\(\int \frac {(A+B x) (a+c x^2)^2}{(d+e x)^9} \, dx\) [60]

Optimal result

Integrand size = 22, antiderivative size = 206 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^9} \, dx=\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{8 e^6 (d+e x)^8}-\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{7 e^6 (d+e x)^7}+\frac {c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{3 e^6 (d+e x)^6}-\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right )}{5 e^6 (d+e x)^5}+\frac {c^2 (5 B d-A e)}{4 e^6 (d+e x)^4}-\frac {B c^2}{3 e^6 (d+e x)^3} \] Output:

1/8*(-A*e+B*d)*(a*e^2+c*d^2)^2/e^6/(e*x+d)^8-1/7*(a*e^2+c*d^2)*(-4*A*c*d*e 
+B*a*e^2+5*B*c*d^2)/e^6/(e*x+d)^7+1/3*c*(-A*a*e^3-3*A*c*d^2*e+3*B*a*d*e^2+ 
5*B*c*d^3)/e^6/(e*x+d)^6-2/5*c*(-2*A*c*d*e+B*a*e^2+5*B*c*d^2)/e^6/(e*x+d)^ 
5+1/4*c^2*(-A*e+5*B*d)/e^6/(e*x+d)^4-1/3*B*c^2/e^6/(e*x+d)^3
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^9} \, dx=-\frac {A e \left (105 a^2 e^4+10 a c e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+3 c^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+B \left (15 a^2 e^4 (d+8 e x)+6 a c e^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+5 c^2 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{840 e^6 (d+e x)^8} \] Input:

Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^9,x]
 

Output:

-1/840*(A*e*(105*a^2*e^4 + 10*a*c*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*c^2 
*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + B*(15*a 
^2*e^4*(d + 8*e*x) + 6*a*c*e^2*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^ 
3) + 5*c^2*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x 
^4 + 56*e^5*x^5)))/(e^6*(d + e*x)^8)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {652, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+c x^2\right )^2 (A+B x)}{(d+e x)^9} \, dx\)

\(\Big \downarrow \) 652

\(\displaystyle \int \left (-\frac {2 c \left (-a B e^2+2 A c d e-5 B c d^2\right )}{e^5 (d+e x)^6}+\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^5 (d+e x)^8}+\frac {\left (a e^2+c d^2\right )^2 (A e-B d)}{e^5 (d+e x)^9}+\frac {2 c \left (a A e^3-3 a B d e^2+3 A c d^2 e-5 B c d^3\right )}{e^5 (d+e x)^7}+\frac {c^2 (A e-5 B d)}{e^5 (d+e x)^5}+\frac {B c^2}{e^5 (d+e x)^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^5}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{7 e^6 (d+e x)^7}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{8 e^6 (d+e x)^8}+\frac {c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^6}+\frac {c^2 (5 B d-A e)}{4 e^6 (d+e x)^4}-\frac {B c^2}{3 e^6 (d+e x)^3}\)

Input:

Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^9,x]
 

Output:

((B*d - A*e)*(c*d^2 + a*e^2)^2)/(8*e^6*(d + e*x)^8) - ((c*d^2 + a*e^2)*(5* 
B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(7*e^6*(d + e*x)^7) + (c*(5*B*c*d^3 - 3*A* 
c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(3*e^6*(d + e*x)^6) - (2*c*(5*B*c*d^2 - 
2*A*c*d*e + a*B*e^2))/(5*e^6*(d + e*x)^5) + (c^2*(5*B*d - A*e))/(4*e^6*(d 
+ e*x)^4) - (B*c^2)/(3*e^6*(d + e*x)^3)
 

Defintions of rubi rules used

rule 652

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ 
)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c 
*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.13

method	result	size
risch	\(\frac {-\frac {B \,c^{2} x^{5}}{3 e}-\frac {c^{2} \left (3 A e +5 B d \right ) x^{4}}{12 e^{2}}-\frac {c \left (3 A c d e +6 B a \,e^{2}+5 B c \,d^{2}\right ) x^{3}}{15 e^{3}}-\frac {c \left (10 A a \,e^{3}+3 A c \,d^{2} e +6 B a d \,e^{2}+5 B c \,d^{3}\right ) x^{2}}{30 e^{4}}-\frac {\left (10 A a c d \,e^{3}+3 A \,c^{2} d^{3} e +15 B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}\right ) x}{105 e^{5}}-\frac {105 A \,a^{2} e^{5}+10 A a c \,d^{2} e^{3}+3 A \,c^{2} d^{4} e +15 B \,a^{2} d \,e^{4}+6 B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{840 e^{6}}}{\left (e x +d \right )^{8}}\)	\(233\)
default	\(-\frac {B \,c^{2}}{3 e^{6} \left (e x +d \right )^{3}}-\frac {c^{2} \left (A e -5 B d \right )}{4 e^{6} \left (e x +d \right )^{4}}-\frac {-4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}}{7 e^{6} \left (e x +d \right )^{7}}+\frac {2 c \left (2 A c d e -B a \,e^{2}-5 B c \,d^{2}\right )}{5 e^{6} \left (e x +d \right )^{5}}-\frac {A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}}{8 e^{6} \left (e x +d \right )^{8}}-\frac {c \left (A a \,e^{3}+3 A c \,d^{2} e -3 B a d \,e^{2}-5 B c \,d^{3}\right )}{3 e^{6} \left (e x +d \right )^{6}}\)	\(249\)
gosper	\(-\frac {280 B \,x^{5} c^{2} e^{5}+210 A \,x^{4} c^{2} e^{5}+350 B \,x^{4} c^{2} d \,e^{4}+168 A \,x^{3} c^{2} d \,e^{4}+336 B \,x^{3} a c \,e^{5}+280 B \,x^{3} c^{2} d^{2} e^{3}+280 A \,x^{2} a c \,e^{5}+84 A \,x^{2} c^{2} d^{2} e^{3}+168 B \,x^{2} a c d \,e^{4}+140 B \,x^{2} c^{2} d^{3} e^{2}+80 A x a c d \,e^{4}+24 A x \,c^{2} d^{3} e^{2}+120 B x \,a^{2} e^{5}+48 B x a c \,d^{2} e^{3}+40 B x \,c^{2} d^{4} e +105 A \,a^{2} e^{5}+10 A a c \,d^{2} e^{3}+3 A \,c^{2} d^{4} e +15 B \,a^{2} d \,e^{4}+6 B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{840 e^{6} \left (e x +d \right )^{8}}\)	\(259\)
orering	\(-\frac {280 B \,x^{5} c^{2} e^{5}+210 A \,x^{4} c^{2} e^{5}+350 B \,x^{4} c^{2} d \,e^{4}+168 A \,x^{3} c^{2} d \,e^{4}+336 B \,x^{3} a c \,e^{5}+280 B \,x^{3} c^{2} d^{2} e^{3}+280 A \,x^{2} a c \,e^{5}+84 A \,x^{2} c^{2} d^{2} e^{3}+168 B \,x^{2} a c d \,e^{4}+140 B \,x^{2} c^{2} d^{3} e^{2}+80 A x a c d \,e^{4}+24 A x \,c^{2} d^{3} e^{2}+120 B x \,a^{2} e^{5}+48 B x a c \,d^{2} e^{3}+40 B x \,c^{2} d^{4} e +105 A \,a^{2} e^{5}+10 A a c \,d^{2} e^{3}+3 A \,c^{2} d^{4} e +15 B \,a^{2} d \,e^{4}+6 B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{840 e^{6} \left (e x +d \right )^{8}}\)	\(259\)
parallelrisch	\(-\frac {280 B \,c^{2} x^{5} e^{7}+210 A \,c^{2} e^{7} x^{4}+350 B \,c^{2} d \,e^{6} x^{4}+168 A \,c^{2} d \,e^{6} x^{3}+336 B a c \,e^{7} x^{3}+280 B \,c^{2} d^{2} e^{5} x^{3}+280 A a c \,e^{7} x^{2}+84 A \,c^{2} d^{2} e^{5} x^{2}+168 B a c d \,e^{6} x^{2}+140 B \,c^{2} d^{3} e^{4} x^{2}+80 A a c d \,e^{6} x +24 A \,c^{2} d^{3} e^{4} x +120 B \,a^{2} e^{7} x +48 B a c \,d^{2} e^{5} x +40 B \,c^{2} d^{4} e^{3} x +105 a^{2} A \,e^{7}+10 A a c \,d^{2} e^{5}+3 A \,c^{2} d^{4} e^{3}+15 B \,a^{2} d \,e^{6}+6 B a c \,d^{3} e^{4}+5 B \,c^{2} d^{5} e^{2}}{840 e^{8} \left (e x +d \right )^{8}}\)	\(266\)
norman	\(\frac {-\frac {B \,c^{2} x^{5}}{3 e}-\frac {\left (3 A \,c^{2} e^{3}+5 B \,c^{2} d \,e^{2}\right ) x^{4}}{12 e^{4}}-\frac {\left (3 A \,c^{2} d \,e^{3}+6 B \,e^{4} a c +5 B \,c^{2} d^{2} e^{2}\right ) x^{3}}{15 e^{5}}-\frac {\left (10 A a c \,e^{5}+3 A \,c^{2} d^{2} e^{3}+6 B a c d \,e^{4}+5 B \,c^{2} d^{3} e^{2}\right ) x^{2}}{30 e^{6}}-\frac {\left (10 A a c d \,e^{5}+3 A \,c^{2} d^{3} e^{3}+15 a^{2} B \,e^{6}+6 B a c \,d^{2} e^{4}+5 B \,c^{2} d^{4} e^{2}\right ) x}{105 e^{7}}-\frac {105 a^{2} A \,e^{7}+10 A a c \,d^{2} e^{5}+3 A \,c^{2} d^{4} e^{3}+15 B \,a^{2} d \,e^{6}+6 B a c \,d^{3} e^{4}+5 B \,c^{2} d^{5} e^{2}}{840 e^{8}}}{\left (e x +d \right )^{8}}\)	\(270\)

Input:

int((B*x+A)*(c*x^2+a)^2/(e*x+d)^9,x,method=_RETURNVERBOSE)
 

Output:

(-1/3*B*c^2*x^5/e-1/12*c^2/e^2*(3*A*e+5*B*d)*x^4-1/15*c/e^3*(3*A*c*d*e+6*B 
*a*e^2+5*B*c*d^2)*x^3-1/30*c/e^4*(10*A*a*e^3+3*A*c*d^2*e+6*B*a*d*e^2+5*B*c 
*d^3)*x^2-1/105/e^5*(10*A*a*c*d*e^3+3*A*c^2*d^3*e+15*B*a^2*e^4+6*B*a*c*d^2 
*e^2+5*B*c^2*d^4)*x-1/840/e^6*(105*A*a^2*e^5+10*A*a*c*d^2*e^3+3*A*c^2*d^4* 
e+15*B*a^2*d*e^4+6*B*a*c*d^3*e^2+5*B*c^2*d^5))/(e*x+d)^8
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.59 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^9} \, dx=-\frac {280 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 3 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} + 10 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} + 105 \, A a^{2} e^{5} + 70 \, {\left (5 \, B c^{2} d e^{4} + 3 \, A c^{2} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B c^{2} d^{2} e^{3} + 3 \, A c^{2} d e^{4} + 6 \, B a c e^{5}\right )} x^{3} + 28 \, {\left (5 \, B c^{2} d^{3} e^{2} + 3 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} + 10 \, A a c e^{5}\right )} x^{2} + 8 \, {\left (5 \, B c^{2} d^{4} e + 3 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 10 \, A a c d e^{4} + 15 \, B a^{2} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^9,x, algorithm="fricas")
 

Output:

-1/840*(280*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 3*A*c^2*d^4*e + 6*B*a*c*d^3*e^2 
+ 10*A*a*c*d^2*e^3 + 15*B*a^2*d*e^4 + 105*A*a^2*e^5 + 70*(5*B*c^2*d*e^4 + 
3*A*c^2*e^5)*x^4 + 56*(5*B*c^2*d^2*e^3 + 3*A*c^2*d*e^4 + 6*B*a*c*e^5)*x^3 
+ 28*(5*B*c^2*d^3*e^2 + 3*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 + 10*A*a*c*e^5)*x^ 
2 + 8*(5*B*c^2*d^4*e + 3*A*c^2*d^3*e^2 + 6*B*a*c*d^2*e^3 + 10*A*a*c*d*e^4 
+ 15*B*a^2*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7 + 28*d^2*e^12*x^6 + 56*d^3*e^1 
1*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + 
d^8*e^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^9} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**9,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.59 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^9} \, dx=-\frac {280 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + 3 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} + 10 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} + 105 \, A a^{2} e^{5} + 70 \, {\left (5 \, B c^{2} d e^{4} + 3 \, A c^{2} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B c^{2} d^{2} e^{3} + 3 \, A c^{2} d e^{4} + 6 \, B a c e^{5}\right )} x^{3} + 28 \, {\left (5 \, B c^{2} d^{3} e^{2} + 3 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} + 10 \, A a c e^{5}\right )} x^{2} + 8 \, {\left (5 \, B c^{2} d^{4} e + 3 \, A c^{2} d^{3} e^{2} + 6 \, B a c d^{2} e^{3} + 10 \, A a c d e^{4} + 15 \, B a^{2} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^9,x, algorithm="maxima")
 

Output:

-1/840*(280*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + 3*A*c^2*d^4*e + 6*B*a*c*d^3*e^2 
+ 10*A*a*c*d^2*e^3 + 15*B*a^2*d*e^4 + 105*A*a^2*e^5 + 70*(5*B*c^2*d*e^4 + 
3*A*c^2*e^5)*x^4 + 56*(5*B*c^2*d^2*e^3 + 3*A*c^2*d*e^4 + 6*B*a*c*e^5)*x^3 
+ 28*(5*B*c^2*d^3*e^2 + 3*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 + 10*A*a*c*e^5)*x^ 
2 + 8*(5*B*c^2*d^4*e + 3*A*c^2*d^3*e^2 + 6*B*a*c*d^2*e^3 + 10*A*a*c*d*e^4 
+ 15*B*a^2*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7 + 28*d^2*e^12*x^6 + 56*d^3*e^1 
1*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + 
d^8*e^6)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.25 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^9} \, dx=-\frac {280 \, B c^{2} e^{5} x^{5} + 350 \, B c^{2} d e^{4} x^{4} + 210 \, A c^{2} e^{5} x^{4} + 280 \, B c^{2} d^{2} e^{3} x^{3} + 168 \, A c^{2} d e^{4} x^{3} + 336 \, B a c e^{5} x^{3} + 140 \, B c^{2} d^{3} e^{2} x^{2} + 84 \, A c^{2} d^{2} e^{3} x^{2} + 168 \, B a c d e^{4} x^{2} + 280 \, A a c e^{5} x^{2} + 40 \, B c^{2} d^{4} e x + 24 \, A c^{2} d^{3} e^{2} x + 48 \, B a c d^{2} e^{3} x + 80 \, A a c d e^{4} x + 120 \, B a^{2} e^{5} x + 5 \, B c^{2} d^{5} + 3 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} + 10 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} + 105 \, A a^{2} e^{5}}{840 \, {\left (e x + d\right )}^{8} e^{6}} \] Input:

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^9,x, algorithm="giac")
 

Output:

-1/840*(280*B*c^2*e^5*x^5 + 350*B*c^2*d*e^4*x^4 + 210*A*c^2*e^5*x^4 + 280* 
B*c^2*d^2*e^3*x^3 + 168*A*c^2*d*e^4*x^3 + 336*B*a*c*e^5*x^3 + 140*B*c^2*d^ 
3*e^2*x^2 + 84*A*c^2*d^2*e^3*x^2 + 168*B*a*c*d*e^4*x^2 + 280*A*a*c*e^5*x^2 
 + 40*B*c^2*d^4*e*x + 24*A*c^2*d^3*e^2*x + 48*B*a*c*d^2*e^3*x + 80*A*a*c*d 
*e^4*x + 120*B*a^2*e^5*x + 5*B*c^2*d^5 + 3*A*c^2*d^4*e + 6*B*a*c*d^3*e^2 + 
 10*A*a*c*d^2*e^3 + 15*B*a^2*d*e^4 + 105*A*a^2*e^5)/((e*x + d)^8*e^6)
 

Mupad [B] (verification not implemented)

Time = 6.31 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^9} \, dx=-\frac {\frac {15\,B\,a^2\,d\,e^4+105\,A\,a^2\,e^5+6\,B\,a\,c\,d^3\,e^2+10\,A\,a\,c\,d^2\,e^3+5\,B\,c^2\,d^5+3\,A\,c^2\,d^4\,e}{840\,e^6}+\frac {x\,\left (15\,B\,a^2\,e^4+6\,B\,a\,c\,d^2\,e^2+10\,A\,a\,c\,d\,e^3+5\,B\,c^2\,d^4+3\,A\,c^2\,d^3\,e\right )}{105\,e^5}+\frac {c\,x^3\,\left (5\,B\,c\,d^2+3\,A\,c\,d\,e+6\,B\,a\,e^2\right )}{15\,e^3}+\frac {c^2\,x^4\,\left (3\,A\,e+5\,B\,d\right )}{12\,e^2}+\frac {c\,x^2\,\left (5\,B\,c\,d^3+3\,A\,c\,d^2\,e+6\,B\,a\,d\,e^2+10\,A\,a\,e^3\right )}{30\,e^4}+\frac {B\,c^2\,x^5}{3\,e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \] Input:

int(((a + c*x^2)^2*(A + B*x))/(d + e*x)^9,x)
 

Output:

-((105*A*a^2*e^5 + 5*B*c^2*d^5 + 15*B*a^2*d*e^4 + 3*A*c^2*d^4*e + 10*A*a*c 
*d^2*e^3 + 6*B*a*c*d^3*e^2)/(840*e^6) + (x*(15*B*a^2*e^4 + 5*B*c^2*d^4 + 3 
*A*c^2*d^3*e + 10*A*a*c*d*e^3 + 6*B*a*c*d^2*e^2))/(105*e^5) + (c*x^3*(6*B* 
a*e^2 + 5*B*c*d^2 + 3*A*c*d*e))/(15*e^3) + (c^2*x^4*(3*A*e + 5*B*d))/(12*e 
^2) + (c*x^2*(10*A*a*e^3 + 5*B*c*d^3 + 6*B*a*d*e^2 + 3*A*c*d^2*e))/(30*e^4 
) + (B*c^2*x^5)/(3*e))/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56* 
d^5*e^3*x^3 + 70*d^4*e^4*x^4 + 56*d^3*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*e*x 
)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.64 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^9} \, dx=\frac {-280 b \,c^{2} e^{5} x^{5}-210 a \,c^{2} e^{5} x^{4}-350 b \,c^{2} d \,e^{4} x^{4}-336 a b c \,e^{5} x^{3}-168 a \,c^{2} d \,e^{4} x^{3}-280 b \,c^{2} d^{2} e^{3} x^{3}-280 a^{2} c \,e^{5} x^{2}-168 a b c d \,e^{4} x^{2}-84 a \,c^{2} d^{2} e^{3} x^{2}-140 b \,c^{2} d^{3} e^{2} x^{2}-120 a^{2} b \,e^{5} x -80 a^{2} c d \,e^{4} x -48 a b c \,d^{2} e^{3} x -24 a \,c^{2} d^{3} e^{2} x -40 b \,c^{2} d^{4} e x -105 a^{3} e^{5}-15 a^{2} b d \,e^{4}-10 a^{2} c \,d^{2} e^{3}-6 a b c \,d^{3} e^{2}-3 a \,c^{2} d^{4} e -5 b \,c^{2} d^{5}}{840 e^{6} \left (e^{8} x^{8}+8 d \,e^{7} x^{7}+28 d^{2} e^{6} x^{6}+56 d^{3} e^{5} x^{5}+70 d^{4} e^{4} x^{4}+56 d^{5} e^{3} x^{3}+28 d^{6} e^{2} x^{2}+8 d^{7} e x +d^{8}\right )} \] Input:

int((B*x+A)*(c*x^2+a)^2/(e*x+d)^9,x)
 

Output:

( - 105*a**3*e**5 - 15*a**2*b*d*e**4 - 120*a**2*b*e**5*x - 10*a**2*c*d**2* 
e**3 - 80*a**2*c*d*e**4*x - 280*a**2*c*e**5*x**2 - 6*a*b*c*d**3*e**2 - 48* 
a*b*c*d**2*e**3*x - 168*a*b*c*d*e**4*x**2 - 336*a*b*c*e**5*x**3 - 3*a*c**2 
*d**4*e - 24*a*c**2*d**3*e**2*x - 84*a*c**2*d**2*e**3*x**2 - 168*a*c**2*d* 
e**4*x**3 - 210*a*c**2*e**5*x**4 - 5*b*c**2*d**5 - 40*b*c**2*d**4*e*x - 14 
0*b*c**2*d**3*e**2*x**2 - 280*b*c**2*d**2*e**3*x**3 - 350*b*c**2*d*e**4*x* 
*4 - 280*b*c**2*e**5*x**5)/(840*e**6*(d**8 + 8*d**7*e*x + 28*d**6*e**2*x** 
2 + 56*d**5*e**3*x**3 + 70*d**4*e**4*x**4 + 56*d**3*e**5*x**5 + 28*d**2*e* 
*6*x**6 + 8*d*e**7*x**7 + e**8*x**8))