^9) - (6*b^2*c - 3*a*b*d + a^2*e)/(6*a^5*x^6) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(3*a^6*x^3) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(6*a^5*(a + b*x^3)^2) + (b*(5*b^3*
c - 4*a*b^2*d + 3*a^2*b*e - 2*a^3*f))/(3*a^6*(a + b*x^3)) + (b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f)*L
og[x])/a^7 - (b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f)*Log[a + b*x^3])/(3*a^7)

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Rubi [A]  time = 0.304493, antiderivative size = 258, 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} \[ \frac{b \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac{b \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{3 a^6 x^3}-\frac{b \log \left (a+b x^3\right ) \left (6 a^2 b e-3 a^3 f-10 a b^2 d+15 b^3 c\right )}{3 a^7}+\frac{b \log (x) \left (6 a^2 b e-3 a^3 f-10 a b^2 d+15 b^3 c\right )}{a^7}-\frac{a^2 e-3 a b d+6 b^2 c}{6 a^5 x^6}+\frac{3 b c-a d}{9 a^4 x^9}-\frac{c}{12 a^3 x^{12}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(12*a^3*x^12) + (3*b*c - a*d)/(9*a^4*x^9) - (6*b^2*c - 3*a*b*d + a^2*e)/(6*a^5*x^6) + (10*b^3*c - 6*a*b^2*d
+ 3*a^2*b*e - a^3*f)/(3*a^6*x^3) + (b*(b^3*c - a*b^2*d + a^2*b*e - a^3*f))/(6*a^5*(a + b*x^3)^2) + (b*(5*b^3*
c - 4*a*b^2*d + 3*a^2*b*e - 2*a^3*f))/(3*a^6*(a + b*x^3)) + (b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f)*L
og[x])/a^7 - (b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f)*Log[a + b*x^3])/(3*a^7)

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 )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{c+d x+e x^2+f x^3}{x^5 (a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{c}{a^3 x^5}+\frac{-3 b c+a d}{a^4 x^4}+\frac{6 b^2 c-3 a b d+a^2 e}{a^5 x^3}+\frac{-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f}{a^6 x^2}-\frac{b \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right )}{a^7 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)^3}+\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)^2}+\frac{b^2 \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right )}{a^7 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{c}{12 a^3 x^{12}}+\frac{3 b c-a d}{9 a^4 x^9}-\frac{6 b^2 c-3 a b d+a^2 e}{6 a^5 x^6}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{3 a^6 x^3}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 a^5 \left (a+b x^3\right )^2}+\frac{b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{3 a^6 \left (a+b x^3\right )}+\frac{b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right ) \log (x)}{a^7}-\frac{b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right ) \log \left (a+b x^3\right )}{3 a^7}\\ \end{align*}

Mathematica [A]  time = 0.179807, size = 238, normalized size = 0.92 \[ \frac{-\frac{a \left (a^3 b^2 x^6 \left (15 c+40 d x^3-108 e x^6+36 f x^9\right )-12 a^2 b^3 x^9 \left (5 c-15 d x^3+6 e x^6\right )-2 a^4 b x^3 \left (3 c+5 d x^3+12 e x^6-27 f x^9\right )+a^5 \left (3 c+4 d x^3+6 e x^6+12 f x^9\right )+30 a b^4 x^{12} \left (4 d x^3-9 c\right )-180 b^5 c x^{15}\right )}{x^{12} \left (a+b x^3\right )^2}+12 b \log \left (a+b x^3\right ) \left (-6 a^2 b e+3 a^3 f+10 a b^2 d-15 b^3 c\right )+36 b \log (x) \left (6 a^2 b e-3 a^3 f-10 a b^2 d+15 b^3 c\right )}{36 a^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a*(-180*b^5*c*x^15 + 30*a*b^4*x^12*(-9*c + 4*d*x^3) - 12*a^2*b^3*x^9*(5*c - 15*d*x^3 + 6*e*x^6) - 2*a^4*b*
x^3*(3*c + 5*d*x^3 + 12*e*x^6 - 27*f*x^9) + a^5*(3*c + 4*d*x^3 + 6*e*x^6 + 12*f*x^9) + a^3*b^2*x^6*(15*c + 40*
d*x^3 - 108*e*x^6 + 36*f*x^9)))/(x^12*(a + b*x^3)^2)) + 36*b*(15*b^3*c - 10*a*b^2*d + 6*a^2*b*e - 3*a^3*f)*Log
[x] + 12*b*(-15*b^3*c + 10*a*b^2*d - 6*a^2*b*e + 3*a^3*f)*Log[a + b*x^3])/(36*a^7)

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Maple [A]  time = 0.019, size = 349, normalized size = 1.4 \begin{align*} -{\frac{d}{9\,{a}^{3}{x}^{9}}}-{\frac{e}{6\,{a}^{3}{x}^{6}}}-{\frac{f}{3\,{a}^{3}{x}^{3}}}-{\frac{c}{12\,{a}^{3}{x}^{12}}}-2\,{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) e}{{a}^{5}}}+{\frac{10\,{b}^{3}\ln \left ( b{x}^{3}+a \right ) d}{3\,{a}^{6}}}+{\frac{bc}{3\,{a}^{4}{x}^{9}}}+{\frac{bd}{2\,{a}^{4}{x}^{6}}}-{\frac{{b}^{2}c}{{a}^{5}{x}^{6}}}+{\frac{be}{{a}^{4}{x}^{3}}}-2\,{\frac{{b}^{2}d}{{a}^{5}{x}^{3}}}+{\frac{10\,{b}^{3}c}{3\,{a}^{6}{x}^{3}}}-3\,{\frac{b\ln \left ( x \right ) f}{{a}^{4}}}+6\,{\frac{{b}^{2}\ln \left ( x \right ) e}{{a}^{5}}}-10\,{\frac{{b}^{3}\ln \left ( x \right ) d}{{a}^{6}}}+15\,{\frac{{b}^{4}\ln \left ( x \right ) c}{{a}^{7}}}-{\frac{fb}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{e{b}^{2}}{6\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{{b}^{3}d}{6\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{c{b}^{4}}{6\,{a}^{5} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) f}{{a}^{4}}}+{\frac{5\,c{b}^{4}}{3\,{a}^{6} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,fb}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-5\,{\frac{{b}^{4}\ln \left ( b{x}^{3}+a \right ) c}{{a}^{7}}}+{\frac{e{b}^{2}}{{a}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{4\,{b}^{3}d}{3\,{a}^{5} \left ( b{x}^{3}+a \right ) }} \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)^3,x)

[Out]

-1/9/a^3/x^9*d-1/6/a^3/x^6*e-1/3/a^3/x^3*f-1/12*c/a^3/x^12-2*b^2/a^5*ln(b*x^3+a)*e+10/3*b^3/a^6*ln(b*x^3+a)*d+
1/3/a^4/x^9*b*c+1/2/a^4/x^6*b*d-1/a^5/x^6*b^2*c+1/a^4/x^3*b*e-2/a^5/x^3*b^2*d+10/3/a^6/x^3*b^3*c-3*b/a^4*ln(x)
*f+6*b^2/a^5*ln(x)*e-10*b^3/a^6*ln(x)*d+15*b^4/a^7*ln(x)*c-1/6*b/a^2/(b*x^3+a)^2*f+1/6*b^2/a^3/(b*x^3+a)^2*e-1
/6*b^3/a^4/(b*x^3+a)^2*d+1/6*b^4/a^5/(b*x^3+a)^2*c+b/a^4*ln(b*x^3+a)*f+5/3*b^4/a^6/(b*x^3+a)*c-2/3*b/a^3/(b*x^
3+a)*f-5*b^4/a^7*ln(b*x^3+a)*c+b^2/a^4/(b*x^3+a)*e-4/3*b^3/a^5/(b*x^3+a)*d

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Maxima [A]  time = 0.988422, size = 378, normalized size = 1.47 \begin{align*} \frac{12 \,{\left (15 \, b^{5} c - 10 \, a b^{4} d + 6 \, a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{15} + 18 \,{\left (15 \, a b^{4} c - 10 \, a^{2} b^{3} d + 6 \, a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{12} + 4 \,{\left (15 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 6 \, a^{4} b e - 3 \, a^{5} f\right )} x^{9} -{\left (15 \, a^{3} b^{2} c - 10 \, a^{4} b d + 6 \, a^{5} e\right )} x^{6} - 3 \, a^{5} c + 2 \,{\left (3 \, a^{4} b c - 2 \, a^{5} d\right )} x^{3}}{36 \,{\left (a^{6} b^{2} x^{18} + 2 \, a^{7} b x^{15} + a^{8} x^{12}\right )}} - \frac{{\left (15 \, b^{4} c - 10 \, a b^{3} d + 6 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{7}} + \frac{{\left (15 \, b^{4} c - 10 \, a b^{3} d + 6 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} \log \left (x^{3}\right )}{3 \, a^{7}} \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)^3,x, algorithm="maxima")

[Out]

1/36*(12*(15*b^5*c - 10*a*b^4*d + 6*a^2*b^3*e - 3*a^3*b^2*f)*x^15 + 18*(15*a*b^4*c - 10*a^2*b^3*d + 6*a^3*b^2*
e - 3*a^4*b*f)*x^12 + 4*(15*a^2*b^3*c - 10*a^3*b^2*d + 6*a^4*b*e - 3*a^5*f)*x^9 - (15*a^3*b^2*c - 10*a^4*b*d +
6*a^5*e)*x^6 - 3*a^5*c + 2*(3*a^4*b*c - 2*a^5*d)*x^3)/(a^6*b^2*x^18 + 2*a^7*b*x^15 + a^8*x^12) - 1/3*(15*b^4*
c - 10*a*b^3*d + 6*a^2*b^2*e - 3*a^3*b*f)*log(b*x^3 + a)/a^7 + 1/3*(15*b^4*c - 10*a*b^3*d + 6*a^2*b^2*e - 3*a^
3*b*f)*log(x^3)/a^7

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Fricas [A]  time = 1.63972, size = 984, normalized size = 3.81 \begin{align*} \frac{12 \,{\left (15 \, a b^{5} c - 10 \, a^{2} b^{4} d + 6 \, a^{3} b^{3} e - 3 \, a^{4} b^{2} f\right )} x^{15} + 18 \,{\left (15 \, a^{2} b^{4} c - 10 \, a^{3} b^{3} d + 6 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{12} + 4 \,{\left (15 \, a^{3} b^{3} c - 10 \, a^{4} b^{2} d + 6 \, a^{5} b e - 3 \, a^{6} f\right )} x^{9} - 3 \, a^{6} c -{\left (15 \, a^{4} b^{2} c - 10 \, a^{5} b d + 6 \, a^{6} e\right )} x^{6} + 2 \,{\left (3 \, a^{5} b c - 2 \, a^{6} d\right )} x^{3} - 12 \,{\left ({\left (15 \, b^{6} c - 10 \, a b^{5} d + 6 \, a^{2} b^{4} e - 3 \, a^{3} b^{3} f\right )} x^{18} + 2 \,{\left (15 \, a b^{5} c - 10 \, a^{2} b^{4} d + 6 \, a^{3} b^{3} e - 3 \, a^{4} b^{2} f\right )} x^{15} +{\left (15 \, a^{2} b^{4} c - 10 \, a^{3} b^{3} d + 6 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{12}\right )} \log \left (b x^{3} + a\right ) + 36 \,{\left ({\left (15 \, b^{6} c - 10 \, a b^{5} d + 6 \, a^{2} b^{4} e - 3 \, a^{3} b^{3} f\right )} x^{18} + 2 \,{\left (15 \, a b^{5} c - 10 \, a^{2} b^{4} d + 6 \, a^{3} b^{3} e - 3 \, a^{4} b^{2} f\right )} x^{15} +{\left (15 \, a^{2} b^{4} c - 10 \, a^{3} b^{3} d + 6 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{12}\right )} \log \left (x\right )}{36 \,{\left (a^{7} b^{2} x^{18} + 2 \, a^{8} b x^{15} + a^{9} 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)^3,x, algorithm="fricas")

[Out]

1/36*(12*(15*a*b^5*c - 10*a^2*b^4*d + 6*a^3*b^3*e - 3*a^4*b^2*f)*x^15 + 18*(15*a^2*b^4*c - 10*a^3*b^3*d + 6*a^
4*b^2*e - 3*a^5*b*f)*x^12 + 4*(15*a^3*b^3*c - 10*a^4*b^2*d + 6*a^5*b*e - 3*a^6*f)*x^9 - 3*a^6*c - (15*a^4*b^2*
c - 10*a^5*b*d + 6*a^6*e)*x^6 + 2*(3*a^5*b*c - 2*a^6*d)*x^3 - 12*((15*b^6*c - 10*a*b^5*d + 6*a^2*b^4*e - 3*a^3
*b^3*f)*x^18 + 2*(15*a*b^5*c - 10*a^2*b^4*d + 6*a^3*b^3*e - 3*a^4*b^2*f)*x^15 + (15*a^2*b^4*c - 10*a^3*b^3*d +
6*a^4*b^2*e - 3*a^5*b*f)*x^12)*log(b*x^3 + a) + 36*((15*b^6*c - 10*a*b^5*d + 6*a^2*b^4*e - 3*a^3*b^3*f)*x^18
+ 2*(15*a*b^5*c - 10*a^2*b^4*d + 6*a^3*b^3*e - 3*a^4*b^2*f)*x^15 + (15*a^2*b^4*c - 10*a^3*b^3*d + 6*a^4*b^2*e
- 3*a^5*b*f)*x^12)*log(x))/(a^7*b^2*x^18 + 2*a^8*b*x^15 + a^9*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)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.07742, size = 513, normalized size = 1.99 \begin{align*} \frac{{\left (15 \, b^{4} c - 10 \, a b^{3} d - 3 \, a^{3} b f + 6 \, a^{2} b^{2} e\right )} \log \left ({\left | x \right |}\right )}{a^{7}} - \frac{{\left (15 \, b^{5} c - 10 \, a b^{4} d - 3 \, a^{3} b^{2} f + 6 \, a^{2} b^{3} e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{7} b} + \frac{45 \, b^{6} c x^{6} - 30 \, a b^{5} d x^{6} - 9 \, a^{3} b^{3} f x^{6} + 18 \, a^{2} b^{4} x^{6} e + 100 \, a b^{5} c x^{3} - 68 \, a^{2} b^{4} d x^{3} - 22 \, a^{4} b^{2} f x^{3} + 42 \, a^{3} b^{3} x^{3} e + 56 \, a^{2} b^{4} c - 39 \, a^{3} b^{3} d - 14 \, a^{5} b f + 25 \, a^{4} b^{2} e}{6 \,{\left (b x^{3} + a\right )}^{2} a^{7}} - \frac{375 \, b^{4} c x^{12} - 250 \, a b^{3} d x^{12} - 75 \, a^{3} b f x^{12} + 150 \, a^{2} b^{2} x^{12} e - 120 \, a b^{3} c x^{9} + 72 \, a^{2} b^{2} d x^{9} + 12 \, a^{4} f x^{9} - 36 \, a^{3} b x^{9} e + 36 \, a^{2} b^{2} c x^{6} - 18 \, a^{3} b d x^{6} + 6 \, a^{4} x^{6} e - 12 \, a^{3} b c x^{3} + 4 \, a^{4} d x^{3} + 3 \, a^{4} c}{36 \, a^{7} 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)^3,x, algorithm="giac")

[Out]

(15*b^4*c - 10*a*b^3*d - 3*a^3*b*f + 6*a^2*b^2*e)*log(abs(x))/a^7 - 1/3*(15*b^5*c - 10*a*b^4*d - 3*a^3*b^2*f +
6*a^2*b^3*e)*log(abs(b*x^3 + a))/(a^7*b) + 1/6*(45*b^6*c*x^6 - 30*a*b^5*d*x^6 - 9*a^3*b^3*f*x^6 + 18*a^2*b^4*
x^6*e + 100*a*b^5*c*x^3 - 68*a^2*b^4*d*x^3 - 22*a^4*b^2*f*x^3 + 42*a^3*b^3*x^3*e + 56*a^2*b^4*c - 39*a^3*b^3*d
- 14*a^5*b*f + 25*a^4*b^2*e)/((b*x^3 + a)^2*a^7) - 1/36*(375*b^4*c*x^12 - 250*a*b^3*d*x^12 - 75*a^3*b*f*x^12
+ 150*a^2*b^2*x^12*e - 120*a*b^3*c*x^9 + 72*a^2*b^2*d*x^9 + 12*a^4*f*x^9 - 36*a^3*b*x^9*e + 36*a^2*b^2*c*x^6 -
18*a^3*b*d*x^6 + 6*a^4*x^6*e - 12*a^3*b*c*x^3 + 4*a^4*d*x^3 + 3*a^4*c)/(a^7*x^12)