\(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^{10}} \, dx\) [77]

Optimal result

Integrand size = 22, antiderivative size = 334 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{9 e^8 (d+e x)^9}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{8 e^8 (d+e x)^8}+\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{7 e^8 (d+e x)^7}+\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{6 e^8 (d+e x)^6}+\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{5 e^8 (d+e x)^5}-\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{4 e^8 (d+e x)^4}+\frac {c^3 (7 B d-A e)}{3 e^8 (d+e x)^3}-\frac {B c^3}{2 e^8 (d+e x)^2} \] Output:

1/9*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^9-1/8*(a*e^2+c*d^2)^2*(-6*A*c*d 
*e+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d)^8+3/7*c*(a*e^2+c*d^2)*(-A*a*e^3-5*A*c*d^ 
2*e+3*B*a*d*e^2+7*B*c*d^3)/e^8/(e*x+d)^7+1/6*c*(4*A*c*d*e*(3*a*e^2+5*c*d^2 
)-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d^4))/e^8/(e*x+d)^6+1/5*c^2*(-3*A*a*e 
^3-15*A*c*d^2*e+15*B*a*d*e^2+35*B*c*d^3)/e^8/(e*x+d)^5-3/4*c^2*(-2*A*c*d*e 
+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d)^4+1/3*c^3*(-A*e+7*B*d)/e^8/(e*x+d)^3-1/2*B 
*c^3/e^8/(e*x+d)^2
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.07 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {2 A e \left (140 a^3 e^6+15 a^2 c e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 c^3 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )+5 B \left (7 a^3 e^6 (d+9 e x)+3 a^2 c e^4 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+3 a c^2 e^2 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+7 c^3 \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )\right )}{2520 e^8 (d+e x)^9} \] Input:

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^10,x]
 

Output:

-1/2520*(2*A*e*(140*a^3*e^6 + 15*a^2*c*e^4*(d^2 + 9*d*e*x + 36*e^2*x^2) + 
6*a*c^2*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4 
) + 5*c^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4 
*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6)) + 5*B*(7*a^3*e^6*(d + 9*e*x) + 3*a^2*c 
*e^4*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 3*a*c^2*e^2*(d^5 + 9* 
d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 
 7*c^3*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^ 
4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^7)))/(e^8*(d + e*x)^9)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 334, 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.

\(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^{10}} \, dx\)

\(\Big \downarrow \) 652

\(\displaystyle \int \left (-\frac {c \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7 (d+e x)^7}-\frac {3 c^2 \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7 (d+e x)^5}+\frac {c^2 \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7 (d+e x)^6}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^9}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^{10}}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)^8}+\frac {c^3 (A e-7 B d)}{e^7 (d+e x)^4}+\frac {B c^3}{e^7 (d+e x)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{6 e^8 (d+e x)^6}-\frac {3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{5 e^8 (d+e x)^5}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{8 e^8 (d+e x)^8}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{9 e^8 (d+e x)^9}+\frac {3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{7 e^8 (d+e x)^7}+\frac {c^3 (7 B d-A e)}{3 e^8 (d+e x)^3}-\frac {B c^3}{2 e^8 (d+e x)^2}\)

Input:

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^10,x]
 

Output:

((B*d - A*e)*(c*d^2 + a*e^2)^3)/(9*e^8*(d + e*x)^9) - ((c*d^2 + a*e^2)^2*( 
7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(8*e^8*(d + e*x)^8) + (3*c*(c*d^2 + a*e^ 
2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(7*e^8*(d + e*x)^7) 
+ (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a 
^2*e^4)))/(6*e^8*(d + e*x)^6) + (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d 
*e^2 - 3*a*A*e^3))/(5*e^8*(d + e*x)^5) - (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a 
*B*e^2))/(4*e^8*(d + e*x)^4) + (c^3*(7*B*d - A*e))/(3*e^8*(d + e*x)^3) - ( 
B*c^3)/(2*e^8*(d + e*x)^2)
 

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 = 0.85 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.27

method	result	size
risch	\(\frac {-\frac {B \,c^{3} x^{7}}{2 e}-\frac {c^{3} \left (2 A e +7 B d \right ) x^{6}}{6 e^{2}}-\frac {c^{2} \left (2 A c d e +3 B a \,e^{2}+7 B c \,d^{2}\right ) x^{5}}{4 e^{3}}-\frac {c^{2} \left (12 A a \,e^{3}+10 A c \,d^{2} e +15 B a d \,e^{2}+35 B c \,d^{3}\right ) x^{4}}{20 e^{4}}-\frac {c \left (12 A a c d \,e^{3}+10 A \,c^{2} d^{3} e +15 B \,e^{4} a^{2}+15 B a c \,d^{2} e^{2}+35 B \,c^{2} d^{4}\right ) x^{3}}{30 e^{5}}-\frac {c \left (30 A \,a^{2} e^{5}+12 A a c \,d^{2} e^{3}+10 A \,c^{2} d^{4} e +15 B \,a^{2} d \,e^{4}+15 B a c \,d^{3} e^{2}+35 B \,c^{2} d^{5}\right ) x^{2}}{70 e^{6}}-\frac {\left (30 A \,a^{2} c d \,e^{5}+12 A a \,c^{2} d^{3} e^{3}+10 A \,c^{3} d^{5} e +35 B \,a^{3} e^{6}+15 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+35 B \,c^{3} d^{6}\right ) x}{280 e^{7}}-\frac {280 A \,a^{3} e^{7}+30 A \,a^{2} c \,d^{2} e^{5}+12 A a \,c^{2} d^{4} e^{3}+10 A \,c^{3} d^{6} e +35 B \,a^{3} d \,e^{6}+15 B \,a^{2} c \,d^{3} e^{4}+15 B a \,c^{2} d^{5} e^{2}+35 B \,c^{3} d^{7}}{2520 e^{8}}}{\left (e x +d \right )^{9}}\)	\(425\)
default	\(-\frac {c^{3} \left (A e -7 B d \right )}{3 e^{8} \left (e x +d \right )^{3}}+\frac {3 c^{2} \left (2 A c d e -B a \,e^{2}-7 B c \,d^{2}\right )}{4 e^{8} \left (e x +d \right )^{4}}-\frac {A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}}{9 e^{8} \left (e x +d \right )^{9}}-\frac {3 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{7 e^{8} \left (e x +d \right )^{7}}-\frac {c^{2} \left (3 A a \,e^{3}+15 A c \,d^{2} e -15 B a d \,e^{2}-35 B c \,d^{3}\right )}{5 e^{8} \left (e x +d \right )^{5}}-\frac {-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}}{8 e^{8} \left (e x +d \right )^{8}}-\frac {B \,c^{3}}{2 e^{8} \left (e x +d \right )^{2}}+\frac {c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{6 e^{8} \left (e x +d \right )^{6}}\)	\(449\)
norman	\(\frac {-\frac {B \,c^{3} x^{7}}{2 e}-\frac {\left (2 A \,c^{3} e^{2}+7 B \,c^{3} d e \right ) x^{6}}{6 e^{3}}-\frac {\left (2 A \,c^{3} d \,e^{2}+3 B \,e^{3} a \,c^{2}+7 B \,c^{3} d^{2} e \right ) x^{5}}{4 e^{4}}-\frac {\left (12 A a \,c^{2} e^{4}+10 A \,c^{3} d^{2} e^{2}+15 B a \,c^{2} d \,e^{3}+35 B \,c^{3} d^{3} e \right ) x^{4}}{20 e^{5}}-\frac {\left (12 A a \,c^{2} d \,e^{4}+10 A \,c^{3} d^{3} e^{2}+15 B \,e^{5} a^{2} c +15 B a \,c^{2} d^{2} e^{3}+35 B \,c^{3} d^{4} e \right ) x^{3}}{30 e^{6}}-\frac {\left (30 a^{2} A c \,e^{6}+12 A a \,c^{2} d^{2} e^{4}+10 A \,c^{3} d^{4} e^{2}+15 B \,a^{2} c d \,e^{5}+15 B a \,c^{2} d^{3} e^{3}+35 B \,c^{3} d^{5} e \right ) x^{2}}{70 e^{7}}-\frac {\left (30 A \,a^{2} c d \,e^{6}+12 A a \,c^{2} d^{3} e^{4}+10 A \,c^{3} d^{5} e^{2}+35 B \,a^{3} e^{7}+15 B \,a^{2} c \,d^{2} e^{5}+15 B a \,c^{2} d^{4} e^{3}+35 B \,c^{3} d^{6} e \right ) x}{280 e^{8}}-\frac {280 a^{3} A \,e^{8}+30 A \,a^{2} c \,d^{2} e^{6}+12 A a \,c^{2} d^{4} e^{4}+10 A \,c^{3} d^{6} e^{2}+35 B \,a^{3} d \,e^{7}+15 B \,a^{2} c \,d^{3} e^{5}+15 B a \,c^{2} d^{5} e^{3}+35 B \,c^{3} d^{7} e}{2520 e^{9}}}{\left (e x +d \right )^{9}}\)	\(469\)
gosper	\(-\frac {1260 B \,x^{7} c^{3} e^{7}+840 A \,x^{6} c^{3} e^{7}+2940 B \,x^{6} c^{3} d \,e^{6}+1260 A \,x^{5} c^{3} d \,e^{6}+1890 B \,x^{5} a \,c^{2} e^{7}+4410 B \,x^{5} c^{3} d^{2} e^{5}+1512 A \,x^{4} a \,c^{2} e^{7}+1260 A \,x^{4} c^{3} d^{2} e^{5}+1890 B \,x^{4} a \,c^{2} d \,e^{6}+4410 B \,x^{4} c^{3} d^{3} e^{4}+1008 A \,x^{3} a \,c^{2} d \,e^{6}+840 A \,x^{3} c^{3} d^{3} e^{4}+1260 B \,x^{3} a^{2} c \,e^{7}+1260 B \,x^{3} a \,c^{2} d^{2} e^{5}+2940 B \,x^{3} c^{3} d^{4} e^{3}+1080 A \,x^{2} a^{2} c \,e^{7}+432 A \,x^{2} a \,c^{2} d^{2} e^{5}+360 A \,x^{2} c^{3} d^{4} e^{3}+540 B \,x^{2} a^{2} c d \,e^{6}+540 B \,x^{2} a \,c^{2} d^{3} e^{4}+1260 B \,x^{2} c^{3} d^{5} e^{2}+270 A x \,a^{2} c d \,e^{6}+108 A x a \,c^{2} d^{3} e^{4}+90 A x \,c^{3} d^{5} e^{2}+315 B x \,a^{3} e^{7}+135 B x \,a^{2} c \,d^{2} e^{5}+135 B x a \,c^{2} d^{4} e^{3}+315 B x \,c^{3} d^{6} e +280 A \,a^{3} e^{7}+30 A \,a^{2} c \,d^{2} e^{5}+12 A a \,c^{2} d^{4} e^{3}+10 A \,c^{3} d^{6} e +35 B \,a^{3} d \,e^{6}+15 B \,a^{2} c \,d^{3} e^{4}+15 B a \,c^{2} d^{5} e^{2}+35 B \,c^{3} d^{7}}{2520 e^{8} \left (e x +d \right )^{9}}\)	\(489\)
orering	\(-\frac {1260 B \,x^{7} c^{3} e^{7}+840 A \,x^{6} c^{3} e^{7}+2940 B \,x^{6} c^{3} d \,e^{6}+1260 A \,x^{5} c^{3} d \,e^{6}+1890 B \,x^{5} a \,c^{2} e^{7}+4410 B \,x^{5} c^{3} d^{2} e^{5}+1512 A \,x^{4} a \,c^{2} e^{7}+1260 A \,x^{4} c^{3} d^{2} e^{5}+1890 B \,x^{4} a \,c^{2} d \,e^{6}+4410 B \,x^{4} c^{3} d^{3} e^{4}+1008 A \,x^{3} a \,c^{2} d \,e^{6}+840 A \,x^{3} c^{3} d^{3} e^{4}+1260 B \,x^{3} a^{2} c \,e^{7}+1260 B \,x^{3} a \,c^{2} d^{2} e^{5}+2940 B \,x^{3} c^{3} d^{4} e^{3}+1080 A \,x^{2} a^{2} c \,e^{7}+432 A \,x^{2} a \,c^{2} d^{2} e^{5}+360 A \,x^{2} c^{3} d^{4} e^{3}+540 B \,x^{2} a^{2} c d \,e^{6}+540 B \,x^{2} a \,c^{2} d^{3} e^{4}+1260 B \,x^{2} c^{3} d^{5} e^{2}+270 A x \,a^{2} c d \,e^{6}+108 A x a \,c^{2} d^{3} e^{4}+90 A x \,c^{3} d^{5} e^{2}+315 B x \,a^{3} e^{7}+135 B x \,a^{2} c \,d^{2} e^{5}+135 B x a \,c^{2} d^{4} e^{3}+315 B x \,c^{3} d^{6} e +280 A \,a^{3} e^{7}+30 A \,a^{2} c \,d^{2} e^{5}+12 A a \,c^{2} d^{4} e^{3}+10 A \,c^{3} d^{6} e +35 B \,a^{3} d \,e^{6}+15 B \,a^{2} c \,d^{3} e^{4}+15 B a \,c^{2} d^{5} e^{2}+35 B \,c^{3} d^{7}}{2520 e^{8} \left (e x +d \right )^{9}}\)	\(489\)
parallelrisch	\(-\frac {1260 B \,c^{3} x^{7} e^{8}+840 A \,c^{3} e^{8} x^{6}+2940 B \,c^{3} d \,e^{7} x^{6}+1260 A \,c^{3} d \,e^{7} x^{5}+1890 B a \,c^{2} e^{8} x^{5}+4410 B \,c^{3} d^{2} e^{6} x^{5}+1512 A a \,c^{2} e^{8} x^{4}+1260 A \,c^{3} d^{2} e^{6} x^{4}+1890 B a \,c^{2} d \,e^{7} x^{4}+4410 B \,c^{3} d^{3} e^{5} x^{4}+1008 A a \,c^{2} d \,e^{7} x^{3}+840 A \,c^{3} d^{3} e^{5} x^{3}+1260 B \,a^{2} c \,e^{8} x^{3}+1260 B a \,c^{2} d^{2} e^{6} x^{3}+2940 B \,c^{3} d^{4} e^{4} x^{3}+1080 A \,a^{2} c \,e^{8} x^{2}+432 A a \,c^{2} d^{2} e^{6} x^{2}+360 A \,c^{3} d^{4} e^{4} x^{2}+540 B \,a^{2} c d \,e^{7} x^{2}+540 B a \,c^{2} d^{3} e^{5} x^{2}+1260 B \,c^{3} d^{5} e^{3} x^{2}+270 A \,a^{2} c d \,e^{7} x +108 A a \,c^{2} d^{3} e^{5} x +90 A \,c^{3} d^{5} e^{3} x +315 B \,a^{3} e^{8} x +135 B \,a^{2} c \,d^{2} e^{6} x +135 B a \,c^{2} d^{4} e^{4} x +315 B \,c^{3} d^{6} e^{2} x +280 a^{3} A \,e^{8}+30 A \,a^{2} c \,d^{2} e^{6}+12 A a \,c^{2} d^{4} e^{4}+10 A \,c^{3} d^{6} e^{2}+35 B \,a^{3} d \,e^{7}+15 B \,a^{2} c \,d^{3} e^{5}+15 B a \,c^{2} d^{5} e^{3}+35 B \,c^{3} d^{7} e}{2520 e^{9} \left (e x +d \right )^{9}}\)	\(494\)

Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^10,x,method=_RETURNVERBOSE)
 

Output:

(-1/2*B*c^3*x^7/e-1/6*c^3/e^2*(2*A*e+7*B*d)*x^6-1/4/e^3*c^2*(2*A*c*d*e+3*B 
*a*e^2+7*B*c*d^2)*x^5-1/20/e^4*c^2*(12*A*a*e^3+10*A*c*d^2*e+15*B*a*d*e^2+3 
5*B*c*d^3)*x^4-1/30*c/e^5*(12*A*a*c*d*e^3+10*A*c^2*d^3*e+15*B*a^2*e^4+15*B 
*a*c*d^2*e^2+35*B*c^2*d^4)*x^3-1/70*c/e^6*(30*A*a^2*e^5+12*A*a*c*d^2*e^3+1 
0*A*c^2*d^4*e+15*B*a^2*d*e^4+15*B*a*c*d^3*e^2+35*B*c^2*d^5)*x^2-1/280/e^7* 
(30*A*a^2*c*d*e^5+12*A*a*c^2*d^3*e^3+10*A*c^3*d^5*e+35*B*a^3*e^6+15*B*a^2* 
c*d^2*e^4+15*B*a*c^2*d^4*e^2+35*B*c^3*d^6)*x-1/2520/e^8*(280*A*a^3*e^7+30* 
A*a^2*c*d^2*e^5+12*A*a*c^2*d^4*e^3+10*A*c^3*d^6*e+35*B*a^3*d*e^6+15*B*a^2* 
c*d^3*e^4+15*B*a*c^2*d^5*e^2+35*B*c^3*d^7))/(e*x+d)^9
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {1260 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 10 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} + 12 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} + 280 \, A a^{3} e^{7} + 420 \, {\left (7 \, B c^{3} d e^{6} + 2 \, A c^{3} e^{7}\right )} x^{6} + 630 \, {\left (7 \, B c^{3} d^{2} e^{5} + 2 \, A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} + 126 \, {\left (35 \, B c^{3} d^{3} e^{4} + 10 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} + 12 \, A a c^{2} e^{7}\right )} x^{4} + 84 \, {\left (35 \, B c^{3} d^{4} e^{3} + 10 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} + 12 \, A a c^{2} d e^{6} + 15 \, B a^{2} c e^{7}\right )} x^{3} + 36 \, {\left (35 \, B c^{3} d^{5} e^{2} + 10 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} + 12 \, A a c^{2} d^{2} e^{5} + 15 \, B a^{2} c d e^{6} + 30 \, A a^{2} c e^{7}\right )} x^{2} + 9 \, {\left (35 \, B c^{3} d^{6} e + 10 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} + 12 \, A a c^{2} d^{3} e^{4} + 15 \, B a^{2} c d^{2} e^{5} + 30 \, A a^{2} c d e^{6} + 35 \, B a^{3} e^{7}\right )} x}{2520 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^10,x, algorithm="fricas")
 

Output:

-1/2520*(1260*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 10*A*c^3*d^6*e + 15*B*a*c^2*d 
^5*e^2 + 12*A*a*c^2*d^4*e^3 + 15*B*a^2*c*d^3*e^4 + 30*A*a^2*c*d^2*e^5 + 35 
*B*a^3*d*e^6 + 280*A*a^3*e^7 + 420*(7*B*c^3*d*e^6 + 2*A*c^3*e^7)*x^6 + 630 
*(7*B*c^3*d^2*e^5 + 2*A*c^3*d*e^6 + 3*B*a*c^2*e^7)*x^5 + 126*(35*B*c^3*d^3 
*e^4 + 10*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 + 12*A*a*c^2*e^7)*x^4 + 84*(35* 
B*c^3*d^4*e^3 + 10*A*c^3*d^3*e^4 + 15*B*a*c^2*d^2*e^5 + 12*A*a*c^2*d*e^6 + 
 15*B*a^2*c*e^7)*x^3 + 36*(35*B*c^3*d^5*e^2 + 10*A*c^3*d^4*e^3 + 15*B*a*c^ 
2*d^3*e^4 + 12*A*a*c^2*d^2*e^5 + 15*B*a^2*c*d*e^6 + 30*A*a^2*c*e^7)*x^2 + 
9*(35*B*c^3*d^6*e + 10*A*c^3*d^5*e^2 + 15*B*a*c^2*d^4*e^3 + 12*A*a*c^2*d^3 
*e^4 + 15*B*a^2*c*d^2*e^5 + 30*A*a^2*c*d*e^6 + 35*B*a^3*e^7)*x)/(e^17*x^9 
+ 9*d*e^16*x^8 + 36*d^2*e^15*x^7 + 84*d^3*e^14*x^6 + 126*d^4*e^13*x^5 + 12 
6*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x + d^9*e^8 
)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**10,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {1260 \, B c^{3} e^{7} x^{7} + 35 \, B c^{3} d^{7} + 10 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} + 12 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} + 280 \, A a^{3} e^{7} + 420 \, {\left (7 \, B c^{3} d e^{6} + 2 \, A c^{3} e^{7}\right )} x^{6} + 630 \, {\left (7 \, B c^{3} d^{2} e^{5} + 2 \, A c^{3} d e^{6} + 3 \, B a c^{2} e^{7}\right )} x^{5} + 126 \, {\left (35 \, B c^{3} d^{3} e^{4} + 10 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} + 12 \, A a c^{2} e^{7}\right )} x^{4} + 84 \, {\left (35 \, B c^{3} d^{4} e^{3} + 10 \, A c^{3} d^{3} e^{4} + 15 \, B a c^{2} d^{2} e^{5} + 12 \, A a c^{2} d e^{6} + 15 \, B a^{2} c e^{7}\right )} x^{3} + 36 \, {\left (35 \, B c^{3} d^{5} e^{2} + 10 \, A c^{3} d^{4} e^{3} + 15 \, B a c^{2} d^{3} e^{4} + 12 \, A a c^{2} d^{2} e^{5} + 15 \, B a^{2} c d e^{6} + 30 \, A a^{2} c e^{7}\right )} x^{2} + 9 \, {\left (35 \, B c^{3} d^{6} e + 10 \, A c^{3} d^{5} e^{2} + 15 \, B a c^{2} d^{4} e^{3} + 12 \, A a c^{2} d^{3} e^{4} + 15 \, B a^{2} c d^{2} e^{5} + 30 \, A a^{2} c d e^{6} + 35 \, B a^{3} e^{7}\right )} x}{2520 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} e^{8}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^10,x, algorithm="maxima")
 

Output:

-1/2520*(1260*B*c^3*e^7*x^7 + 35*B*c^3*d^7 + 10*A*c^3*d^6*e + 15*B*a*c^2*d 
^5*e^2 + 12*A*a*c^2*d^4*e^3 + 15*B*a^2*c*d^3*e^4 + 30*A*a^2*c*d^2*e^5 + 35 
*B*a^3*d*e^6 + 280*A*a^3*e^7 + 420*(7*B*c^3*d*e^6 + 2*A*c^3*e^7)*x^6 + 630 
*(7*B*c^3*d^2*e^5 + 2*A*c^3*d*e^6 + 3*B*a*c^2*e^7)*x^5 + 126*(35*B*c^3*d^3 
*e^4 + 10*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 + 12*A*a*c^2*e^7)*x^4 + 84*(35* 
B*c^3*d^4*e^3 + 10*A*c^3*d^3*e^4 + 15*B*a*c^2*d^2*e^5 + 12*A*a*c^2*d*e^6 + 
 15*B*a^2*c*e^7)*x^3 + 36*(35*B*c^3*d^5*e^2 + 10*A*c^3*d^4*e^3 + 15*B*a*c^ 
2*d^3*e^4 + 12*A*a*c^2*d^2*e^5 + 15*B*a^2*c*d*e^6 + 30*A*a^2*c*e^7)*x^2 + 
9*(35*B*c^3*d^6*e + 10*A*c^3*d^5*e^2 + 15*B*a*c^2*d^4*e^3 + 12*A*a*c^2*d^3 
*e^4 + 15*B*a^2*c*d^2*e^5 + 30*A*a^2*c*d*e^6 + 35*B*a^3*e^7)*x)/(e^17*x^9 
+ 9*d*e^16*x^8 + 36*d^2*e^15*x^7 + 84*d^3*e^14*x^6 + 126*d^4*e^13*x^5 + 12 
6*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x + d^9*e^8 
)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {1260 \, B c^{3} e^{7} x^{7} + 2940 \, B c^{3} d e^{6} x^{6} + 840 \, A c^{3} e^{7} x^{6} + 4410 \, B c^{3} d^{2} e^{5} x^{5} + 1260 \, A c^{3} d e^{6} x^{5} + 1890 \, B a c^{2} e^{7} x^{5} + 4410 \, B c^{3} d^{3} e^{4} x^{4} + 1260 \, A c^{3} d^{2} e^{5} x^{4} + 1890 \, B a c^{2} d e^{6} x^{4} + 1512 \, A a c^{2} e^{7} x^{4} + 2940 \, B c^{3} d^{4} e^{3} x^{3} + 840 \, A c^{3} d^{3} e^{4} x^{3} + 1260 \, B a c^{2} d^{2} e^{5} x^{3} + 1008 \, A a c^{2} d e^{6} x^{3} + 1260 \, B a^{2} c e^{7} x^{3} + 1260 \, B c^{3} d^{5} e^{2} x^{2} + 360 \, A c^{3} d^{4} e^{3} x^{2} + 540 \, B a c^{2} d^{3} e^{4} x^{2} + 432 \, A a c^{2} d^{2} e^{5} x^{2} + 540 \, B a^{2} c d e^{6} x^{2} + 1080 \, A a^{2} c e^{7} x^{2} + 315 \, B c^{3} d^{6} e x + 90 \, A c^{3} d^{5} e^{2} x + 135 \, B a c^{2} d^{4} e^{3} x + 108 \, A a c^{2} d^{3} e^{4} x + 135 \, B a^{2} c d^{2} e^{5} x + 270 \, A a^{2} c d e^{6} x + 315 \, B a^{3} e^{7} x + 35 \, B c^{3} d^{7} + 10 \, A c^{3} d^{6} e + 15 \, B a c^{2} d^{5} e^{2} + 12 \, A a c^{2} d^{4} e^{3} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a^{2} c d^{2} e^{5} + 35 \, B a^{3} d e^{6} + 280 \, A a^{3} e^{7}}{2520 \, {\left (e x + d\right )}^{9} e^{8}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^10,x, algorithm="giac")
 

Output:

-1/2520*(1260*B*c^3*e^7*x^7 + 2940*B*c^3*d*e^6*x^6 + 840*A*c^3*e^7*x^6 + 4 
410*B*c^3*d^2*e^5*x^5 + 1260*A*c^3*d*e^6*x^5 + 1890*B*a*c^2*e^7*x^5 + 4410 
*B*c^3*d^3*e^4*x^4 + 1260*A*c^3*d^2*e^5*x^4 + 1890*B*a*c^2*d*e^6*x^4 + 151 
2*A*a*c^2*e^7*x^4 + 2940*B*c^3*d^4*e^3*x^3 + 840*A*c^3*d^3*e^4*x^3 + 1260* 
B*a*c^2*d^2*e^5*x^3 + 1008*A*a*c^2*d*e^6*x^3 + 1260*B*a^2*c*e^7*x^3 + 1260 
*B*c^3*d^5*e^2*x^2 + 360*A*c^3*d^4*e^3*x^2 + 540*B*a*c^2*d^3*e^4*x^2 + 432 
*A*a*c^2*d^2*e^5*x^2 + 540*B*a^2*c*d*e^6*x^2 + 1080*A*a^2*c*e^7*x^2 + 315* 
B*c^3*d^6*e*x + 90*A*c^3*d^5*e^2*x + 135*B*a*c^2*d^4*e^3*x + 108*A*a*c^2*d 
^3*e^4*x + 135*B*a^2*c*d^2*e^5*x + 270*A*a^2*c*d*e^6*x + 315*B*a^3*e^7*x + 
 35*B*c^3*d^7 + 10*A*c^3*d^6*e + 15*B*a*c^2*d^5*e^2 + 12*A*a*c^2*d^4*e^3 + 
 15*B*a^2*c*d^3*e^4 + 30*A*a^2*c*d^2*e^5 + 35*B*a^3*d*e^6 + 280*A*a^3*e^7) 
/((e*x + d)^9*e^8)
 

Mupad [B] (verification not implemented)

Time = 6.02 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {\frac {35\,B\,a^3\,d\,e^6+280\,A\,a^3\,e^7+15\,B\,a^2\,c\,d^3\,e^4+30\,A\,a^2\,c\,d^2\,e^5+15\,B\,a\,c^2\,d^5\,e^2+12\,A\,a\,c^2\,d^4\,e^3+35\,B\,c^3\,d^7+10\,A\,c^3\,d^6\,e}{2520\,e^8}+\frac {x\,\left (35\,B\,a^3\,e^6+15\,B\,a^2\,c\,d^2\,e^4+30\,A\,a^2\,c\,d\,e^5+15\,B\,a\,c^2\,d^4\,e^2+12\,A\,a\,c^2\,d^3\,e^3+35\,B\,c^3\,d^6+10\,A\,c^3\,d^5\,e\right )}{280\,e^7}+\frac {c^2\,x^4\,\left (35\,B\,c\,d^3+10\,A\,c\,d^2\,e+15\,B\,a\,d\,e^2+12\,A\,a\,e^3\right )}{20\,e^4}+\frac {c\,x^3\,\left (15\,B\,a^2\,e^4+15\,B\,a\,c\,d^2\,e^2+12\,A\,a\,c\,d\,e^3+35\,B\,c^2\,d^4+10\,A\,c^2\,d^3\,e\right )}{30\,e^5}+\frac {c^3\,x^6\,\left (2\,A\,e+7\,B\,d\right )}{6\,e^2}+\frac {c^2\,x^5\,\left (7\,B\,c\,d^2+2\,A\,c\,d\,e+3\,B\,a\,e^2\right )}{4\,e^3}+\frac {c\,x^2\,\left (15\,B\,a^2\,d\,e^4+30\,A\,a^2\,e^5+15\,B\,a\,c\,d^3\,e^2+12\,A\,a\,c\,d^2\,e^3+35\,B\,c^2\,d^5+10\,A\,c^2\,d^4\,e\right )}{70\,e^6}+\frac {B\,c^3\,x^7}{2\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \] Input:

int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^10,x)
 

Output:

-((280*A*a^3*e^7 + 35*B*c^3*d^7 + 35*B*a^3*d*e^6 + 10*A*c^3*d^6*e + 12*A*a 
*c^2*d^4*e^3 + 30*A*a^2*c*d^2*e^5 + 15*B*a*c^2*d^5*e^2 + 15*B*a^2*c*d^3*e^ 
4)/(2520*e^8) + (x*(35*B*a^3*e^6 + 35*B*c^3*d^6 + 10*A*c^3*d^5*e + 12*A*a* 
c^2*d^3*e^3 + 15*B*a*c^2*d^4*e^2 + 15*B*a^2*c*d^2*e^4 + 30*A*a^2*c*d*e^5)) 
/(280*e^7) + (c^2*x^4*(12*A*a*e^3 + 35*B*c*d^3 + 15*B*a*d*e^2 + 10*A*c*d^2 
*e))/(20*e^4) + (c*x^3*(15*B*a^2*e^4 + 35*B*c^2*d^4 + 10*A*c^2*d^3*e + 12* 
A*a*c*d*e^3 + 15*B*a*c*d^2*e^2))/(30*e^5) + (c^3*x^6*(2*A*e + 7*B*d))/(6*e 
^2) + (c^2*x^5*(3*B*a*e^2 + 7*B*c*d^2 + 2*A*c*d*e))/(4*e^3) + (c*x^2*(30*A 
*a^2*e^5 + 35*B*c^2*d^5 + 15*B*a^2*d*e^4 + 10*A*c^2*d^4*e + 12*A*a*c*d^2*e 
^3 + 15*B*a*c*d^3*e^2))/(70*e^6) + (B*c^3*x^7)/(2*e))/(d^9 + e^9*x^9 + 9*d 
*e^8*x^8 + 36*d^7*e^2*x^2 + 84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e^5 
*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.73 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=\frac {-1260 b \,c^{3} e^{7} x^{7}-840 a \,c^{3} e^{7} x^{6}-2940 b \,c^{3} d \,e^{6} x^{6}-1890 a b \,c^{2} e^{7} x^{5}-1260 a \,c^{3} d \,e^{6} x^{5}-4410 b \,c^{3} d^{2} e^{5} x^{5}-1512 a^{2} c^{2} e^{7} x^{4}-1890 a b \,c^{2} d \,e^{6} x^{4}-1260 a \,c^{3} d^{2} e^{5} x^{4}-4410 b \,c^{3} d^{3} e^{4} x^{4}-1260 a^{2} b c \,e^{7} x^{3}-1008 a^{2} c^{2} d \,e^{6} x^{3}-1260 a b \,c^{2} d^{2} e^{5} x^{3}-840 a \,c^{3} d^{3} e^{4} x^{3}-2940 b \,c^{3} d^{4} e^{3} x^{3}-1080 a^{3} c \,e^{7} x^{2}-540 a^{2} b c d \,e^{6} x^{2}-432 a^{2} c^{2} d^{2} e^{5} x^{2}-540 a b \,c^{2} d^{3} e^{4} x^{2}-360 a \,c^{3} d^{4} e^{3} x^{2}-1260 b \,c^{3} d^{5} e^{2} x^{2}-315 a^{3} b \,e^{7} x -270 a^{3} c d \,e^{6} x -135 a^{2} b c \,d^{2} e^{5} x -108 a^{2} c^{2} d^{3} e^{4} x -135 a b \,c^{2} d^{4} e^{3} x -90 a \,c^{3} d^{5} e^{2} x -315 b \,c^{3} d^{6} e x -280 a^{4} e^{7}-35 a^{3} b d \,e^{6}-30 a^{3} c \,d^{2} e^{5}-15 a^{2} b c \,d^{3} e^{4}-12 a^{2} c^{2} d^{4} e^{3}-15 a b \,c^{2} d^{5} e^{2}-10 a \,c^{3} d^{6} e -35 b \,c^{3} d^{7}}{2520 e^{8} \left (e^{9} x^{9}+9 d \,e^{8} x^{8}+36 d^{2} e^{7} x^{7}+84 d^{3} e^{6} x^{6}+126 d^{4} e^{5} x^{5}+126 d^{5} e^{4} x^{4}+84 d^{6} e^{3} x^{3}+36 d^{7} e^{2} x^{2}+9 d^{8} e x +d^{9}\right )} \] Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^10,x)
 

Output:

( - 280*a**4*e**7 - 35*a**3*b*d*e**6 - 315*a**3*b*e**7*x - 30*a**3*c*d**2* 
e**5 - 270*a**3*c*d*e**6*x - 1080*a**3*c*e**7*x**2 - 15*a**2*b*c*d**3*e**4 
 - 135*a**2*b*c*d**2*e**5*x - 540*a**2*b*c*d*e**6*x**2 - 1260*a**2*b*c*e** 
7*x**3 - 12*a**2*c**2*d**4*e**3 - 108*a**2*c**2*d**3*e**4*x - 432*a**2*c** 
2*d**2*e**5*x**2 - 1008*a**2*c**2*d*e**6*x**3 - 1512*a**2*c**2*e**7*x**4 - 
 15*a*b*c**2*d**5*e**2 - 135*a*b*c**2*d**4*e**3*x - 540*a*b*c**2*d**3*e**4 
*x**2 - 1260*a*b*c**2*d**2*e**5*x**3 - 1890*a*b*c**2*d*e**6*x**4 - 1890*a* 
b*c**2*e**7*x**5 - 10*a*c**3*d**6*e - 90*a*c**3*d**5*e**2*x - 360*a*c**3*d 
**4*e**3*x**2 - 840*a*c**3*d**3*e**4*x**3 - 1260*a*c**3*d**2*e**5*x**4 - 1 
260*a*c**3*d*e**6*x**5 - 840*a*c**3*e**7*x**6 - 35*b*c**3*d**7 - 315*b*c** 
3*d**6*e*x - 1260*b*c**3*d**5*e**2*x**2 - 2940*b*c**3*d**4*e**3*x**3 - 441 
0*b*c**3*d**3*e**4*x**4 - 4410*b*c**3*d**2*e**5*x**5 - 2940*b*c**3*d*e**6* 
x**6 - 1260*b*c**3*e**7*x**7)/(2520*e**8*(d**9 + 9*d**8*e*x + 36*d**7*e**2 
*x**2 + 84*d**6*e**3*x**3 + 126*d**5*e**4*x**4 + 126*d**4*e**5*x**5 + 84*d 
**3*e**6*x**6 + 36*d**2*e**7*x**7 + 9*d*e**8*x**8 + e**9*x**9))