3.251 \(\int \frac {x^{11} (c+d x^3+e x^6+f x^9)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=220 \[ \frac {x^9 \left (3 a^2 f-2 a b e+b^2 d\right )}{9 b^4}+\frac {a^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^7 \left (a+b x^3\right )}+\frac {a^2 \log \left (a+b x^3\right ) \left (-6 a^3 f+5 a^2 b e-4 a b^2 d+3 b^3 c\right )}{3 b^7}-\frac {a x^3 \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac {x^6 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{6 b^5}+\frac {x^{12} (b e-2 a f)}{12 b^3}+\frac {f x^{15}}{15 b^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.34, antiderivative size = 220, normalized size of antiderivative = 1.00, 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} \[ \frac {x^6 \left (3 a^2 b e-4 a^3 f-2 a b^2 d+b^3 c\right )}{6 b^5}-\frac {a x^3 \left (4 a^2 b e-5 a^3 f-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac {a^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^7 \left (a+b x^3\right )}+\frac {a^2 \log \left (a+b x^3\right ) \left (5 a^2 b e-6 a^3 f-4 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac {x^9 \left (3 a^2 f-2 a b e+b^2 d\right )}{9 b^4}+\frac {x^{12} (b e-2 a f)}{12 b^3}+\frac {f x^{15}}{15 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

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

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]

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]]

Rubi steps

\begin {align*} \int \frac {x^{11} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{(a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right )}{b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x}{b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^2}{b^4}+\frac {(b e-2 a f) x^3}{b^3}+\frac {f x^4}{b^2}+\frac {a^3 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^6 (a+b x)^2}-\frac {a^2 \left (-3 b^3 c+4 a b^2 d-5 a^2 b e+6 a^3 f\right )}{b^6 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) x^3}{3 b^6}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) x^6}{6 b^5}+\frac {\left (b^2 d-2 a b e+3 a^2 f\right ) x^9}{9 b^4}+\frac {(b e-2 a f) x^{12}}{12 b^3}+\frac {f x^{15}}{15 b^2}+\frac {a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 b^7 \left (a+b x^3\right )}+\frac {a^2 \left (3 b^3 c-4 a b^2 d+5 a^2 b e-6 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.21, size = 205, normalized size = 0.93 \[ \frac {20 b^3 x^9 \left (3 a^2 f-2 a b e+b^2 d\right )+30 b^2 x^6 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )+60 a b x^3 \left (5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right )-\frac {60 a^3 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a+b x^3}+60 a^2 \log \left (a+b x^3\right ) \left (-6 a^3 f+5 a^2 b e-4 a b^2 d+3 b^3 c\right )+15 b^4 x^{12} (b e-2 a f)+12 b^5 f x^{15}}{180 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]

[Out]

(60*a*b*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x^3 + 30*b^2*(b^3*c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*x^
6 + 20*b^3*(b^2*d - 2*a*b*e + 3*a^2*f)*x^9 + 15*b^4*(b*e - 2*a*f)*x^12 + 12*b^5*f*x^15 - (60*a^3*(-(b^3*c) + a
*b^2*d - a^2*b*e + a^3*f))/(a + b*x^3) + 60*a^2*(3*b^3*c - 4*a*b^2*d + 5*a^2*b*e - 6*a^3*f)*Log[a + b*x^3])/(1
80*b^7)

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 303, normalized size = 1.38 \[ \frac {12 \, b^{6} f x^{18} + 3 \, {\left (5 \, b^{6} e - 6 \, a b^{5} f\right )} x^{15} + 5 \, {\left (4 \, b^{6} d - 5 \, a b^{5} e + 6 \, a^{2} b^{4} f\right )} x^{12} + 10 \, {\left (3 \, b^{6} c - 4 \, a b^{5} d + 5 \, a^{2} b^{4} e - 6 \, a^{3} b^{3} f\right )} x^{9} + 60 \, a^{3} b^{3} c - 60 \, a^{4} b^{2} d + 60 \, a^{5} b e - 60 \, a^{6} f - 30 \, {\left (3 \, a b^{5} c - 4 \, a^{2} b^{4} d + 5 \, a^{3} b^{3} e - 6 \, a^{4} b^{2} f\right )} x^{6} - 60 \, {\left (2 \, a^{2} b^{4} c - 3 \, a^{3} b^{3} d + 4 \, a^{4} b^{2} e - 5 \, a^{5} b f\right )} x^{3} + 60 \, {\left (3 \, a^{3} b^{3} c - 4 \, a^{4} b^{2} d + 5 \, a^{5} b e - 6 \, a^{6} f + {\left (3 \, a^{2} b^{4} c - 4 \, a^{3} b^{3} d + 5 \, a^{4} b^{2} e - 6 \, a^{5} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{180 \, {\left (b^{8} x^{3} + a b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/180*(12*b^6*f*x^18 + 3*(5*b^6*e - 6*a*b^5*f)*x^15 + 5*(4*b^6*d - 5*a*b^5*e + 6*a^2*b^4*f)*x^12 + 10*(3*b^6*c
- 4*a*b^5*d + 5*a^2*b^4*e - 6*a^3*b^3*f)*x^9 + 60*a^3*b^3*c - 60*a^4*b^2*d + 60*a^5*b*e - 60*a^6*f - 30*(3*a*
b^5*c - 4*a^2*b^4*d + 5*a^3*b^3*e - 6*a^4*b^2*f)*x^6 - 60*(2*a^2*b^4*c - 3*a^3*b^3*d + 4*a^4*b^2*e - 5*a^5*b*f
)*x^3 + 60*(3*a^3*b^3*c - 4*a^4*b^2*d + 5*a^5*b*e - 6*a^6*f + (3*a^2*b^4*c - 4*a^3*b^3*d + 5*a^4*b^2*e - 6*a^5
*b*f)*x^3)*log(b*x^3 + a))/(b^8*x^3 + a*b^7)

________________________________________________________________________________________

giac [A]  time = 0.25, size = 300, normalized size = 1.36 \[ \frac {{\left (3 \, a^{2} b^{3} c - 4 \, a^{3} b^{2} d - 6 \, a^{5} f + 5 \, a^{4} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{7}} - \frac {3 \, a^{2} b^{4} c x^{3} - 4 \, a^{3} b^{3} d x^{3} - 6 \, a^{5} b f x^{3} + 5 \, a^{4} b^{2} x^{3} e + 2 \, a^{3} b^{3} c - 3 \, a^{4} b^{2} d - 5 \, a^{6} f + 4 \, a^{5} b e}{3 \, {\left (b x^{3} + a\right )} b^{7}} + \frac {12 \, b^{8} f x^{15} - 30 \, a b^{7} f x^{12} + 15 \, b^{8} x^{12} e + 20 \, b^{8} d x^{9} + 60 \, a^{2} b^{6} f x^{9} - 40 \, a b^{7} x^{9} e + 30 \, b^{8} c x^{6} - 60 \, a b^{7} d x^{6} - 120 \, a^{3} b^{5} f x^{6} + 90 \, a^{2} b^{6} x^{6} e - 120 \, a b^{7} c x^{3} + 180 \, a^{2} b^{6} d x^{3} + 300 \, a^{4} b^{4} f x^{3} - 240 \, a^{3} b^{5} x^{3} e}{180 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/3*(3*a^2*b^3*c - 4*a^3*b^2*d - 6*a^5*f + 5*a^4*b*e)*log(abs(b*x^3 + a))/b^7 - 1/3*(3*a^2*b^4*c*x^3 - 4*a^3*b
^3*d*x^3 - 6*a^5*b*f*x^3 + 5*a^4*b^2*x^3*e + 2*a^3*b^3*c - 3*a^4*b^2*d - 5*a^6*f + 4*a^5*b*e)/((b*x^3 + a)*b^7
) + 1/180*(12*b^8*f*x^15 - 30*a*b^7*f*x^12 + 15*b^8*x^12*e + 20*b^8*d*x^9 + 60*a^2*b^6*f*x^9 - 40*a*b^7*x^9*e
+ 30*b^8*c*x^6 - 60*a*b^7*d*x^6 - 120*a^3*b^5*f*x^6 + 90*a^2*b^6*x^6*e - 120*a*b^7*c*x^3 + 180*a^2*b^6*d*x^3 +
300*a^4*b^4*f*x^3 - 240*a^3*b^5*x^3*e)/b^10

________________________________________________________________________________________

maple [A]  time = 0.06, size = 288, normalized size = 1.31 \[ \frac {f \,x^{15}}{15 b^{2}}-\frac {a f \,x^{12}}{6 b^{3}}+\frac {e \,x^{12}}{12 b^{2}}+\frac {a^{2} f \,x^{9}}{3 b^{4}}-\frac {2 a e \,x^{9}}{9 b^{3}}+\frac {d \,x^{9}}{9 b^{2}}-\frac {2 a^{3} f \,x^{6}}{3 b^{5}}+\frac {a^{2} e \,x^{6}}{2 b^{4}}-\frac {a d \,x^{6}}{3 b^{3}}+\frac {c \,x^{6}}{6 b^{2}}+\frac {5 a^{4} f \,x^{3}}{3 b^{6}}-\frac {4 a^{3} e \,x^{3}}{3 b^{5}}+\frac {a^{2} d \,x^{3}}{b^{4}}-\frac {2 a c \,x^{3}}{3 b^{3}}-\frac {a^{6} f}{3 \left (b \,x^{3}+a \right ) b^{7}}+\frac {a^{5} e}{3 \left (b \,x^{3}+a \right ) b^{6}}-\frac {2 a^{5} f \ln \left (b \,x^{3}+a \right )}{b^{7}}-\frac {a^{4} d}{3 \left (b \,x^{3}+a \right ) b^{5}}+\frac {5 a^{4} e \ln \left (b \,x^{3}+a \right )}{3 b^{6}}+\frac {a^{3} c}{3 \left (b \,x^{3}+a \right ) b^{4}}-\frac {4 a^{3} d \ln \left (b \,x^{3}+a \right )}{3 b^{5}}+\frac {a^{2} c \ln \left (b \,x^{3}+a \right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.30, size = 222, normalized size = 1.01 \[ \frac {a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f}{3 \, {\left (b^{8} x^{3} + a b^{7}\right )}} + \frac {12 \, b^{4} f x^{15} + 15 \, {\left (b^{4} e - 2 \, a b^{3} f\right )} x^{12} + 20 \, {\left (b^{4} d - 2 \, a b^{3} e + 3 \, a^{2} b^{2} f\right )} x^{9} + 30 \, {\left (b^{4} c - 2 \, a b^{3} d + 3 \, a^{2} b^{2} e - 4 \, a^{3} b f\right )} x^{6} - 60 \, {\left (2 \, a b^{3} c - 3 \, a^{2} b^{2} d + 4 \, a^{3} b e - 5 \, a^{4} f\right )} x^{3}}{180 \, b^{6}} + \frac {{\left (3 \, a^{2} b^{3} c - 4 \, a^{3} b^{2} d + 5 \, a^{4} b e - 6 \, a^{5} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/3*(a^3*b^3*c - a^4*b^2*d + a^5*b*e - a^6*f)/(b^8*x^3 + a*b^7) + 1/180*(12*b^4*f*x^15 + 15*(b^4*e - 2*a*b^3*f
)*x^12 + 20*(b^4*d - 2*a*b^3*e + 3*a^2*b^2*f)*x^9 + 30*(b^4*c - 2*a*b^3*d + 3*a^2*b^2*e - 4*a^3*b*f)*x^6 - 60*
(2*a*b^3*c - 3*a^2*b^2*d + 4*a^3*b*e - 5*a^4*f)*x^3)/b^6 + 1/3*(3*a^2*b^3*c - 4*a^3*b^2*d + 5*a^4*b*e - 6*a^5*
f)*log(b*x^3 + a)/b^7

________________________________________________________________________________________

mupad [B]  time = 4.99, size = 356, normalized size = 1.62 \[ x^{12}\,\left (\frac {e}{12\,b^2}-\frac {a\,f}{6\,b^3}\right )-x^3\,\left (\frac {2\,a\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )}{3\,b}-\frac {a^2\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{3\,b^2}\right )-x^9\,\left (\frac {a^2\,f}{9\,b^4}-\frac {d}{9\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{9\,b}\right )+x^6\,\left (\frac {c}{6\,b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{6\,b^2}+\frac {a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{3\,b}\right )-\frac {\ln \left (b\,x^3+a\right )\,\left (6\,f\,a^5-5\,e\,a^4\,b+4\,d\,a^3\,b^2-3\,c\,a^2\,b^3\right )}{3\,b^7}+\frac {f\,x^{15}}{15\,b^2}-\frac {f\,a^6-e\,a^5\,b+d\,a^4\,b^2-c\,a^3\,b^3}{3\,b\,\left (b^7\,x^3+a\,b^6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^11*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 14.42, size = 236, normalized size = 1.07 \[ - \frac {a^{2} \left (6 a^{3} f - 5 a^{2} b e + 4 a b^{2} d - 3 b^{3} c\right ) \log {\left (a + b x^{3} \right )}}{3 b^{7}} + x^{12} \left (- \frac {a f}{6 b^{3}} + \frac {e}{12 b^{2}}\right ) + x^{9} \left (\frac {a^{2} f}{3 b^{4}} - \frac {2 a e}{9 b^{3}} + \frac {d}{9 b^{2}}\right ) + x^{6} \left (- \frac {2 a^{3} f}{3 b^{5}} + \frac {a^{2} e}{2 b^{4}} - \frac {a d}{3 b^{3}} + \frac {c}{6 b^{2}}\right ) + x^{3} \left (\frac {5 a^{4} f}{3 b^{6}} - \frac {4 a^{3} e}{3 b^{5}} + \frac {a^{2} d}{b^{4}} - \frac {2 a c}{3 b^{3}}\right ) + \frac {- a^{6} f + a^{5} b e - a^{4} b^{2} d + a^{3} b^{3} c}{3 a b^{7} + 3 b^{8} x^{3}} + \frac {f x^{15}}{15 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)

[Out]

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

________________________________________________________________________________________