\(\int \frac {c+d x^3+e x^6+f x^9}{x^{13} (a+b x^3)^3} \, dx\) [285]

Optimal result

Integrand size = 30, antiderivative size = 258 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^3} \, dx=-\frac {c}{12 a^3 x^{12}}+\frac {3 b c-a d}{9 a^4 x^9}-\frac {6 b^2 c-3 a b d+a^2 e}{6 a^5 x^6}+\frac {10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{3 a^6 x^3}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 a^5 \left (a+b x^3\right )^2}+\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{3 a^6 \left (a+b x^3\right )}+\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right ) \log (x)}{a^7}-\frac {b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right ) \log \left (a+b x^3\right )}{3 a^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1835, 1634} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^3} \, dx=\frac {3 b c-a d}{9 a^4 x^9}-\frac {c}{12 a^3 x^{12}}-\frac {a^2 e-3 a b d+6 b^2 c}{6 a^5 x^6}-\frac {b \log \left (a+b x^3\right ) \left (-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right )}{3 a^7}+\frac {b \log (x) \left (-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right )}{a^7}+\frac {b \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac {a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{3 a^6 x^3}+\frac {b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^5 \left (a+b x^3\right )^2} \]

[In]

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

[Out]

-1/12*c/(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)
*Log[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 1634

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]

Rule 1835

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]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {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} \text {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] (verified)

Time = 0.20 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^3} \, dx=\frac {-\frac {a \left (-180 b^5 c x^{15}+30 a b^4 x^{12} \left (-9 c+4 d x^3\right )-12 a^2 b^3 x^9 \left (5 c-15 d x^3+6 e x^6\right )-2 a^4 b x^3 \left (3 c+5 d x^3+12 e x^6-27 f x^9\right )+a^5 \left (3 c+4 d x^3+6 e x^6+12 f x^9\right )+a^3 b^2 x^6 \left (15 c+40 d x^3-108 e x^6+36 f x^9\right )\right )}{x^{12} \left (a+b x^3\right )^2}+36 b \left (15 b^3 c-10 a b^2 d+6 a^2 b e-3 a^3 f\right ) \log (x)+12 b \left (-15 b^3 c+10 a b^2 d-6 a^2 b e+3 a^3 f\right ) \log \left (a+b x^3\right )}{36 a^7} \]

[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)

Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {c}{12 a^{3} x^{12}}-\frac {a d -3 b c}{9 a^{4} x^{9}}-\frac {a^{2} e -3 a b d +6 b^{2} c}{6 a^{5} x^{6}}-\frac {f \,a^{3}-3 a^{2} b e +6 a \,b^{2} d -10 b^{3} c}{3 a^{6} x^{3}}-\frac {b \left (3 f \,a^{3}-6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right ) \ln \left (x \right )}{a^{7}}+\frac {b^{2} \left (\frac {\left (3 f \,a^{3}-6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{b}-\frac {a^{2} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{2 b \left (b \,x^{3}+a \right )^{2}}-\frac {a \left (2 f \,a^{3}-3 a^{2} b e +4 a \,b^{2} d -5 b^{3} c \right )}{b \left (b \,x^{3}+a \right )}\right )}{3 a^{7}}\)	\(253\)
norman	\(\frac {-\frac {c}{12 a}-\frac {\left (2 a d -3 b c \right ) x^{3}}{18 a^{2}}-\frac {\left (6 a^{2} e -10 a b d +15 b^{2} c \right ) x^{6}}{36 a^{3}}-\frac {\left (3 f \,a^{3}-6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right ) x^{9}}{9 a^{4}}+\frac {\left (-3 a^{3} b^{3} f +6 a^{2} b^{4} e -10 a \,b^{5} d +15 b^{6} c \right ) x^{12}}{2 a^{5} b^{2}}+\frac {\left (-3 a^{3} b^{3} f +6 a^{2} b^{4} e -10 a \,b^{5} d +15 b^{6} c \right ) x^{15}}{3 a^{6} b}}{x^{12} \left (b \,x^{3}+a \right )^{2}}-\frac {b \left (3 f \,a^{3}-6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right ) \ln \left (x \right )}{a^{7}}+\frac {b \left (3 f \,a^{3}-6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{7}}\)	\(262\)
risch	\(\frac {-\frac {c}{12 a}-\frac {\left (2 a d -3 b c \right ) x^{3}}{18 a^{2}}-\frac {\left (6 a^{2} e -10 a b d +15 b^{2} c \right ) x^{6}}{36 a^{3}}-\frac {\left (3 f \,a^{3}-6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right ) x^{9}}{9 a^{4}}-\frac {b \left (3 f \,a^{3}-6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right ) x^{12}}{2 a^{5}}-\frac {b^{2} \left (3 f \,a^{3}-6 a^{2} b e +10 a \,b^{2} d -15 b^{3} c \right ) x^{15}}{3 a^{6}}}{x^{12} \left (b \,x^{3}+a \right )^{2}}-\frac {3 b \ln \left (x \right ) f}{a^{4}}+\frac {6 b^{2} \ln \left (x \right ) e}{a^{5}}-\frac {10 b^{3} \ln \left (x \right ) d}{a^{6}}+\frac {15 b^{4} \ln \left (x \right ) c}{a^{7}}+\frac {b \ln \left (-b \,x^{3}-a \right ) f}{a^{4}}-\frac {2 b^{2} \ln \left (-b \,x^{3}-a \right ) e}{a^{5}}+\frac {10 b^{3} \ln \left (-b \,x^{3}-a \right ) d}{3 a^{6}}-\frac {5 b^{4} \ln \left (-b \,x^{3}-a \right ) c}{a^{7}}\)	\(293\)
parallelrisch	\(-\frac {-60 x^{9} a^{3} b^{5} c +6 x^{6} a^{6} b^{2} e -10 x^{6} a^{5} b^{3} d +15 x^{6} a^{4} b^{4} c +4 x^{3} a^{6} b^{2} d -6 x^{3} a^{5} b^{3} c -540 \ln \left (x \right ) x^{18} b^{8} c +180 \ln \left (b \,x^{3}+a \right ) x^{18} b^{8} c +3 a^{6} b^{2} c +108 \ln \left (x \right ) x^{12} a^{5} b^{3} f -216 \ln \left (x \right ) x^{12} a^{4} b^{4} e +360 \ln \left (x \right ) x^{12} a^{3} b^{5} d -540 \ln \left (x \right ) x^{12} a^{2} b^{6} c -36 \ln \left (b \,x^{3}+a \right ) x^{12} a^{5} b^{3} f +72 \ln \left (b \,x^{3}+a \right ) x^{12} a^{4} b^{4} e -120 \ln \left (b \,x^{3}+a \right ) x^{12} a^{3} b^{5} d +180 \ln \left (b \,x^{3}+a \right ) x^{12} a^{2} b^{6} c +36 x^{15} a^{4} b^{4} f -72 x^{15} a^{3} b^{5} e +120 x^{15} a^{2} b^{6} d -180 x^{15} a \,b^{7} c +54 x^{12} a^{5} b^{3} f -108 x^{12} a^{4} b^{4} e +180 x^{12} a^{3} b^{5} d -270 x^{12} a^{2} b^{6} c +12 x^{9} a^{6} b^{2} f -24 x^{9} a^{5} b^{3} e +40 x^{9} a^{4} b^{4} d +108 \ln \left (x \right ) x^{18} a^{3} b^{5} f -216 \ln \left (x \right ) x^{18} a^{2} b^{6} e +360 \ln \left (x \right ) x^{18} a \,b^{7} d -36 \ln \left (b \,x^{3}+a \right ) x^{18} a^{3} b^{5} f +72 \ln \left (b \,x^{3}+a \right ) x^{18} a^{2} b^{6} e -120 \ln \left (b \,x^{3}+a \right ) x^{18} a \,b^{7} d +216 \ln \left (x \right ) x^{15} a^{4} b^{4} f -432 \ln \left (x \right ) x^{15} a^{3} b^{5} e +720 \ln \left (x \right ) x^{15} a^{2} b^{6} d -1080 \ln \left (x \right ) x^{15} a \,b^{7} c -72 \ln \left (b \,x^{3}+a \right ) x^{15} a^{4} b^{4} f +144 \ln \left (b \,x^{3}+a \right ) x^{15} a^{3} b^{5} e -240 \ln \left (b \,x^{3}+a \right ) x^{15} a^{2} b^{6} d +360 \ln \left (b \,x^{3}+a \right ) x^{15} a \,b^{7} c}{36 a^{7} b^{2} x^{12} \left (b \,x^{3}+a \right )^{2}}\)	\(627\)

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^13/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.74 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^3} \, dx=\frac {12 \, {\left (15 \, a b^{5} c - 10 \, a^{2} b^{4} d + 6 \, a^{3} b^{3} e - 3 \, a^{4} b^{2} f\right )} x^{15} + 18 \, {\left (15 \, a^{2} b^{4} c - 10 \, a^{3} b^{3} d + 6 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{12} + 4 \, {\left (15 \, a^{3} b^{3} c - 10 \, a^{4} b^{2} d + 6 \, a^{5} b e - 3 \, a^{6} f\right )} x^{9} - 3 \, a^{6} c - {\left (15 \, a^{4} b^{2} c - 10 \, a^{5} b d + 6 \, a^{6} e\right )} x^{6} + 2 \, {\left (3 \, a^{5} b c - 2 \, a^{6} d\right )} x^{3} - 12 \, {\left ({\left (15 \, b^{6} c - 10 \, a b^{5} d + 6 \, a^{2} b^{4} e - 3 \, a^{3} b^{3} f\right )} x^{18} + 2 \, {\left (15 \, a b^{5} c - 10 \, a^{2} b^{4} d + 6 \, a^{3} b^{3} e - 3 \, a^{4} b^{2} f\right )} x^{15} + {\left (15 \, a^{2} b^{4} c - 10 \, a^{3} b^{3} d + 6 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{12}\right )} \log \left (b x^{3} + a\right ) + 36 \, {\left ({\left (15 \, b^{6} c - 10 \, a b^{5} d + 6 \, a^{2} b^{4} e - 3 \, a^{3} b^{3} f\right )} x^{18} + 2 \, {\left (15 \, a b^{5} c - 10 \, a^{2} b^{4} d + 6 \, a^{3} b^{3} e - 3 \, a^{4} b^{2} f\right )} x^{15} + {\left (15 \, a^{2} b^{4} c - 10 \, a^{3} b^{3} d + 6 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{12}\right )} \log \left (x\right )}{36 \, {\left (a^{7} b^{2} x^{18} + 2 \, a^{8} b x^{15} + a^{9} x^{12}\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((f*x**9+e*x**6+d*x**3+c)/x**13/(b*x**3+a)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.09 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^3} \, dx=\frac {12 \, {\left (15 \, b^{5} c - 10 \, a b^{4} d + 6 \, a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{15} + 18 \, {\left (15 \, a b^{4} c - 10 \, a^{2} b^{3} d + 6 \, a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{12} + 4 \, {\left (15 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 6 \, a^{4} b e - 3 \, a^{5} f\right )} x^{9} - {\left (15 \, a^{3} b^{2} c - 10 \, a^{4} b d + 6 \, a^{5} e\right )} x^{6} - 3 \, a^{5} c + 2 \, {\left (3 \, a^{4} b c - 2 \, a^{5} d\right )} x^{3}}{36 \, {\left (a^{6} b^{2} x^{18} + 2 \, a^{7} b x^{15} + a^{8} x^{12}\right )}} - \frac {{\left (15 \, b^{4} c - 10 \, a b^{3} d + 6 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{7}} + \frac {{\left (15 \, b^{4} c - 10 \, a b^{3} d + 6 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} \log \left (x^{3}\right )}{3 \, a^{7}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.44 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^3} \, dx=\frac {{\left (15 \, b^{4} c - 10 \, a b^{3} d + 6 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} \log \left ({\left | x \right |}\right )}{a^{7}} - \frac {{\left (15 \, b^{5} c - 10 \, a b^{4} d + 6 \, a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{7} b} + \frac {45 \, b^{6} c x^{6} - 30 \, a b^{5} d x^{6} + 18 \, a^{2} b^{4} e x^{6} - 9 \, a^{3} b^{3} f x^{6} + 100 \, a b^{5} c x^{3} - 68 \, a^{2} b^{4} d x^{3} + 42 \, a^{3} b^{3} e x^{3} - 22 \, a^{4} b^{2} f x^{3} + 56 \, a^{2} b^{4} c - 39 \, a^{3} b^{3} d + 25 \, a^{4} b^{2} e - 14 \, a^{5} b f}{6 \, {\left (b x^{3} + a\right )}^{2} a^{7}} - \frac {375 \, b^{4} c x^{12} - 250 \, a b^{3} d x^{12} + 150 \, a^{2} b^{2} e x^{12} - 75 \, a^{3} b f x^{12} - 120 \, a b^{3} c x^{9} + 72 \, a^{2} b^{2} d x^{9} - 36 \, a^{3} b e x^{9} + 12 \, a^{4} f x^{9} + 36 \, a^{2} b^{2} c x^{6} - 18 \, a^{3} b d x^{6} + 6 \, a^{4} e x^{6} - 12 \, a^{3} b c x^{3} + 4 \, a^{4} d x^{3} + 3 \, a^{4} c}{36 \, a^{7} x^{12}} \]

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

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^3} \, dx=\frac {\ln \left (x\right )\,\left (-3\,f\,a^3\,b+6\,e\,a^2\,b^2-10\,d\,a\,b^3+15\,c\,b^4\right )}{a^7}-\frac {\ln \left (b\,x^3+a\right )\,\left (-3\,f\,a^3\,b+6\,e\,a^2\,b^2-10\,d\,a\,b^3+15\,c\,b^4\right )}{3\,a^7}-\frac {\frac {c}{12\,a}-\frac {x^9\,\left (-3\,f\,a^3+6\,e\,a^2\,b-10\,d\,a\,b^2+15\,c\,b^3\right )}{9\,a^4}+\frac {x^3\,\left (2\,a\,d-3\,b\,c\right )}{18\,a^2}+\frac {x^6\,\left (6\,e\,a^2-10\,d\,a\,b+15\,c\,b^2\right )}{36\,a^3}-\frac {b\,x^{12}\,\left (-3\,f\,a^3+6\,e\,a^2\,b-10\,d\,a\,b^2+15\,c\,b^3\right )}{2\,a^5}-\frac {b^2\,x^{15}\,\left (-3\,f\,a^3+6\,e\,a^2\,b-10\,d\,a\,b^2+15\,c\,b^3\right )}{3\,a^6}}{a^2\,x^{12}+2\,a\,b\,x^{15}+b^2\,x^{18}} \]

[In]

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

[Out]

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