\(\int \frac {(A+B x) (a+b x+c x^2)^2}{(d+e x)^7} \, dx\) [729]

Optimal result

Integrand size = 25, antiderivative size = 300 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{6 e^6 (d+e x)^6}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{5 e^6 (d+e x)^5}+\frac {B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{4 e^6 (d+e x)^4}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{3 e^6 (d+e x)^3}+\frac {c (5 B c d-2 b B e-A c e)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (d+e x)} \] Output:

1/6*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^2/e^6/(e*x+d)^6-1/5*(a*e^2-b*d*e+c*d^2) 
*(5*B*c*d^2-B*e*(-a*e+3*b*d)-2*A*e*(-b*e+2*c*d))/e^6/(e*x+d)^5+1/4*(B*(10* 
c^2*d^3+b*e^2*(-2*a*e+3*b*d)-6*c*d*e*(-a*e+2*b*d))-A*e*(6*c^2*d^2+b^2*e^2- 
2*c*e*(-a*e+3*b*d)))/e^6/(e*x+d)^4+1/3*(2*A*c*e*(-b*e+2*c*d)-B*(10*c^2*d^2 
+b^2*e^2-2*c*e*(-a*e+4*b*d)))/e^6/(e*x+d)^3+1/2*c*(-A*c*e-2*B*b*e+5*B*c*d) 
/e^6/(e*x+d)^2-B*c^2/e^6/(e*x+d)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.24 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {A e \left (2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+e^2 \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 c e \left (a e \left (d^2+6 d e x+15 e^2 x^2\right )+b \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )\right )+B \left (10 c^2 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+e^2 \left (2 a^2 e^2 (d+6 e x)+2 a b e \left (d^2+6 d e x+15 e^2 x^2\right )+b^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+2 c e \left (a e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 b \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )\right )}{60 e^6 (d+e x)^6} \] Input:

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^7,x]
 

Output:

-1/60*(A*e*(2*c^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^ 
4*x^4) + e^2*(10*a^2*e^2 + 4*a*b*e*(d + 6*e*x) + b^2*(d^2 + 6*d*e*x + 15*e 
^2*x^2)) + 2*c*e*(a*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b*(d^3 + 6*d^2*e*x + 
15*d*e^2*x^2 + 20*e^3*x^3))) + B*(10*c^2*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 
 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) + e^2*(2*a^2*e^2*(d + 6*e*x) 
 + 2*a*b*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b^2*(d^3 + 6*d^2*e*x + 15*d*e^2* 
x^2 + 20*e^3*x^3)) + 2*c*e*(a*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x 
^3) + 2*b*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4))) 
)/(e^6*(d + e*x)^6)
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1195, 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.

\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (\frac {B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )-2 A c e (2 c d-b e)}{e^5 (d+e x)^4}+\frac {A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )}{e^5 (d+e x)^5}+\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^5 (d+e x)^6}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )^2}{e^5 (d+e x)^7}+\frac {c (A c e+2 b B e-5 B c d)}{e^5 (d+e x)^3}+\frac {B c^2}{e^5 (d+e x)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac {B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{4 e^6 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{5 e^6 (d+e x)^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{6 e^6 (d+e x)^6}+\frac {c (-A c e-2 b B e+5 B c d)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (d+e x)}\)

Input:

Int[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^7,x]
 

Output:

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(6*e^6*(d + e*x)^6) - ((c*d^2 - b* 
d*e + a*e^2)*(5*B*c*d^2 - B*e*(3*b*d - a*e) - 2*A*e*(2*c*d - b*e)))/(5*e^6 
*(d + e*x)^5) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d - 
a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(4*e^6*(d + e*x)^ 
4) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e 
)))/(3*e^6*(d + e*x)^3) + (c*(5*B*c*d - 2*b*B*e - A*c*e))/(2*e^6*(d + e*x) 
^2) - (B*c^2)/(e^6*(d + e*x))
 

Defintions of rubi rules used

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, 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 = 1.48 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.44

method	result	size
risch	\(\frac {-\frac {B \,c^{2} x^{5}}{e}-\frac {c \left (A c e +2 B b e +5 B c d \right ) x^{4}}{2 e^{2}}-\frac {\left (2 A b c \,e^{2}+2 A \,c^{2} d e +2 B \,e^{2} a c +B \,e^{2} b^{2}+4 B b c d e +10 B \,c^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {\left (2 A a c \,e^{3}+A \,b^{2} e^{3}+2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +2 B a b \,e^{3}+2 B a d \,e^{2} c +B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right ) x^{2}}{4 e^{4}}-\frac {\left (4 A a b \,e^{4}+2 A a c d \,e^{3}+A \,b^{2} d \,e^{3}+2 A b c \,d^{2} e^{2}+2 A \,c^{2} d^{3} e +2 B \,e^{4} a^{2}+2 B a b d \,e^{3}+2 B a c \,d^{2} e^{2}+B \,b^{2} d^{2} e^{2}+4 B b c \,d^{3} e +10 B \,c^{2} d^{4}\right ) x}{10 e^{5}}-\frac {10 A \,a^{2} e^{5}+4 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}+2 B a c \,d^{3} e^{2}+B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{60 e^{6}}}{\left (e x +d \right )^{6}}\)	\(431\)
norman	\(\frac {-\frac {B \,c^{2} x^{5}}{e}-\frac {\left (A \,c^{2} e +2 B e b c +5 B \,c^{2} d \right ) x^{4}}{2 e^{2}}-\frac {\left (2 A b c \,e^{2}+2 A \,c^{2} d e +2 B \,e^{2} a c +B \,e^{2} b^{2}+4 B b c d e +10 B \,c^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {\left (2 A a c \,e^{3}+A \,b^{2} e^{3}+2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +2 B a b \,e^{3}+2 B a d \,e^{2} c +B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right ) x^{2}}{4 e^{4}}-\frac {\left (4 A a b \,e^{4}+2 A a c d \,e^{3}+A \,b^{2} d \,e^{3}+2 A b c \,d^{2} e^{2}+2 A \,c^{2} d^{3} e +2 B \,e^{4} a^{2}+2 B a b d \,e^{3}+2 B a c \,d^{2} e^{2}+B \,b^{2} d^{2} e^{2}+4 B b c \,d^{3} e +10 B \,c^{2} d^{4}\right ) x}{10 e^{5}}-\frac {10 A \,a^{2} e^{5}+4 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}+2 B a c \,d^{3} e^{2}+B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{60 e^{6}}}{\left (e x +d \right )^{6}}\)	\(435\)
default	\(-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +2 B \,e^{2} a c +B \,e^{2} b^{2}-8 B b c d e +10 B \,c^{2} d^{2}}{3 e^{6} \left (e x +d \right )^{3}}-\frac {2 A a c \,e^{3}+A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e +2 B a b \,e^{3}-6 B a d \,e^{2} c -3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}}{4 e^{6} \left (e x +d \right )^{4}}-\frac {2 A a b \,e^{4}-4 A a c d \,e^{3}-2 A \,b^{2} d \,e^{3}+6 A b c \,d^{2} e^{2}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}-4 B a b d \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}-8 B b c \,d^{3} e +5 B \,c^{2} d^{4}}{5 e^{6} \left (e x +d \right )^{5}}-\frac {B \,c^{2}}{e^{6} \left (e x +d \right )}-\frac {c \left (A c e +2 B b e -5 B c d \right )}{2 e^{6} \left (e x +d \right )^{2}}-\frac {A \,a^{2} e^{5}-2 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}-2 A b c \,d^{3} e^{2}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}-2 B a c \,d^{3} e^{2}-B \,b^{2} d^{3} e^{2}+2 B b c \,d^{4} e -B \,c^{2} d^{5}}{6 e^{6} \left (e x +d \right )^{6}}\)	\(453\)
gosper	\(-\frac {60 B \,x^{5} c^{2} e^{5}+30 A \,x^{4} c^{2} e^{5}+60 B \,x^{4} b c \,e^{5}+150 B \,x^{4} c^{2} d \,e^{4}+40 A \,x^{3} b c \,e^{5}+40 A \,x^{3} c^{2} d \,e^{4}+40 B \,x^{3} a c \,e^{5}+20 B \,x^{3} b^{2} e^{5}+80 B \,x^{3} b c d \,e^{4}+200 B \,x^{3} c^{2} d^{2} e^{3}+30 A \,x^{2} a c \,e^{5}+15 A \,x^{2} b^{2} e^{5}+30 A \,x^{2} b c d \,e^{4}+30 A \,x^{2} c^{2} d^{2} e^{3}+30 B \,x^{2} a b \,e^{5}+30 B \,x^{2} a c d \,e^{4}+15 B \,x^{2} b^{2} d \,e^{4}+60 B \,x^{2} b c \,d^{2} e^{3}+150 B \,x^{2} c^{2} d^{3} e^{2}+24 A x a b \,e^{5}+12 A x a c d \,e^{4}+6 A x \,b^{2} d \,e^{4}+12 A x b c \,d^{2} e^{3}+12 A x \,c^{2} d^{3} e^{2}+12 B x \,a^{2} e^{5}+12 B x a b d \,e^{4}+12 B x a c \,d^{2} e^{3}+6 B x \,b^{2} d^{2} e^{3}+24 B x b c \,d^{3} e^{2}+60 B x \,c^{2} d^{4} e +10 A \,a^{2} e^{5}+4 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}+2 B a c \,d^{3} e^{2}+B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{60 \left (e x +d \right )^{6} e^{6}}\)	\(496\)
parallelrisch	\(-\frac {60 B \,x^{5} c^{2} e^{5}+30 A \,x^{4} c^{2} e^{5}+60 B \,x^{4} b c \,e^{5}+150 B \,x^{4} c^{2} d \,e^{4}+40 A \,x^{3} b c \,e^{5}+40 A \,x^{3} c^{2} d \,e^{4}+40 B \,x^{3} a c \,e^{5}+20 B \,x^{3} b^{2} e^{5}+80 B \,x^{3} b c d \,e^{4}+200 B \,x^{3} c^{2} d^{2} e^{3}+30 A \,x^{2} a c \,e^{5}+15 A \,x^{2} b^{2} e^{5}+30 A \,x^{2} b c d \,e^{4}+30 A \,x^{2} c^{2} d^{2} e^{3}+30 B \,x^{2} a b \,e^{5}+30 B \,x^{2} a c d \,e^{4}+15 B \,x^{2} b^{2} d \,e^{4}+60 B \,x^{2} b c \,d^{2} e^{3}+150 B \,x^{2} c^{2} d^{3} e^{2}+24 A x a b \,e^{5}+12 A x a c d \,e^{4}+6 A x \,b^{2} d \,e^{4}+12 A x b c \,d^{2} e^{3}+12 A x \,c^{2} d^{3} e^{2}+12 B x \,a^{2} e^{5}+12 B x a b d \,e^{4}+12 B x a c \,d^{2} e^{3}+6 B x \,b^{2} d^{2} e^{3}+24 B x b c \,d^{3} e^{2}+60 B x \,c^{2} d^{4} e +10 A \,a^{2} e^{5}+4 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}+2 B a c \,d^{3} e^{2}+B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{60 \left (e x +d \right )^{6} e^{6}}\)	\(496\)
orering	\(-\frac {60 B \,x^{5} c^{2} e^{5}+30 A \,x^{4} c^{2} e^{5}+60 B \,x^{4} b c \,e^{5}+150 B \,x^{4} c^{2} d \,e^{4}+40 A \,x^{3} b c \,e^{5}+40 A \,x^{3} c^{2} d \,e^{4}+40 B \,x^{3} a c \,e^{5}+20 B \,x^{3} b^{2} e^{5}+80 B \,x^{3} b c d \,e^{4}+200 B \,x^{3} c^{2} d^{2} e^{3}+30 A \,x^{2} a c \,e^{5}+15 A \,x^{2} b^{2} e^{5}+30 A \,x^{2} b c d \,e^{4}+30 A \,x^{2} c^{2} d^{2} e^{3}+30 B \,x^{2} a b \,e^{5}+30 B \,x^{2} a c d \,e^{4}+15 B \,x^{2} b^{2} d \,e^{4}+60 B \,x^{2} b c \,d^{2} e^{3}+150 B \,x^{2} c^{2} d^{3} e^{2}+24 A x a b \,e^{5}+12 A x a c d \,e^{4}+6 A x \,b^{2} d \,e^{4}+12 A x b c \,d^{2} e^{3}+12 A x \,c^{2} d^{3} e^{2}+12 B x \,a^{2} e^{5}+12 B x a b d \,e^{4}+12 B x a c \,d^{2} e^{3}+6 B x \,b^{2} d^{2} e^{3}+24 B x b c \,d^{3} e^{2}+60 B x \,c^{2} d^{4} e +10 A \,a^{2} e^{5}+4 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}+2 B a c \,d^{3} e^{2}+B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{60 \left (e x +d \right )^{6} e^{6}}\)	\(496\)

Input:

int((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^7,x,method=_RETURNVERBOSE)
 

Output:

(-B*c^2*x^5/e-1/2*c*(A*c*e+2*B*b*e+5*B*c*d)/e^2*x^4-1/3*(2*A*b*c*e^2+2*A*c 
^2*d*e+2*B*a*c*e^2+B*b^2*e^2+4*B*b*c*d*e+10*B*c^2*d^2)/e^3*x^3-1/4*(2*A*a* 
c*e^3+A*b^2*e^3+2*A*b*c*d*e^2+2*A*c^2*d^2*e+2*B*a*b*e^3+2*B*a*c*d*e^2+B*b^ 
2*d*e^2+4*B*b*c*d^2*e+10*B*c^2*d^3)/e^4*x^2-1/10*(4*A*a*b*e^4+2*A*a*c*d*e^ 
3+A*b^2*d*e^3+2*A*b*c*d^2*e^2+2*A*c^2*d^3*e+2*B*a^2*e^4+2*B*a*b*d*e^3+2*B* 
a*c*d^2*e^2+B*b^2*d^2*e^2+4*B*b*c*d^3*e+10*B*c^2*d^4)/e^5*x-1/60*(10*A*a^2 
*e^5+4*A*a*b*d*e^4+2*A*a*c*d^2*e^3+A*b^2*d^2*e^3+2*A*b*c*d^3*e^2+2*A*c^2*d 
^4*e+2*B*a^2*d*e^4+2*B*a*b*d^2*e^3+2*B*a*c*d^3*e^2+B*b^2*d^3*e^2+4*B*b*c*d 
^4*e+10*B*c^2*d^5)/e^6)/(e*x+d)^6
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {60 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 10 \, A a^{2} e^{5} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 30 \, {\left (5 \, B c^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \, {\left (10 \, B c^{2} d^{2} e^{3} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 15 \, {\left (10 \, B c^{2} d^{3} e^{2} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 6 \, {\left (10 \, B c^{2} d^{4} e + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{60 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^7,x, algorithm="fricas")
 

Output:

-1/60*(60*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 10*A*a^2*e^5 + 2*(2*B*b*c + A*c^2 
)*d^4*e + (B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + (2*B*a*b + A*b^2 + 2*A*a*c)* 
d^2*e^3 + 2*(B*a^2 + 2*A*a*b)*d*e^4 + 30*(5*B*c^2*d*e^4 + (2*B*b*c + A*c^2 
)*e^5)*x^4 + 20*(10*B*c^2*d^2*e^3 + 2*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2 
*(B*a + A*b)*c)*e^5)*x^3 + 15*(10*B*c^2*d^3*e^2 + 2*(2*B*b*c + A*c^2)*d^2* 
e^3 + (B*b^2 + 2*(B*a + A*b)*c)*d*e^4 + (2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x 
^2 + 6*(10*B*c^2*d^4*e + 2*(2*B*b*c + A*c^2)*d^3*e^2 + (B*b^2 + 2*(B*a + A 
*b)*c)*d^2*e^3 + (2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + 2*(B*a^2 + 2*A*a*b)*e 
^5)*x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^ 
4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(c*x**2+b*x+a)**2/(e*x+d)**7,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {60 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 10 \, A a^{2} e^{5} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 30 \, {\left (5 \, B c^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \, {\left (10 \, B c^{2} d^{2} e^{3} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 15 \, {\left (10 \, B c^{2} d^{3} e^{2} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 6 \, {\left (10 \, B c^{2} d^{4} e + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{60 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^7,x, algorithm="maxima")
 

Output:

-1/60*(60*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 10*A*a^2*e^5 + 2*(2*B*b*c + A*c^2 
)*d^4*e + (B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + (2*B*a*b + A*b^2 + 2*A*a*c)* 
d^2*e^3 + 2*(B*a^2 + 2*A*a*b)*d*e^4 + 30*(5*B*c^2*d*e^4 + (2*B*b*c + A*c^2 
)*e^5)*x^4 + 20*(10*B*c^2*d^2*e^3 + 2*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2 
*(B*a + A*b)*c)*e^5)*x^3 + 15*(10*B*c^2*d^3*e^2 + 2*(2*B*b*c + A*c^2)*d^2* 
e^3 + (B*b^2 + 2*(B*a + A*b)*c)*d*e^4 + (2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x 
^2 + 6*(10*B*c^2*d^4*e + 2*(2*B*b*c + A*c^2)*d^3*e^2 + (B*b^2 + 2*(B*a + A 
*b)*c)*d^2*e^3 + (2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + 2*(B*a^2 + 2*A*a*b)*e 
^5)*x)/(e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^ 
4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.65 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {60 \, B c^{2} e^{5} x^{5} + 150 \, B c^{2} d e^{4} x^{4} + 60 \, B b c e^{5} x^{4} + 30 \, A c^{2} e^{5} x^{4} + 200 \, B c^{2} d^{2} e^{3} x^{3} + 80 \, B b c d e^{4} x^{3} + 40 \, A c^{2} d e^{4} x^{3} + 20 \, B b^{2} e^{5} x^{3} + 40 \, B a c e^{5} x^{3} + 40 \, A b c e^{5} x^{3} + 150 \, B c^{2} d^{3} e^{2} x^{2} + 60 \, B b c d^{2} e^{3} x^{2} + 30 \, A c^{2} d^{2} e^{3} x^{2} + 15 \, B b^{2} d e^{4} x^{2} + 30 \, B a c d e^{4} x^{2} + 30 \, A b c d e^{4} x^{2} + 30 \, B a b e^{5} x^{2} + 15 \, A b^{2} e^{5} x^{2} + 30 \, A a c e^{5} x^{2} + 60 \, B c^{2} d^{4} e x + 24 \, B b c d^{3} e^{2} x + 12 \, A c^{2} d^{3} e^{2} x + 6 \, B b^{2} d^{2} e^{3} x + 12 \, B a c d^{2} e^{3} x + 12 \, A b c d^{2} e^{3} x + 12 \, B a b d e^{4} x + 6 \, A b^{2} d e^{4} x + 12 \, A a c d e^{4} x + 12 \, B a^{2} e^{5} x + 24 \, A a b e^{5} x + 10 \, B c^{2} d^{5} + 4 \, B b c d^{4} e + 2 \, A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, B a c d^{3} e^{2} + 2 \, A b c d^{3} e^{2} + 2 \, B a b d^{2} e^{3} + A b^{2} d^{2} e^{3} + 2 \, A a c d^{2} e^{3} + 2 \, B a^{2} d e^{4} + 4 \, A a b d e^{4} + 10 \, A a^{2} e^{5}}{60 \, {\left (e x + d\right )}^{6} e^{6}} \] Input:

integrate((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^7,x, algorithm="giac")
 

Output:

-1/60*(60*B*c^2*e^5*x^5 + 150*B*c^2*d*e^4*x^4 + 60*B*b*c*e^5*x^4 + 30*A*c^ 
2*e^5*x^4 + 200*B*c^2*d^2*e^3*x^3 + 80*B*b*c*d*e^4*x^3 + 40*A*c^2*d*e^4*x^ 
3 + 20*B*b^2*e^5*x^3 + 40*B*a*c*e^5*x^3 + 40*A*b*c*e^5*x^3 + 150*B*c^2*d^3 
*e^2*x^2 + 60*B*b*c*d^2*e^3*x^2 + 30*A*c^2*d^2*e^3*x^2 + 15*B*b^2*d*e^4*x^ 
2 + 30*B*a*c*d*e^4*x^2 + 30*A*b*c*d*e^4*x^2 + 30*B*a*b*e^5*x^2 + 15*A*b^2* 
e^5*x^2 + 30*A*a*c*e^5*x^2 + 60*B*c^2*d^4*e*x + 24*B*b*c*d^3*e^2*x + 12*A* 
c^2*d^3*e^2*x + 6*B*b^2*d^2*e^3*x + 12*B*a*c*d^2*e^3*x + 12*A*b*c*d^2*e^3* 
x + 12*B*a*b*d*e^4*x + 6*A*b^2*d*e^4*x + 12*A*a*c*d*e^4*x + 12*B*a^2*e^5*x 
 + 24*A*a*b*e^5*x + 10*B*c^2*d^5 + 4*B*b*c*d^4*e + 2*A*c^2*d^4*e + B*b^2*d 
^3*e^2 + 2*B*a*c*d^3*e^2 + 2*A*b*c*d^3*e^2 + 2*B*a*b*d^2*e^3 + A*b^2*d^2*e 
^3 + 2*A*a*c*d^2*e^3 + 2*B*a^2*d*e^4 + 4*A*a*b*d*e^4 + 10*A*a^2*e^5)/((e*x 
 + d)^6*e^6)
 

Mupad [B] (verification not implemented)

Time = 11.52 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.62 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\frac {2\,B\,a^2\,d\,e^4+10\,A\,a^2\,e^5+2\,B\,a\,b\,d^2\,e^3+4\,A\,a\,b\,d\,e^4+2\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3+B\,b^2\,d^3\,e^2+A\,b^2\,d^2\,e^3+4\,B\,b\,c\,d^4\,e+2\,A\,b\,c\,d^3\,e^2+10\,B\,c^2\,d^5+2\,A\,c^2\,d^4\,e}{60\,e^6}+\frac {x^3\,\left (B\,b^2\,e^2+4\,B\,b\,c\,d\,e+2\,A\,b\,c\,e^2+10\,B\,c^2\,d^2+2\,A\,c^2\,d\,e+2\,B\,a\,c\,e^2\right )}{3\,e^3}+\frac {x^2\,\left (B\,b^2\,d\,e^2+A\,b^2\,e^3+4\,B\,b\,c\,d^2\,e+2\,A\,b\,c\,d\,e^2+2\,B\,a\,b\,e^3+10\,B\,c^2\,d^3+2\,A\,c^2\,d^2\,e+2\,B\,a\,c\,d\,e^2+2\,A\,a\,c\,e^3\right )}{4\,e^4}+\frac {x\,\left (2\,B\,a^2\,e^4+2\,B\,a\,b\,d\,e^3+4\,A\,a\,b\,e^4+2\,B\,a\,c\,d^2\,e^2+2\,A\,a\,c\,d\,e^3+B\,b^2\,d^2\,e^2+A\,b^2\,d\,e^3+4\,B\,b\,c\,d^3\,e+2\,A\,b\,c\,d^2\,e^2+10\,B\,c^2\,d^4+2\,A\,c^2\,d^3\,e\right )}{10\,e^5}+\frac {c\,x^4\,\left (A\,c\,e+2\,B\,b\,e+5\,B\,c\,d\right )}{2\,e^2}+\frac {B\,c^2\,x^5}{e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \] Input:

int(((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^7,x)
 

Output:

-((10*A*a^2*e^5 + 10*B*c^2*d^5 + 2*B*a^2*d*e^4 + 2*A*c^2*d^4*e + A*b^2*d^2 
*e^3 + B*b^2*d^3*e^2 + 4*A*a*b*d*e^4 + 4*B*b*c*d^4*e + 2*A*a*c*d^2*e^3 + 2 
*B*a*b*d^2*e^3 + 2*A*b*c*d^3*e^2 + 2*B*a*c*d^3*e^2)/(60*e^6) + (x^3*(B*b^2 
*e^2 + 10*B*c^2*d^2 + 2*A*b*c*e^2 + 2*B*a*c*e^2 + 2*A*c^2*d*e + 4*B*b*c*d* 
e))/(3*e^3) + (x^2*(A*b^2*e^3 + 10*B*c^2*d^3 + 2*A*a*c*e^3 + 2*B*a*b*e^3 + 
 2*A*c^2*d^2*e + B*b^2*d*e^2 + 2*A*b*c*d*e^2 + 2*B*a*c*d*e^2 + 4*B*b*c*d^2 
*e))/(4*e^4) + (x*(2*B*a^2*e^4 + 10*B*c^2*d^4 + 4*A*a*b*e^4 + A*b^2*d*e^3 
+ 2*A*c^2*d^3*e + B*b^2*d^2*e^2 + 2*A*a*c*d*e^3 + 2*B*a*b*d*e^3 + 4*B*b*c* 
d^3*e + 2*A*b*c*d^2*e^2 + 2*B*a*c*d^2*e^2))/(10*e^5) + (c*x^4*(A*c*e + 2*B 
*b*e + 5*B*c*d))/(2*e^2) + (B*c^2*x^5)/e)/(d^6 + e^6*x^6 + 6*d*e^5*x^5 + 1 
5*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e*x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.37 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {10 b \,c^{2} e^{5} x^{6}-30 a \,c^{2} d \,e^{4} x^{4}-60 b^{2} c d \,e^{4} x^{4}-80 a b c d \,e^{4} x^{3}-40 a \,c^{2} d^{2} e^{3} x^{3}-20 b^{3} d \,e^{4} x^{3}-80 b^{2} c \,d^{2} e^{3} x^{3}-30 a^{2} c d \,e^{4} x^{2}-45 a \,b^{2} d \,e^{4} x^{2}-60 a b c \,d^{2} e^{3} x^{2}-30 a \,c^{2} d^{3} e^{2} x^{2}-15 b^{3} d^{2} e^{3} x^{2}-60 b^{2} c \,d^{3} e^{2} x^{2}-36 a^{2} b d \,e^{4} x -12 a^{2} c \,d^{2} e^{3} x -18 a \,b^{2} d^{2} e^{3} x -24 a b c \,d^{3} e^{2} x -12 a \,c^{2} d^{4} e x -6 b^{3} d^{3} e^{2} x -24 b^{2} c \,d^{4} e x -10 a^{3} d \,e^{4}-6 a^{2} b \,d^{2} e^{3}-2 a^{2} c \,d^{3} e^{2}-3 a \,b^{2} d^{3} e^{2}-4 a b c \,d^{4} e -2 a \,c^{2} d^{5}-b^{3} d^{4} e -4 b^{2} c \,d^{5}}{60 d \,e^{5} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:

int((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^7,x)
 

Output:

( - 10*a**3*d*e**4 - 6*a**2*b*d**2*e**3 - 36*a**2*b*d*e**4*x - 2*a**2*c*d* 
*3*e**2 - 12*a**2*c*d**2*e**3*x - 30*a**2*c*d*e**4*x**2 - 3*a*b**2*d**3*e* 
*2 - 18*a*b**2*d**2*e**3*x - 45*a*b**2*d*e**4*x**2 - 4*a*b*c*d**4*e - 24*a 
*b*c*d**3*e**2*x - 60*a*b*c*d**2*e**3*x**2 - 80*a*b*c*d*e**4*x**3 - 2*a*c* 
*2*d**5 - 12*a*c**2*d**4*e*x - 30*a*c**2*d**3*e**2*x**2 - 40*a*c**2*d**2*e 
**3*x**3 - 30*a*c**2*d*e**4*x**4 - b**3*d**4*e - 6*b**3*d**3*e**2*x - 15*b 
**3*d**2*e**3*x**2 - 20*b**3*d*e**4*x**3 - 4*b**2*c*d**5 - 24*b**2*c*d**4* 
e*x - 60*b**2*c*d**3*e**2*x**2 - 80*b**2*c*d**2*e**3*x**3 - 60*b**2*c*d*e* 
*4*x**4 + 10*b*c**2*e**5*x**6)/(60*d*e**5*(d**6 + 6*d**5*e*x + 15*d**4*e** 
2*x**2 + 20*d**3*e**3*x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x**5 + e**6*x**6 
))