\(\int \frac {(A+B x) (a+b x+c x^2)^2}{d+e x} \, dx\) [723]

Optimal result

Integrand size = 25, antiderivative size = 257 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=-\frac {\left (A e (c d-b e) \left (c d^2-e (b d-2 a e)\right )-B \left (c d^2-e (b d-a e)\right )^2\right ) x}{e^5}-\frac {\left (B (c d-b e) \left (c d^2-e (b d-2 a e)\right )-A e \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right )\right ) x^2}{2 e^4}-\frac {\left (A c e (c d-2 b e)-B \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right )\right ) x^3}{3 e^3}-\frac {c (B c d-2 b B e-A c e) x^4}{4 e^2}+\frac {B c^2 x^5}{5 e}-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2 \log (d+e x)}{e^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.16 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {e x \left (5 A e \left (6 b e^2 (-2 b d+4 a e+b e x)+c^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+4 c e \left (3 a e (-2 d+e x)+b \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )\right )+B \left (c^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+10 e^2 \left (6 a^2 e^2+6 a b e (-2 d+e x)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+10 c e \left (2 a e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )\right )\right )-60 (B d-A e) \left (c d^2+e (-b d+a e)\right )^2 \log (d+e x)}{60 e^6} \] Input:

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

Output:

(e*x*(5*A*e*(6*b*e^2*(-2*b*d + 4*a*e + b*e*x) + c^2*(-12*d^3 + 6*d^2*e*x - 
 4*d*e^2*x^2 + 3*e^3*x^3) + 4*c*e*(3*a*e*(-2*d + e*x) + b*(6*d^2 - 3*d*e*x 
 + 2*e^2*x^2))) + B*(c^2*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3* 
x^3 + 12*e^4*x^4) + 10*e^2*(6*a^2*e^2 + 6*a*b*e*(-2*d + e*x) + b^2*(6*d^2 
- 3*d*e*x + 2*e^2*x^2)) + 10*c*e*(2*a*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + b* 
(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)))) - 60*(B*d - A*e)*(c*d^2 
 + e*(-(b*d) + a*e))^2*Log[d + e*x])/(60*e^6)
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 257, 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.

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

\(\Big \downarrow \) 1195

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 1.49 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.65

method	result	size
norman	\(\frac {\left (2 A a b \,e^{4}-2 A a c d \,e^{3}-A \,b^{2} d \,e^{3}+2 A b c \,d^{2} e^{2}-A \,c^{2} d^{3} e +B \,e^{4} a^{2}-2 B a b d \,e^{3}+2 B a c \,d^{2} e^{2}+B \,b^{2} d^{2} e^{2}-2 B b c \,d^{3} e +B \,c^{2} d^{4}\right ) x}{e^{5}}+\frac {\left (2 A a c \,e^{3}+A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e +2 B a b \,e^{3}-2 B a d \,e^{2} c -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right ) x^{2}}{2 e^{4}}+\frac {\left (2 A b c \,e^{2}-A \,c^{2} d e +2 B \,e^{2} a c +B \,e^{2} b^{2}-2 B b c d e +B \,c^{2} d^{2}\right ) x^{3}}{3 e^{3}}+\frac {B \,c^{2} x^{5}}{5 e}+\frac {c \left (A c e +2 B b e -B c d \right ) x^{4}}{4 e^{2}}+\frac {\left (A \,a^{2} e^{5}-2 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}-2 A b c \,d^{3} e^{2}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}-2 B a c \,d^{3} e^{2}-B \,b^{2} d^{3} e^{2}+2 B b c \,d^{4} e -B \,c^{2} d^{5}\right ) \ln \left (e x +d \right )}{e^{6}}\)	\(424\)
default	\(\frac {-\frac {2}{3} B b c d \,e^{3} x^{3}-A b c d \,e^{3} x^{2}+\frac {1}{4} A \,c^{2} e^{4} x^{4}+\frac {2}{3} A b c \,e^{4} x^{3}-\frac {1}{4} B \,c^{2} d \,e^{3} x^{4}-2 B b c \,d^{3} e x +2 A b c \,d^{2} e^{2} x -A \,c^{2} d^{3} e x -2 A a c d \,e^{3} x +\frac {2}{3} B a c \,e^{4} x^{3}+\frac {1}{3} B \,c^{2} d^{2} e^{2} x^{3}+A a c \,e^{4} x^{2}+\frac {1}{2} A \,c^{2} d^{2} e^{2} x^{2}-\frac {1}{2} B \,c^{2} d^{3} e \,x^{2}+\frac {1}{2} B b c \,e^{4} x^{4}+B \,b^{2} d^{2} e^{2} x -A \,b^{2} d \,e^{3} x -\frac {1}{2} B \,b^{2} d \,e^{3} x^{2}-\frac {1}{3} A \,c^{2} d \,e^{3} x^{3}+B a b \,e^{4} x^{2}-B a c d \,e^{3} x^{2}+2 B a c \,d^{2} e^{2} x +B \,c^{2} d^{4} x +B b c \,d^{2} e^{2} x^{2}+2 A a b \,e^{4} x -2 B a b d \,e^{3} x +\frac {1}{3} B \,b^{2} e^{4} x^{3}+\frac {1}{5} B \,c^{2} x^{5} e^{4}+\frac {1}{2} A \,b^{2} e^{4} x^{2}+B \,a^{2} e^{4} x}{e^{5}}+\frac {\left (A \,a^{2} e^{5}-2 A a b d \,e^{4}+2 A a c \,d^{2} e^{3}+A \,b^{2} d^{2} e^{3}-2 A b c \,d^{3} e^{2}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}+2 B a b \,d^{2} e^{3}-2 B a c \,d^{3} e^{2}-B \,b^{2} d^{3} e^{2}+2 B b c \,d^{4} e -B \,c^{2} d^{5}\right ) \ln \left (e x +d \right )}{e^{6}}\)	\(486\)
risch	\(\frac {A \,c^{2} x^{4}}{4 e}+\frac {\ln \left (e x +d \right ) a^{2} A}{e}-\frac {A \,c^{2} d^{3} x}{e^{4}}+\frac {2 B a c \,x^{3}}{3 e}+\frac {B \,c^{2} d^{2} x^{3}}{3 e^{3}}-\frac {2 B b c d \,x^{3}}{3 e^{2}}-\frac {A b c d \,x^{2}}{e^{2}}-\frac {2 B b c \,d^{3} x}{e^{4}}+\frac {2 A b c \,d^{2} x}{e^{3}}-\frac {2 A a c d x}{e^{2}}-\frac {B a c d \,x^{2}}{e^{2}}+\frac {\ln \left (e x +d \right ) A \,b^{2} d^{2}}{e^{3}}-\frac {\ln \left (e x +d \right ) B \,a^{2} d}{e^{2}}-\frac {\ln \left (e x +d \right ) B \,b^{2} d^{3}}{e^{4}}-\frac {2 B a b d x}{e^{2}}+\frac {B a b \,x^{2}}{e}-\frac {B \,b^{2} d \,x^{2}}{2 e^{2}}+\frac {2 \ln \left (e x +d \right ) B a b \,d^{2}}{e^{3}}-\frac {\ln \left (e x +d \right ) B \,c^{2} d^{5}}{e^{6}}+\frac {2 A a b x}{e}+\frac {A a c \,x^{2}}{e}+\frac {A \,c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {B \,c^{2} d^{3} x^{2}}{2 e^{4}}+\frac {B b c \,x^{4}}{2 e}-\frac {A \,c^{2} d \,x^{3}}{3 e^{2}}+\frac {B \,c^{2} d^{4} x}{e^{5}}+\frac {\ln \left (e x +d \right ) A \,c^{2} d^{4}}{e^{5}}-\frac {2 \ln \left (e x +d \right ) A a b d}{e^{2}}-\frac {A \,b^{2} d x}{e^{2}}+\frac {B \,b^{2} d^{2} x}{e^{3}}+\frac {A \,b^{2} x^{2}}{2 e}+\frac {B \,a^{2} x}{e}+\frac {B \,b^{2} x^{3}}{3 e}+\frac {2 A b c \,x^{3}}{3 e}-\frac {B \,c^{2} d \,x^{4}}{4 e^{2}}+\frac {2 \ln \left (e x +d \right ) A a c \,d^{2}}{e^{3}}-\frac {2 \ln \left (e x +d \right ) A b c \,d^{3}}{e^{4}}-\frac {2 \ln \left (e x +d \right ) B a c \,d^{3}}{e^{4}}+\frac {B \,c^{2} x^{5}}{5 e}+\frac {2 B a c \,d^{2} x}{e^{3}}+\frac {B b c \,d^{2} x^{2}}{e^{3}}+\frac {2 \ln \left (e x +d \right ) B b c \,d^{4}}{e^{5}}\)	\(558\)
parallelrisch	\(\frac {-60 B \ln \left (e x +d \right ) b^{2} d^{3} e^{2}+30 B \,x^{4} b c \,e^{5}-15 B \,x^{4} c^{2} d \,e^{4}+40 A \,x^{3} b c \,e^{5}-20 A \,x^{3} c^{2} d \,e^{4}+40 B \,x^{3} a c \,e^{5}+20 B \,x^{3} c^{2} d^{2} e^{3}+60 A \,x^{2} a c \,e^{5}+30 A \,x^{2} c^{2} d^{2} e^{3}+60 B \,x^{2} a b \,e^{5}-30 B \,x^{2} b^{2} d \,e^{4}-30 B \,x^{2} c^{2} d^{3} e^{2}+120 A x a b \,e^{5}-60 A x \,b^{2} d \,e^{4}-60 A x \,c^{2} d^{3} e^{2}+60 B x \,b^{2} d^{2} e^{3}+60 B x \,c^{2} d^{4} e +60 A \ln \left (e x +d \right ) b^{2} d^{2} e^{3}+60 A \ln \left (e x +d \right ) c^{2} d^{4} e -60 B \ln \left (e x +d \right ) a^{2} d \,e^{4}+12 B \,x^{5} c^{2} e^{5}+60 A \ln \left (e x +d \right ) a^{2} e^{5}-60 B \ln \left (e x +d \right ) c^{2} d^{5}+15 A \,x^{4} c^{2} e^{5}+20 B \,x^{3} b^{2} e^{5}+30 A \,x^{2} b^{2} e^{5}+60 B x \,a^{2} e^{5}-120 A \ln \left (e x +d \right ) b c \,d^{3} e^{2}+120 B \ln \left (e x +d \right ) a b \,d^{2} e^{3}+60 B \,x^{2} b c \,d^{2} e^{3}-60 B \,x^{2} a c d \,e^{4}-120 A x a c d \,e^{4}+120 A x b c \,d^{2} e^{3}-40 B \,x^{3} b c d \,e^{4}-60 A \,x^{2} b c d \,e^{4}-120 B \ln \left (e x +d \right ) a c \,d^{3} e^{2}+120 B \ln \left (e x +d \right ) b c \,d^{4} e -120 B x a b d \,e^{4}+120 B x a c \,d^{2} e^{3}-120 B x b c \,d^{3} e^{2}-120 A \ln \left (e x +d \right ) a b d \,e^{4}+120 A \ln \left (e x +d \right ) a c \,d^{2} e^{3}}{60 e^{6}}\)	\(563\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.47 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {12 \, B c^{2} e^{5} x^{5} - 15 \, {\left (B c^{2} d e^{4} - {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \, {\left (B c^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} - 30 \, {\left (B c^{2} d^{3} e^{2} - {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 60 \, {\left (B c^{2} d^{4} e - {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x - 60 \, {\left (B c^{2} d^{5} - A a^{2} e^{5} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.45 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {B c^{2} x^{5}}{5 e} + x^{4} \left (\frac {A c^{2}}{4 e} + \frac {B b c}{2 e} - \frac {B c^{2} d}{4 e^{2}}\right ) + x^{3} \cdot \left (\frac {2 A b c}{3 e} - \frac {A c^{2} d}{3 e^{2}} + \frac {2 B a c}{3 e} + \frac {B b^{2}}{3 e} - \frac {2 B b c d}{3 e^{2}} + \frac {B c^{2} d^{2}}{3 e^{3}}\right ) + x^{2} \left (\frac {A a c}{e} + \frac {A b^{2}}{2 e} - \frac {A b c d}{e^{2}} + \frac {A c^{2} d^{2}}{2 e^{3}} + \frac {B a b}{e} - \frac {B a c d}{e^{2}} - \frac {B b^{2} d}{2 e^{2}} + \frac {B b c d^{2}}{e^{3}} - \frac {B c^{2} d^{3}}{2 e^{4}}\right ) + x \left (\frac {2 A a b}{e} - \frac {2 A a c d}{e^{2}} - \frac {A b^{2} d}{e^{2}} + \frac {2 A b c d^{2}}{e^{3}} - \frac {A c^{2} d^{3}}{e^{4}} + \frac {B a^{2}}{e} - \frac {2 B a b d}{e^{2}} + \frac {2 B a c d^{2}}{e^{3}} + \frac {B b^{2} d^{2}}{e^{3}} - \frac {2 B b c d^{3}}{e^{4}} + \frac {B c^{2} d^{4}}{e^{5}}\right ) - \frac {\left (- A e + B d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \log {\left (d + e x \right )}}{e^{6}} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.47 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {12 \, B c^{2} e^{4} x^{5} - 15 \, {\left (B c^{2} d e^{3} - {\left (2 \, B b c + A c^{2}\right )} e^{4}\right )} x^{4} + 20 \, {\left (B c^{2} d^{2} e^{2} - {\left (2 \, B b c + A c^{2}\right )} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{4}\right )} x^{3} - 30 \, {\left (B c^{2} d^{3} e - {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{2} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{3} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{4}\right )} x^{2} + 60 \, {\left (B c^{2} d^{4} - {\left (2 \, B b c + A c^{2}\right )} d^{3} e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{2} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x}{60 \, e^{5}} - \frac {{\left (B c^{2} d^{5} - A a^{2} e^{5} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} - {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.92 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {12 \, B c^{2} e^{4} x^{5} - 15 \, B c^{2} d e^{3} x^{4} + 30 \, B b c e^{4} x^{4} + 15 \, A c^{2} e^{4} x^{4} + 20 \, B c^{2} d^{2} e^{2} x^{3} - 40 \, B b c d e^{3} x^{3} - 20 \, A c^{2} d e^{3} x^{3} + 20 \, B b^{2} e^{4} x^{3} + 40 \, B a c e^{4} x^{3} + 40 \, A b c e^{4} x^{3} - 30 \, B c^{2} d^{3} e x^{2} + 60 \, B b c d^{2} e^{2} x^{2} + 30 \, A c^{2} d^{2} e^{2} x^{2} - 30 \, B b^{2} d e^{3} x^{2} - 60 \, B a c d e^{3} x^{2} - 60 \, A b c d e^{3} x^{2} + 60 \, B a b e^{4} x^{2} + 30 \, A b^{2} e^{4} x^{2} + 60 \, A a c e^{4} x^{2} + 60 \, B c^{2} d^{4} x - 120 \, B b c d^{3} e x - 60 \, A c^{2} d^{3} e x + 60 \, B b^{2} d^{2} e^{2} x + 120 \, B a c d^{2} e^{2} x + 120 \, A b c d^{2} e^{2} x - 120 \, B a b d e^{3} x - 60 \, A b^{2} d e^{3} x - 120 \, A a c d e^{3} x + 60 \, B a^{2} e^{4} x + 120 \, A a b e^{4} x}{60 \, e^{5}} - \frac {{\left (B c^{2} d^{5} - 2 \, B b c d^{4} e - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, B a c d^{3} e^{2} + 2 \, A b c d^{3} e^{2} - 2 \, B a b d^{2} e^{3} - A b^{2} d^{2} e^{3} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} + 2 \, A a b d e^{4} - A a^{2} e^{5}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{6}} \] Input:

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

Output:

1/60*(12*B*c^2*e^4*x^5 - 15*B*c^2*d*e^3*x^4 + 30*B*b*c*e^4*x^4 + 15*A*c^2* 
e^4*x^4 + 20*B*c^2*d^2*e^2*x^3 - 40*B*b*c*d*e^3*x^3 - 20*A*c^2*d*e^3*x^3 + 
 20*B*b^2*e^4*x^3 + 40*B*a*c*e^4*x^3 + 40*A*b*c*e^4*x^3 - 30*B*c^2*d^3*e*x 
^2 + 60*B*b*c*d^2*e^2*x^2 + 30*A*c^2*d^2*e^2*x^2 - 30*B*b^2*d*e^3*x^2 - 60 
*B*a*c*d*e^3*x^2 - 60*A*b*c*d*e^3*x^2 + 60*B*a*b*e^4*x^2 + 30*A*b^2*e^4*x^ 
2 + 60*A*a*c*e^4*x^2 + 60*B*c^2*d^4*x - 120*B*b*c*d^3*e*x - 60*A*c^2*d^3*e 
*x + 60*B*b^2*d^2*e^2*x + 120*B*a*c*d^2*e^2*x + 120*A*b*c*d^2*e^2*x - 120* 
B*a*b*d*e^3*x - 60*A*b^2*d*e^3*x - 120*A*a*c*d*e^3*x + 60*B*a^2*e^4*x + 12 
0*A*a*b*e^4*x)/e^5 - (B*c^2*d^5 - 2*B*b*c*d^4*e - 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 + B*a^2*d*e^4 + 2*A*a*b*d*e^4 - A*a^2*e^5)*log(abs(e*x + 
 d))/e^6
 

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.65 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=x^3\,\left (\frac {B\,b^2+2\,A\,c\,b+2\,B\,a\,c}{3\,e}-\frac {d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e}-\frac {B\,c^2\,d}{e^2}\right )}{3\,e}\right )+x\,\left (\frac {B\,a^2+2\,A\,b\,a}{e}-\frac {d\,\left (\frac {A\,b^2+2\,B\,a\,b+2\,A\,a\,c}{e}-\frac {d\,\left (\frac {B\,b^2+2\,A\,c\,b+2\,B\,a\,c}{e}-\frac {d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e}-\frac {B\,c^2\,d}{e^2}\right )}{e}\right )}{e}\right )}{e}\right )+x^4\,\left (\frac {A\,c^2+2\,B\,b\,c}{4\,e}-\frac {B\,c^2\,d}{4\,e^2}\right )+x^2\,\left (\frac {A\,b^2+2\,B\,a\,b+2\,A\,a\,c}{2\,e}-\frac {d\,\left (\frac {B\,b^2+2\,A\,c\,b+2\,B\,a\,c}{e}-\frac {d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e}-\frac {B\,c^2\,d}{e^2}\right )}{e}\right )}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (-B\,a^2\,d\,e^4+A\,a^2\,e^5+2\,B\,a\,b\,d^2\,e^3-2\,A\,a\,b\,d\,e^4-2\,B\,a\,c\,d^3\,e^2+2\,A\,a\,c\,d^2\,e^3-B\,b^2\,d^3\,e^2+A\,b^2\,d^2\,e^3+2\,B\,b\,c\,d^4\,e-2\,A\,b\,c\,d^3\,e^2-B\,c^2\,d^5+A\,c^2\,d^4\,e\right )}{e^6}+\frac {B\,c^2\,x^5}{5\,e} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.75 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {-180 \,\mathrm {log}\left (e x +d \right ) a^{2} b d \,e^{4}+120 \,\mathrm {log}\left (e x +d \right ) a^{2} c \,d^{2} e^{3}-120 a^{2} c d \,e^{4} x -240 \,\mathrm {log}\left (e x +d \right ) a b c \,d^{3} e^{2}+240 a b c \,d^{2} e^{3} x -120 a b c d \,e^{4} x^{2}+60 \,\mathrm {log}\left (e x +d \right ) a^{3} e^{5}-60 \,\mathrm {log}\left (e x +d \right ) b^{3} d^{3} e^{2}-60 \,\mathrm {log}\left (e x +d \right ) b \,c^{2} d^{5}+90 a \,b^{2} e^{5} x^{2}+15 a \,c^{2} e^{5} x^{4}+60 b^{3} d^{2} e^{3} x -30 b^{3} d \,e^{4} x^{2}+30 b^{2} c \,e^{5} x^{4}+12 b \,c^{2} e^{5} x^{5}+20 b^{3} e^{5} x^{3}+180 a^{2} b \,e^{5} x +60 a^{2} c \,e^{5} x^{2}-15 b \,c^{2} d \,e^{4} x^{4}+180 \,\mathrm {log}\left (e x +d \right ) a \,b^{2} d^{2} e^{3}+60 \,\mathrm {log}\left (e x +d \right ) a \,c^{2} d^{4} e +120 \,\mathrm {log}\left (e x +d \right ) b^{2} c \,d^{4} e -180 a \,b^{2} d \,e^{4} x +80 a b c \,e^{5} x^{3}-60 a \,c^{2} d^{3} e^{2} x +30 a \,c^{2} d^{2} e^{3} x^{2}-20 a \,c^{2} d \,e^{4} x^{3}-120 b^{2} c \,d^{3} e^{2} x +60 b^{2} c \,d^{2} e^{3} x^{2}-40 b^{2} c d \,e^{4} x^{3}+60 b \,c^{2} d^{4} e x -30 b \,c^{2} d^{3} e^{2} x^{2}+20 b \,c^{2} d^{2} e^{3} x^{3}}{60 e^{6}} \] Input:

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

Output:

(60*log(d + e*x)*a**3*e**5 - 180*log(d + e*x)*a**2*b*d*e**4 + 120*log(d + 
e*x)*a**2*c*d**2*e**3 + 180*log(d + e*x)*a*b**2*d**2*e**3 - 240*log(d + e* 
x)*a*b*c*d**3*e**2 + 60*log(d + e*x)*a*c**2*d**4*e - 60*log(d + e*x)*b**3* 
d**3*e**2 + 120*log(d + e*x)*b**2*c*d**4*e - 60*log(d + e*x)*b*c**2*d**5 + 
 180*a**2*b*e**5*x - 120*a**2*c*d*e**4*x + 60*a**2*c*e**5*x**2 - 180*a*b** 
2*d*e**4*x + 90*a*b**2*e**5*x**2 + 240*a*b*c*d**2*e**3*x - 120*a*b*c*d*e** 
4*x**2 + 80*a*b*c*e**5*x**3 - 60*a*c**2*d**3*e**2*x + 30*a*c**2*d**2*e**3* 
x**2 - 20*a*c**2*d*e**4*x**3 + 15*a*c**2*e**5*x**4 + 60*b**3*d**2*e**3*x - 
 30*b**3*d*e**4*x**2 + 20*b**3*e**5*x**3 - 120*b**2*c*d**3*e**2*x + 60*b** 
2*c*d**2*e**3*x**2 - 40*b**2*c*d*e**4*x**3 + 30*b**2*c*e**5*x**4 + 60*b*c* 
*2*d**4*e*x - 30*b*c**2*d**3*e**2*x**2 + 20*b*c**2*d**2*e**3*x**3 - 15*b*c 
**2*d*e**4*x**4 + 12*b*c**2*e**5*x**5)/(60*e**6)