\(\int \frac {(A+B x) (b x+c x^2)^2}{(d+e x)^7} \, dx\) [20]

Optimal result

Integrand size = 24, antiderivative size = 253 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {d^2 (B d-A e) (c d-b e)^2}{6 e^6 (d+e x)^6}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6 (d+e x)^5}-\frac {A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{4 e^6 (d+e x)^4}+\frac {2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )}{3 e^6 (d+e x)^3}+\frac {c (5 B c d-2 b B e-A c e)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (d+e x)} \] Output:

1/6*d^2*(-A*e+B*d)*(-b*e+c*d)^2/e^6/(e*x+d)^6-1/5*d*(-b*e+c*d)*(B*d*(-3*b* 
e+5*c*d)-2*A*e*(-b*e+2*c*d))/e^6/(e*x+d)^5-1/4*(A*e*(b^2*e^2-6*b*c*d*e+6*c 
^2*d^2)-B*d*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2))/e^6/(e*x+d)^4+1/3*(2*A*c*e* 
(-b*e+2*c*d)-B*(b^2*e^2-8*b*c*d*e+10*c^2*d^2))/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 = 0.13 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.02 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {A e \left (b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 b c e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+B \left (b^2 e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+4 b c e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+10 c^2 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{60 e^6 (d+e x)^6} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.

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

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{3 e^6 (d+e x)^3}-\frac {A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}+\frac {d^2 (B d-A e) (c d-b e)^2}{6 e^6 (d+e x)^6}+\frac {c (-A c e-2 b B e+5 B c d)}{2 e^6 (d+e x)^2}-\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6 (d+e x)^5}-\frac {B c^2}{e^6 (d+e x)}\)

Input:

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

Output:

(d^2*(B*d - A*e)*(c*d - b*e)^2)/(6*e^6*(d + e*x)^6) - (d*(c*d - b*e)*(B*d* 
(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e)))/(5*e^6*(d + e*x)^5) - (A*e*(6*c^2* 
d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))/(4 
*e^6*(d + e*x)^4) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b 
^2*e^2))/(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 = 0.85 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.12

method	result	size
risch	\(\frac {-\frac {B \,c^{2} x^{5}}{e}-\frac {c \left (A c e +2 B b e +5 B c d \right ) x^{4}}{2 e^{2}}-\frac {\left (2 A b c \,e^{2}+2 A \,c^{2} d e +B \,e^{2} b^{2}+4 B b c d e +10 B \,c^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {\left (A \,b^{2} e^{3}+2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right ) x^{2}}{4 e^{4}}-\frac {d \left (A \,b^{2} e^{3}+2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right ) x}{10 e^{5}}-\frac {d^{2} \left (A \,b^{2} e^{3}+2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right )}{60 e^{6}}}{\left (e x +d \right )^{6}}\)	\(283\)
norman	\(\frac {-\frac {B \,c^{2} x^{5}}{e}-\frac {\left (A \,c^{2} e +2 B e b c +5 B \,c^{2} d \right ) x^{4}}{2 e^{2}}-\frac {\left (2 A b c \,e^{2}+2 A \,c^{2} d e +B \,e^{2} b^{2}+4 B b c d e +10 B \,c^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {\left (A \,b^{2} e^{3}+2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right ) x^{2}}{4 e^{4}}-\frac {d \left (A \,b^{2} e^{3}+2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right ) x}{10 e^{5}}-\frac {d^{2} \left (A \,b^{2} e^{3}+2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e +10 B \,c^{2} d^{3}\right )}{60 e^{6}}}{\left (e x +d \right )^{6}}\)	\(287\)
default	\(-\frac {2 A b c \,e^{2}-4 A \,c^{2} d e +B \,e^{2} b^{2}-8 B b c d e +10 B \,c^{2} d^{2}}{3 e^{6} \left (e x +d \right )^{3}}-\frac {A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}}{4 e^{6} \left (e x +d \right )^{4}}+\frac {d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right )}{5 e^{6} \left (e x +d \right )^{5}}-\frac {B \,c^{2}}{e^{6} \left (e x +d \right )}-\frac {c \left (A c e +2 B b e -5 B c d \right )}{2 e^{6} \left (e x +d \right )^{2}}-\frac {d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{6 e^{6} \left (e x +d \right )^{6}}\)	\(307\)
gosper	\(-\frac {60 B \,x^{5} c^{2} e^{5}+30 A \,x^{4} c^{2} e^{5}+60 B \,x^{4} b c \,e^{5}+150 B \,x^{4} c^{2} d \,e^{4}+40 A \,x^{3} b c \,e^{5}+40 A \,x^{3} c^{2} d \,e^{4}+20 B \,x^{3} b^{2} e^{5}+80 B \,x^{3} b c d \,e^{4}+200 B \,x^{3} c^{2} d^{2} e^{3}+15 A \,x^{2} b^{2} e^{5}+30 A \,x^{2} b c d \,e^{4}+30 A \,x^{2} c^{2} d^{2} e^{3}+15 B \,x^{2} b^{2} d \,e^{4}+60 B \,x^{2} b c \,d^{2} e^{3}+150 B \,x^{2} c^{2} d^{3} e^{2}+6 A x \,b^{2} d \,e^{4}+12 A x b c \,d^{2} e^{3}+12 A x \,c^{2} d^{3} e^{2}+6 B x \,b^{2} d^{2} e^{3}+24 B x b c \,d^{3} e^{2}+60 B x \,c^{2} d^{4} e +A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{60 e^{6} \left (e x +d \right )^{6}}\)	\(339\)
parallelrisch	\(-\frac {60 B \,x^{5} c^{2} e^{5}+30 A \,x^{4} c^{2} e^{5}+60 B \,x^{4} b c \,e^{5}+150 B \,x^{4} c^{2} d \,e^{4}+40 A \,x^{3} b c \,e^{5}+40 A \,x^{3} c^{2} d \,e^{4}+20 B \,x^{3} b^{2} e^{5}+80 B \,x^{3} b c d \,e^{4}+200 B \,x^{3} c^{2} d^{2} e^{3}+15 A \,x^{2} b^{2} e^{5}+30 A \,x^{2} b c d \,e^{4}+30 A \,x^{2} c^{2} d^{2} e^{3}+15 B \,x^{2} b^{2} d \,e^{4}+60 B \,x^{2} b c \,d^{2} e^{3}+150 B \,x^{2} c^{2} d^{3} e^{2}+6 A x \,b^{2} d \,e^{4}+12 A x b c \,d^{2} e^{3}+12 A x \,c^{2} d^{3} e^{2}+6 B x \,b^{2} d^{2} e^{3}+24 B x b c \,d^{3} e^{2}+60 B x \,c^{2} d^{4} e +A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}}{60 e^{6} \left (e x +d \right )^{6}}\)	\(339\)
orering	\(-\frac {\left (60 B \,x^{5} c^{2} e^{5}+30 A \,x^{4} c^{2} e^{5}+60 B \,x^{4} b c \,e^{5}+150 B \,x^{4} c^{2} d \,e^{4}+40 A \,x^{3} b c \,e^{5}+40 A \,x^{3} c^{2} d \,e^{4}+20 B \,x^{3} b^{2} e^{5}+80 B \,x^{3} b c d \,e^{4}+200 B \,x^{3} c^{2} d^{2} e^{3}+15 A \,x^{2} b^{2} e^{5}+30 A \,x^{2} b c d \,e^{4}+30 A \,x^{2} c^{2} d^{2} e^{3}+15 B \,x^{2} b^{2} d \,e^{4}+60 B \,x^{2} b c \,d^{2} e^{3}+150 B \,x^{2} c^{2} d^{3} e^{2}+6 A x \,b^{2} d \,e^{4}+12 A x b c \,d^{2} e^{3}+12 A x \,c^{2} d^{3} e^{2}+6 B x \,b^{2} d^{2} e^{3}+24 B x b c \,d^{3} e^{2}+60 B x \,c^{2} d^{4} e +A \,b^{2} d^{2} e^{3}+2 A b c \,d^{3} e^{2}+2 A \,c^{2} d^{4} e +B \,b^{2} d^{3} e^{2}+4 B b c \,d^{4} e +10 B \,c^{2} d^{5}\right ) \left (c \,x^{2}+b x \right )^{2}}{60 e^{6} \left (c x +b \right )^{2} \left (e x +d \right )^{6} x^{2}}\)	\(360\)

Input:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {60 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 30 \, {\left (5 \, B c^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \, {\left (10 \, B c^{2} d^{2} e^{3} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 15 \, {\left (10 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 6 \, {\left (10 \, B c^{2} d^{4} e + A b^{2} d e^{4} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{60 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^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 + A*b^2*d^2*e^3 + 2*(2*B*b*c + A*c^ 
2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2 + 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*A*b*c)*e^5)*x^3 + 15*(10*B*c^2*d^3*e^2 + A*b^2*e^5 + 2*(2*B*b*c + A*c^2) 
*d^2*e^3 + (B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 6*(10*B*c^2*d^4*e + A*b^2*d*e^4 
+ 2*(2*B*b*c + A*c^2)*d^3*e^2 + (B*b^2 + 2*A*b*c)*d^2*e^3)*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. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {60 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 30 \, {\left (5 \, B c^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \, {\left (10 \, B c^{2} d^{2} e^{3} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 15 \, {\left (10 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 6 \, {\left (10 \, B c^{2} d^{4} e + A b^{2} d e^{4} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{60 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^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 + A*b^2*d^2*e^3 + 2*(2*B*b*c + A*c^ 
2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2 + 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*A*b*c)*e^5)*x^3 + 15*(10*B*c^2*d^3*e^2 + A*b^2*e^5 + 2*(2*B*b*c + A*c^2) 
*d^2*e^3 + (B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 6*(10*B*c^2*d^4*e + A*b^2*d*e^4 
+ 2*(2*B*b*c + A*c^2)*d^3*e^2 + (B*b^2 + 2*A*b*c)*d^2*e^3)*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 = 0.29 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {60 \, B c^{2} e^{5} x^{5} + 150 \, B c^{2} d e^{4} x^{4} + 60 \, B b c e^{5} x^{4} + 30 \, A c^{2} e^{5} x^{4} + 200 \, B c^{2} d^{2} e^{3} x^{3} + 80 \, B b c d e^{4} x^{3} + 40 \, A c^{2} d e^{4} x^{3} + 20 \, B b^{2} e^{5} x^{3} + 40 \, A b c e^{5} x^{3} + 150 \, B c^{2} d^{3} e^{2} x^{2} + 60 \, B b c d^{2} e^{3} x^{2} + 30 \, A c^{2} d^{2} e^{3} x^{2} + 15 \, B b^{2} d e^{4} x^{2} + 30 \, A b c d e^{4} x^{2} + 15 \, A b^{2} e^{5} x^{2} + 60 \, B c^{2} d^{4} e x + 24 \, B b c d^{3} e^{2} x + 12 \, A c^{2} d^{3} e^{2} x + 6 \, B b^{2} d^{2} e^{3} x + 12 \, A b c d^{2} e^{3} x + 6 \, A b^{2} d e^{4} x + 10 \, B c^{2} d^{5} + 4 \, B b c d^{4} e + 2 \, A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, A b c d^{3} e^{2} + A b^{2} d^{2} e^{3}}{60 \, {\left (e x + d\right )}^{6} e^{6}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^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*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*A*b*c*d*e^4* 
x^2 + 15*A*b^2*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*A*b*c*d^2*e^3*x + 6*A*b^2*d*e^4*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*A*b*c*d^3*e^ 
2 + A*b^2*d^2*e^3)/((e*x + d)^6*e^6)
 

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\frac {x^3\,\left (B\,b^2\,e^2+4\,B\,b\,c\,d\,e+2\,A\,b\,c\,e^2+10\,B\,c^2\,d^2+2\,A\,c^2\,d\,e\right )}{3\,e^3}+\frac {d^2\,\left (B\,b^2\,d\,e^2+A\,b^2\,e^3+4\,B\,b\,c\,d^2\,e+2\,A\,b\,c\,d\,e^2+10\,B\,c^2\,d^3+2\,A\,c^2\,d^2\,e\right )}{60\,e^6}+\frac {x^2\,\left (B\,b^2\,d\,e^2+A\,b^2\,e^3+4\,B\,b\,c\,d^2\,e+2\,A\,b\,c\,d\,e^2+10\,B\,c^2\,d^3+2\,A\,c^2\,d^2\,e\right )}{4\,e^4}+\frac {d\,x\,\left (B\,b^2\,d\,e^2+A\,b^2\,e^3+4\,B\,b\,c\,d^2\,e+2\,A\,b\,c\,d\,e^2+10\,B\,c^2\,d^3+2\,A\,c^2\,d^2\,e\right )}{10\,e^5}+\frac {c\,x^4\,\left (A\,c\,e+2\,B\,b\,e+5\,B\,c\,d\right )}{2\,e^2}+\frac {B\,c^2\,x^5}{e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \] Input:

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

Output:

-((x^3*(B*b^2*e^2 + 10*B*c^2*d^2 + 2*A*b*c*e^2 + 2*A*c^2*d*e + 4*B*b*c*d*e 
))/(3*e^3) + (d^2*(A*b^2*e^3 + 10*B*c^2*d^3 + 2*A*c^2*d^2*e + B*b^2*d*e^2 
+ 2*A*b*c*d*e^2 + 4*B*b*c*d^2*e))/(60*e^6) + (x^2*(A*b^2*e^3 + 10*B*c^2*d^ 
3 + 2*A*c^2*d^2*e + B*b^2*d*e^2 + 2*A*b*c*d*e^2 + 4*B*b*c*d^2*e))/(4*e^4) 
+ (d*x*(A*b^2*e^3 + 10*B*c^2*d^3 + 2*A*c^2*d^2*e + B*b^2*d*e^2 + 2*A*b*c*d 
*e^2 + 4*B*b*c*d^2*e))/(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 + 15*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 = 0.25 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {10 b \,c^{2} e^{5} x^{6}-30 a \,c^{2} d \,e^{4} x^{4}-60 b^{2} c d \,e^{4} x^{4}-40 a b c d \,e^{4} x^{3}-40 a \,c^{2} d^{2} e^{3} x^{3}-20 b^{3} d \,e^{4} x^{3}-80 b^{2} c \,d^{2} e^{3} x^{3}-15 a \,b^{2} d \,e^{4} x^{2}-30 a b c \,d^{2} e^{3} x^{2}-30 a \,c^{2} d^{3} e^{2} x^{2}-15 b^{3} d^{2} e^{3} x^{2}-60 b^{2} c \,d^{3} e^{2} x^{2}-6 a \,b^{2} d^{2} e^{3} x -12 a b c \,d^{3} e^{2} x -12 a \,c^{2} d^{4} e x -6 b^{3} d^{3} e^{2} x -24 b^{2} c \,d^{4} e x -a \,b^{2} d^{3} e^{2}-2 a b c \,d^{4} e -2 a \,c^{2} d^{5}-b^{3} d^{4} e -4 b^{2} c \,d^{5}}{60 d \,e^{5} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:

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

Output:

( - a*b**2*d**3*e**2 - 6*a*b**2*d**2*e**3*x - 15*a*b**2*d*e**4*x**2 - 2*a* 
b*c*d**4*e - 12*a*b*c*d**3*e**2*x - 30*a*b*c*d**2*e**3*x**2 - 40*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))