3.24.26 \(\int \frac {(A+B x) (a+b x+c x^2)^2}{(d+e x)^3} \, dx\) [2326]

3.24.26.1 Optimal result

Integrand size = 25, antiderivative size = 281 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=-\frac {\left (A c e (3 c d-2 b e)-B \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) x}{e^5}-\frac {c (3 B c d-2 b B e-A c e) x^2}{2 e^4}+\frac {B c^2 x^3}{3 e^3}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{2 e^6 (d+e x)^2}+\frac {\left (c d^2-b d e+a e^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{e^6 (d+e x)}-\frac {\left (B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) \log (d+e x)}{e^6} \]

output

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

3.24.26.2 Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {6 e \left (A c e (-3 c d+2 b e)+B \left (6 c^2 d^2+b^2 e^2+2 c e (-3 b d+a e)\right )\right ) x+3 c e^2 (-3 B c d+2 b B e+A c e) x^2+2 B c^2 e^3 x^3+\frac {3 (B d-A e) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^2}-\frac {6 \left (c d^2+e (-b d+a e)\right ) \left (5 B c d^2+B e (-3 b d+a e)+2 A e (-2 c d+b e)\right )}{d+e x}+6 \left (A e \left (6 c^2 d^2+b^2 e^2+2 c e (-3 b d+a e)\right )+B \left (-10 c^2 d^3+6 c d e (2 b d-a e)+b e^2 (-3 b d+2 a e)\right )\right ) \log (d+e x)}{6 e^6} \]

input

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

output

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

3.24.26.3 Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 280, 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.

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

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.24.26.3.1 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]
 

3.24.26.4 Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.56

method	result	size
norman	\(\frac {-\frac {A \,a^{2} e^{5}+2 A a b d \,e^{4}-6 A a c \,d^{2} e^{3}-3 A \,b^{2} d^{2} e^{3}+18 A b c \,d^{3} e^{2}-18 A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}-6 B a b \,d^{2} e^{3}+18 B a c \,d^{3} e^{2}+9 B \,b^{2} d^{3} e^{2}-36 B b c \,d^{4} e +30 B \,c^{2} d^{5}}{2 e^{6}}+\frac {\left (6 A b c \,e^{2}-6 A \,c^{2} d e +6 B \,e^{2} a c +3 B \,b^{2} e^{2}-12 B b c d e +10 B \,c^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {\left (2 A a b \,e^{4}-4 A a c d \,e^{3}-2 A \,b^{2} d \,e^{3}+12 A b c \,d^{2} e^{2}-12 A \,c^{2} d^{3} e +B \,e^{4} a^{2}-4 B a b d \,e^{3}+12 B a c \,d^{2} e^{2}+6 B \,b^{2} d^{2} e^{2}-24 B b c \,d^{3} e +20 B \,c^{2} d^{4}\right ) x}{e^{5}}+\frac {B \,c^{2} x^{5}}{3 e}+\frac {c \left (3 A c e +6 B b e -5 B c d \right ) x^{4}}{6 e^{2}}}{\left (e x +d \right )^{2}}+\frac {\left (2 A a c \,e^{3}+A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e +2 B a b \,e^{3}-6 B a c d \,e^{2}-3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{6}}\)	\(438\)
default	\(\frac {\frac {1}{3} B \,c^{2} x^{3} e^{2}+\frac {1}{2} A \,c^{2} e^{2} x^{2}+B b c \,e^{2} x^{2}-\frac {3}{2} B \,c^{2} d e \,x^{2}+2 A b c \,e^{2} x -3 A \,c^{2} d e x +2 B a c \,e^{2} x +B \,b^{2} e^{2} x -6 B b c d e x +6 B \,c^{2} d^{2} x}{e^{5}}-\frac {2 A a b \,e^{4}-4 A a c d \,e^{3}-2 A \,b^{2} d \,e^{3}+6 A b c \,d^{2} e^{2}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}-4 B a b d \,e^{3}+6 B a c \,d^{2} e^{2}+3 B \,b^{2} d^{2} e^{2}-8 B b c \,d^{3} e +5 B \,c^{2} d^{4}}{e^{6} \left (e x +d \right )}+\frac {\left (2 A a c \,e^{3}+A \,b^{2} e^{3}-6 A b c d \,e^{2}+6 A \,c^{2} d^{2} e +2 B a b \,e^{3}-6 B a c d \,e^{2}-3 B \,b^{2} d \,e^{2}+12 B b c \,d^{2} e -10 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{6}}-\frac {A \,a^{2} e^{5}-2 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}-2 A b c \,d^{3} e^{2}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}-2 B a c \,d^{3} e^{2}-B \,b^{2} d^{3} e^{2}+2 B b c \,d^{4} e -B \,c^{2} d^{5}}{2 e^{6} \left (e x +d \right )^{2}}\)	\(450\)
risch	\(\frac {B \,c^{2} x^{3}}{3 e^{3}}+\frac {A \,c^{2} x^{2}}{2 e^{3}}+\frac {B b c \,x^{2}}{e^{3}}-\frac {3 B \,c^{2} d \,x^{2}}{2 e^{4}}+\frac {2 A b c x}{e^{3}}-\frac {3 A \,c^{2} d x}{e^{4}}+\frac {2 B a c x}{e^{3}}+\frac {b^{2} B x}{e^{3}}-\frac {6 B b c d x}{e^{4}}+\frac {6 B \,c^{2} d^{2} x}{e^{5}}+\frac {\left (-2 A a b \,e^{4}+4 A a c d \,e^{3}+2 A \,b^{2} d \,e^{3}-6 A b c \,d^{2} e^{2}+4 A \,c^{2} d^{3} e -B \,e^{4} a^{2}+4 B a b d \,e^{3}-6 B a c \,d^{2} e^{2}-3 B \,b^{2} d^{2} e^{2}+8 B b c \,d^{3} e -5 B \,c^{2} d^{4}\right ) x -\frac {A \,a^{2} e^{5}+2 A a b d \,e^{4}-6 A a c \,d^{2} e^{3}-3 A \,b^{2} d^{2} e^{3}+10 A b c \,d^{3} e^{2}-7 A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}-6 B a b \,d^{2} e^{3}+10 B a c \,d^{3} e^{2}+5 B \,b^{2} d^{3} e^{2}-14 B b c \,d^{4} e +9 B \,c^{2} d^{5}}{2 e}}{e^{5} \left (e x +d \right )^{2}}+\frac {2 \ln \left (e x +d \right ) A a c}{e^{3}}+\frac {\ln \left (e x +d \right ) A \,b^{2}}{e^{3}}-\frac {6 \ln \left (e x +d \right ) A b c d}{e^{4}}+\frac {6 \ln \left (e x +d \right ) A \,c^{2} d^{2}}{e^{5}}+\frac {2 \ln \left (e x +d \right ) B a b}{e^{3}}-\frac {6 \ln \left (e x +d \right ) B a c d}{e^{4}}-\frac {3 \ln \left (e x +d \right ) B \,b^{2} d}{e^{4}}+\frac {12 \ln \left (e x +d \right ) B b c \,d^{2}}{e^{5}}-\frac {10 \ln \left (e x +d \right ) B \,c^{2} d^{3}}{e^{6}}\)	\(501\)
parallelrisch	\(\frac {6 A \ln \left (e x +d \right ) b^{2} d^{2} e^{3}-18 B \ln \left (e x +d \right ) b^{2} d^{3} e^{2}-3 A \,a^{2} e^{5}+6 A \ln \left (e x +d \right ) x^{2} b^{2} e^{5}-36 A \ln \left (e x +d \right ) b c \,d^{3} e^{2}+12 B \ln \left (e x +d \right ) a b \,d^{2} e^{3}+72 B \ln \left (e x +d \right ) b c \,d^{4} e +12 A \ln \left (e x +d \right ) x^{2} a c \,e^{5}+12 B \ln \left (e x +d \right ) x^{2} a b \,e^{5}-18 B \ln \left (e x +d \right ) x^{2} b^{2} d \,e^{4}+24 B x a b d \,e^{4}-12 A x a b \,e^{5}-90 B \,c^{2} d^{5}-72 B x a c \,d^{2} e^{3}+12 A \ln \left (e x +d \right ) a c \,d^{2} e^{3}+24 A x a c d \,e^{4}-36 B \ln \left (e x +d \right ) a c \,d^{3} e^{2}-6 B x \,a^{2} e^{5}-120 B \ln \left (e x +d \right ) x \,c^{2} d^{4} e +72 A \ln \left (e x +d \right ) x \,c^{2} d^{3} e^{2}+36 A \ln \left (e x +d \right ) x^{2} c^{2} d^{2} e^{3}-60 B \ln \left (e x +d \right ) x^{2} c^{2} d^{3} e^{2}-60 B \ln \left (e x +d \right ) c^{2} d^{5}+3 A \,x^{4} c^{2} e^{5}+6 B \,x^{3} b^{2} e^{5}+2 B \,x^{5} c^{2} e^{5}+144 B x b c \,d^{3} e^{2}-72 A x b c \,d^{2} e^{3}-24 B \,x^{3} b c d \,e^{4}-3 B \,a^{2} d \,e^{4}+9 A \,b^{2} d^{2} e^{3}+54 A \,c^{2} d^{4} e -27 B \,b^{2} d^{3} e^{2}+144 B \ln \left (e x +d \right ) x b c \,d^{3} e^{2}-72 A \ln \left (e x +d \right ) x b c \,d^{2} e^{3}+24 B \ln \left (e x +d \right ) x a b d \,e^{4}-54 A b c \,d^{3} e^{2}+108 B b c \,d^{4} e -36 B \ln \left (e x +d \right ) x^{2} a c d \,e^{4}+24 A \ln \left (e x +d \right ) x a c d \,e^{4}+12 B \,x^{3} a c \,e^{5}-36 A \ln \left (e x +d \right ) x^{2} b c d \,e^{4}+72 B \ln \left (e x +d \right ) x^{2} b c \,d^{2} e^{3}+18 A a c \,d^{2} e^{3}-54 B a c \,d^{3} e^{2}+18 B a b \,d^{2} e^{3}-72 B \ln \left (e x +d \right ) x a c \,d^{2} e^{3}+12 A \,x^{3} b c \,e^{5}-12 A \,x^{3} c^{2} d \,e^{4}+20 B \,x^{3} c^{2} d^{2} e^{3}+12 A x \,b^{2} d \,e^{4}+72 A x \,c^{2} d^{3} e^{2}-36 B x \,b^{2} d^{2} e^{3}-120 B x \,c^{2} d^{4} e +36 A \ln \left (e x +d \right ) c^{2} d^{4} e +6 B \,x^{4} b c \,e^{5}-5 B \,x^{4} c^{2} d \,e^{4}-6 A a b d \,e^{4}-36 B \ln \left (e x +d \right ) x \,b^{2} d^{2} e^{3}+12 A \ln \left (e x +d \right ) x \,b^{2} d \,e^{4}}{6 e^{6} \left (e x +d \right )^{2}}\)	\(860\)

input

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

output

(-1/2*(A*a^2*e^5+2*A*a*b*d*e^4-6*A*a*c*d^2*e^3-3*A*b^2*d^2*e^3+18*A*b*c*d^ 
3*e^2-18*A*c^2*d^4*e+B*a^2*d*e^4-6*B*a*b*d^2*e^3+18*B*a*c*d^3*e^2+9*B*b^2* 
d^3*e^2-36*B*b*c*d^4*e+30*B*c^2*d^5)/e^6+1/3*(6*A*b*c*e^2-6*A*c^2*d*e+6*B* 
a*c*e^2+3*B*b^2*e^2-12*B*b*c*d*e+10*B*c^2*d^2)/e^3*x^3-(2*A*a*b*e^4-4*A*a* 
c*d*e^3-2*A*b^2*d*e^3+12*A*b*c*d^2*e^2-12*A*c^2*d^3*e+B*a^2*e^4-4*B*a*b*d* 
e^3+12*B*a*c*d^2*e^2+6*B*b^2*d^2*e^2-24*B*b*c*d^3*e+20*B*c^2*d^4)/e^5*x+1/ 
3*B*c^2*x^5/e+1/6*c*(3*A*c*e+6*B*b*e-5*B*c*d)/e^2*x^4)/(e*x+d)^2+1/e^6*(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*c*d*e^2- 
3*B*b^2*d*e^2+12*B*b*c*d^2*e-10*B*c^2*d^3)*ln(e*x+d)
 

3.24.26.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (274) = 548\).

Time = 0.38 (sec) , antiderivative size = 620, normalized size of antiderivative = 2.21 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {2 \, B c^{2} e^{5} x^{5} - 27 \, B c^{2} d^{5} - 3 \, A a^{2} e^{5} + 21 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e - 15 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 9 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4} - {\left (5 \, B c^{2} d e^{4} - 3 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \, {\left (10 \, B c^{2} d^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 3 \, {\left (21 \, B c^{2} d^{3} e^{2} - 11 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4}\right )} x^{2} + 6 \, {\left (B c^{2} d^{4} e + {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 2 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} - {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x - 6 \, {\left (10 \, B c^{2} d^{5} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + {\left (10 \, B c^{2} d^{3} e^{2} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 2 \, {\left (10 \, B c^{2} d^{4} e - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \]

input

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

output

1/6*(2*B*c^2*e^5*x^5 - 27*B*c^2*d^5 - 3*A*a^2*e^5 + 21*(2*B*b*c + A*c^2)*d 
^4*e - 15*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + 9*(2*B*a*b + A*b^2 + 2*A*a*c 
)*d^2*e^3 - 3*(B*a^2 + 2*A*a*b)*d*e^4 - (5*B*c^2*d*e^4 - 3*(2*B*b*c + A*c^ 
2)*e^5)*x^4 + 2*(10*B*c^2*d^2*e^3 - 6*(2*B*b*c + A*c^2)*d*e^4 + 3*(B*b^2 + 
 2*(B*a + A*b)*c)*e^5)*x^3 + 3*(21*B*c^2*d^3*e^2 - 11*(2*B*b*c + A*c^2)*d^ 
2*e^3 + 4*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4)*x^2 + 6*(B*c^2*d^4*e + (2*B*b*c 
 + A*c^2)*d^3*e^2 - 2*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 + 2*(2*B*a*b + A*b 
^2 + 2*A*a*c)*d*e^4 - (B*a^2 + 2*A*a*b)*e^5)*x - 6*(10*B*c^2*d^5 - 6*(2*B* 
b*c + A*c^2)*d^4*e + 3*(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 + (10*B*c^2*d^3*e^2 - 6*(2*B*b*c + A*c^2)*d^2*e^3 + 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 + 
2*(10*B*c^2*d^4*e - 6*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(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)*x)*log(e*x + d))/(e^8*x^2 
 + 2*d*e^7*x + d^2*e^6)
 

3.24.26.6 Sympy [A] (verification not implemented)

Time = 9.34 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.90 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {B c^{2} x^{3}}{3 e^{3}} + x^{2} \left (\frac {A c^{2}}{2 e^{3}} + \frac {B b c}{e^{3}} - \frac {3 B c^{2} d}{2 e^{4}}\right ) + x \left (\frac {2 A b c}{e^{3}} - \frac {3 A c^{2} d}{e^{4}} + \frac {2 B a c}{e^{3}} + \frac {B b^{2}}{e^{3}} - \frac {6 B b c d}{e^{4}} + \frac {6 B c^{2} d^{2}}{e^{5}}\right ) + \frac {- A a^{2} e^{5} - 2 A a b d e^{4} + 6 A a c d^{2} e^{3} + 3 A b^{2} d^{2} e^{3} - 10 A b c d^{3} e^{2} + 7 A c^{2} d^{4} e - B a^{2} d e^{4} + 6 B a b d^{2} e^{3} - 10 B a c d^{3} e^{2} - 5 B b^{2} d^{3} e^{2} + 14 B b c d^{4} e - 9 B c^{2} d^{5} + x \left (- 4 A a b e^{5} + 8 A a c d e^{4} + 4 A b^{2} d e^{4} - 12 A b c d^{2} e^{3} + 8 A c^{2} d^{3} e^{2} - 2 B a^{2} e^{5} + 8 B a b d e^{4} - 12 B a c d^{2} e^{3} - 6 B b^{2} d^{2} e^{3} + 16 B b c d^{3} e^{2} - 10 B c^{2} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} + \frac {\left (2 A a c e^{3} + A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e + 2 B a b e^{3} - 6 B a c d e^{2} - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right ) \log {\left (d + e x \right )}}{e^{6}} \]

input

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

output

B*c**2*x**3/(3*e**3) + x**2*(A*c**2/(2*e**3) + B*b*c/e**3 - 3*B*c**2*d/(2* 
e**4)) + x*(2*A*b*c/e**3 - 3*A*c**2*d/e**4 + 2*B*a*c/e**3 + B*b**2/e**3 - 
6*B*b*c*d/e**4 + 6*B*c**2*d**2/e**5) + (-A*a**2*e**5 - 2*A*a*b*d*e**4 + 6* 
A*a*c*d**2*e**3 + 3*A*b**2*d**2*e**3 - 10*A*b*c*d**3*e**2 + 7*A*c**2*d**4* 
e - B*a**2*d*e**4 + 6*B*a*b*d**2*e**3 - 10*B*a*c*d**3*e**2 - 5*B*b**2*d**3 
*e**2 + 14*B*b*c*d**4*e - 9*B*c**2*d**5 + x*(-4*A*a*b*e**5 + 8*A*a*c*d*e** 
4 + 4*A*b**2*d*e**4 - 12*A*b*c*d**2*e**3 + 8*A*c**2*d**3*e**2 - 2*B*a**2*e 
**5 + 8*B*a*b*d*e**4 - 12*B*a*c*d**2*e**3 - 6*B*b**2*d**2*e**3 + 16*B*b*c* 
d**3*e**2 - 10*B*c**2*d**4*e))/(2*d**2*e**6 + 4*d*e**7*x + 2*e**8*x**2) + 
(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*c*d*e**2 - 3*B*b**2*d*e**2 + 12*B*b*c*d**2*e - 10*B*c**2*d**3) 
*log(d + e*x)/e**6
 

3.24.26.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=-\frac {9 \, B c^{2} d^{5} + A a^{2} e^{5} - 7 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 5 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - 3 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + {\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 2 \, {\left (5 \, B c^{2} d^{4} e - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{2 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {2 \, B c^{2} e^{2} x^{3} - 3 \, {\left (3 \, B c^{2} d e - {\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (6 \, B c^{2} d^{2} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac {{\left (10 \, B c^{2} d^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{2} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{6}} \]

input

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

output

-1/2*(9*B*c^2*d^5 + A*a^2*e^5 - 7*(2*B*b*c + A*c^2)*d^4*e + 5*(B*b^2 + 2*( 
B*a + A*b)*c)*d^3*e^2 - 3*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + (B*a^2 + 2 
*A*a*b)*d*e^4 + 2*(5*B*c^2*d^4*e - 4*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 
+ 2*(B*a + A*b)*c)*d^2*e^3 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + (B*a^2 
+ 2*A*a*b)*e^5)*x)/(e^8*x^2 + 2*d*e^7*x + d^2*e^6) + 1/6*(2*B*c^2*e^2*x^3 
- 3*(3*B*c^2*d*e - (2*B*b*c + A*c^2)*e^2)*x^2 + 6*(6*B*c^2*d^2 - 3*(2*B*b* 
c + A*c^2)*d*e + (B*b^2 + 2*(B*a + A*b)*c)*e^2)*x)/e^5 - (10*B*c^2*d^3 - 6 
*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*(B*a + A*b)*c)*d*e^2 - (2*B*a*b + 
A*b^2 + 2*A*a*c)*e^3)*log(e*x + d)/e^6
 

3.24.26.8 Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.64 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=-\frac {{\left (10 \, B c^{2} d^{3} - 12 \, B b c d^{2} e - 6 \, A c^{2} d^{2} e + 3 \, B b^{2} d e^{2} + 6 \, B a c d e^{2} + 6 \, A b c d e^{2} - 2 \, B a b e^{3} - A b^{2} e^{3} - 2 \, A a c e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{6}} - \frac {9 \, B c^{2} d^{5} - 14 \, B b c d^{4} e - 7 \, A c^{2} d^{4} e + 5 \, B b^{2} d^{3} e^{2} + 10 \, B a c d^{3} e^{2} + 10 \, A b c d^{3} e^{2} - 6 \, B a b d^{2} e^{3} - 3 \, A b^{2} d^{2} e^{3} - 6 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + 2 \, A a b d e^{4} + A a^{2} e^{5} + 2 \, {\left (5 \, B c^{2} d^{4} e - 8 \, B b c d^{3} e^{2} - 4 \, A c^{2} d^{3} e^{2} + 3 \, B b^{2} d^{2} e^{3} + 6 \, B a c d^{2} e^{3} + 6 \, A b c d^{2} e^{3} - 4 \, B a b d e^{4} - 2 \, A b^{2} d e^{4} - 4 \, A a c d e^{4} + B a^{2} e^{5} + 2 \, A a b e^{5}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{6}} + \frac {2 \, B c^{2} e^{6} x^{3} - 9 \, B c^{2} d e^{5} x^{2} + 6 \, B b c e^{6} x^{2} + 3 \, A c^{2} e^{6} x^{2} + 36 \, B c^{2} d^{2} e^{4} x - 36 \, B b c d e^{5} x - 18 \, A c^{2} d e^{5} x + 6 \, B b^{2} e^{6} x + 12 \, B a c e^{6} x + 12 \, A b c e^{6} x}{6 \, e^{9}} \]

input

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

output

-(10*B*c^2*d^3 - 12*B*b*c*d^2*e - 6*A*c^2*d^2*e + 3*B*b^2*d*e^2 + 6*B*a*c* 
d*e^2 + 6*A*b*c*d*e^2 - 2*B*a*b*e^3 - A*b^2*e^3 - 2*A*a*c*e^3)*log(abs(e*x 
 + d))/e^6 - 1/2*(9*B*c^2*d^5 - 14*B*b*c*d^4*e - 7*A*c^2*d^4*e + 5*B*b^2*d 
^3*e^2 + 10*B*a*c*d^3*e^2 + 10*A*b*c*d^3*e^2 - 6*B*a*b*d^2*e^3 - 3*A*b^2*d 
^2*e^3 - 6*A*a*c*d^2*e^3 + B*a^2*d*e^4 + 2*A*a*b*d*e^4 + A*a^2*e^5 + 2*(5* 
B*c^2*d^4*e - 8*B*b*c*d^3*e^2 - 4*A*c^2*d^3*e^2 + 3*B*b^2*d^2*e^3 + 6*B*a* 
c*d^2*e^3 + 6*A*b*c*d^2*e^3 - 4*B*a*b*d*e^4 - 2*A*b^2*d*e^4 - 4*A*a*c*d*e^ 
4 + B*a^2*e^5 + 2*A*a*b*e^5)*x)/((e*x + d)^2*e^6) + 1/6*(2*B*c^2*e^6*x^3 - 
 9*B*c^2*d*e^5*x^2 + 6*B*b*c*e^6*x^2 + 3*A*c^2*e^6*x^2 + 36*B*c^2*d^2*e^4* 
x - 36*B*b*c*d*e^5*x - 18*A*c^2*d*e^5*x + 6*B*b^2*e^6*x + 12*B*a*c*e^6*x + 
 12*A*b*c*e^6*x)/e^9
 

3.24.26.9 Mupad [B] (verification not implemented)

Time = 11.00 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.67 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^3} \, dx=x^2\,\left (\frac {A\,c^2+2\,B\,b\,c}{2\,e^3}-\frac {3\,B\,c^2\,d}{2\,e^4}\right )-\frac {x\,\left (B\,a^2\,e^4-4\,B\,a\,b\,d\,e^3+2\,A\,a\,b\,e^4+6\,B\,a\,c\,d^2\,e^2-4\,A\,a\,c\,d\,e^3+3\,B\,b^2\,d^2\,e^2-2\,A\,b^2\,d\,e^3-8\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2+5\,B\,c^2\,d^4-4\,A\,c^2\,d^3\,e\right )+\frac {B\,a^2\,d\,e^4+A\,a^2\,e^5-6\,B\,a\,b\,d^2\,e^3+2\,A\,a\,b\,d\,e^4+10\,B\,a\,c\,d^3\,e^2-6\,A\,a\,c\,d^2\,e^3+5\,B\,b^2\,d^3\,e^2-3\,A\,b^2\,d^2\,e^3-14\,B\,b\,c\,d^4\,e+10\,A\,b\,c\,d^3\,e^2+9\,B\,c^2\,d^5-7\,A\,c^2\,d^4\,e}{2\,e}}{d^2\,e^5+2\,d\,e^6\,x+e^7\,x^2}-x\,\left (\frac {3\,d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^3}-\frac {3\,B\,c^2\,d}{e^4}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b+2\,B\,a\,c}{e^3}+\frac {3\,B\,c^2\,d^2}{e^5}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-3\,B\,b^2\,d\,e^2+A\,b^2\,e^3+12\,B\,b\,c\,d^2\,e-6\,A\,b\,c\,d\,e^2+2\,B\,a\,b\,e^3-10\,B\,c^2\,d^3+6\,A\,c^2\,d^2\,e-6\,B\,a\,c\,d\,e^2+2\,A\,a\,c\,e^3\right )}{e^6}+\frac {B\,c^2\,x^3}{3\,e^3} \]

input

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

output

x^2*((A*c^2 + 2*B*b*c)/(2*e^3) - (3*B*c^2*d)/(2*e^4)) - (x*(B*a^2*e^4 + 5* 
B*c^2*d^4 + 2*A*a*b*e^4 - 2*A*b^2*d*e^3 - 4*A*c^2*d^3*e + 3*B*b^2*d^2*e^2 
- 4*A*a*c*d*e^3 - 4*B*a*b*d*e^3 - 8*B*b*c*d^3*e + 6*A*b*c*d^2*e^2 + 6*B*a* 
c*d^2*e^2) + (A*a^2*e^5 + 9*B*c^2*d^5 + B*a^2*d*e^4 - 7*A*c^2*d^4*e - 3*A* 
b^2*d^2*e^3 + 5*B*b^2*d^3*e^2 + 2*A*a*b*d*e^4 - 14*B*b*c*d^4*e - 6*A*a*c*d 
^2*e^3 - 6*B*a*b*d^2*e^3 + 10*A*b*c*d^3*e^2 + 10*B*a*c*d^3*e^2)/(2*e))/(d^ 
2*e^5 + e^7*x^2 + 2*d*e^6*x) - x*((3*d*((A*c^2 + 2*B*b*c)/e^3 - (3*B*c^2*d 
)/e^4))/e - (B*b^2 + 2*A*b*c + 2*B*a*c)/e^3 + (3*B*c^2*d^2)/e^5) + (log(d 
+ e*x)*(A*b^2*e^3 - 10*B*c^2*d^3 + 2*A*a*c*e^3 + 2*B*a*b*e^3 + 6*A*c^2*d^2 
*e - 3*B*b^2*d*e^2 - 6*A*b*c*d*e^2 - 6*B*a*c*d*e^2 + 12*B*b*c*d^2*e))/e^6 
+ (B*c^2*x^3)/(3*e^3)