3.12.57 \(\int \frac {(A+B x) (d+e x)^2}{(b x+c x^2)^3} \, dx\) [1157]

3.12.57.1 Optimal result

Integrand size = 24, antiderivative size = 198 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=-\frac {A d^2}{2 b^3 x^2}-\frac {d (b B d-3 A c d+2 A b e)}{b^4 x}-\frac {(b B-A c) (c d-b e)^2}{2 b^3 c (b+c x)^2}-\frac {(c d-b e) (2 b B d-3 A c d+A b e)}{b^4 (b+c x)}+\frac {\left (6 A c^2 d^2+b^2 e (2 B d+A e)-3 b c d (B d+2 A e)\right ) \log (x)}{b^5}-\frac {\left (6 A c^2 d^2+b^2 e (2 B d+A e)-3 b c d (B d+2 A e)\right ) \log (b+c x)}{b^5} \]

output

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

3.12.57.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {A b^2 d^2}{x^2}+\frac {2 b d (b B d-3 A c d+2 A b e)}{x}+\frac {b^2 (b B-A c) (c d-b e)^2}{c (b+c x)^2}-\frac {2 b (-c d+b e) (2 b B d-3 A c d+A b e)}{b+c x}-2 \left (6 A c^2 d^2+b^2 e (2 B d+A e)-3 b c d (B d+2 A e)\right ) \log (x)+2 \left (6 A c^2 d^2+b^2 e (2 B d+A e)-3 b c d (B d+2 A e)\right ) \log (b+c x)}{2 b^5} \]

input

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

output

-1/2*((A*b^2*d^2)/x^2 + (2*b*d*(b*B*d - 3*A*c*d + 2*A*b*e))/x + (b^2*(b*B 
- A*c)*(c*d - b*e)^2)/(c*(b + c*x)^2) - (2*b*(-(c*d) + b*e)*(2*b*B*d - 3*A 
*c*d + A*b*e))/(b + c*x) - 2*(6*A*c^2*d^2 + b^2*e*(2*B*d + A*e) - 3*b*c*d* 
(B*d + 2*A*e))*Log[x] + 2*(6*A*c^2*d^2 + b^2*e*(2*B*d + A*e) - 3*b*c*d*(B* 
d + 2*A*e))*Log[b + c*x])/b^5
 

3.12.57.3 Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 198, 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 = {1206, 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.

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

\(\Big \downarrow \) 1206

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.12.57.3.1 Defintions of rubi rules used

rule 1206

Int[((d_) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((b_.)*(x_) + (c_.) 
*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[x^p*(d + e*x)^m*(f + g*x)^n 
*(b + c*x)^p, x], x] /; FreeQ[{b, c, d, e, f, g}, x] && ILtQ[p, -1] && Inte 
gersQ[m, n]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.12.57.4 Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.31

method	result	size
default	\(\frac {\left (A \,b^{2} e^{2}-6 A b c d e +6 A \,c^{2} d^{2}+2 B \,b^{2} d e -3 B b c \,d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {A \,d^{2}}{2 b^{3} x^{2}}-\frac {d \left (2 A b e -3 A c d +B b d \right )}{b^{4} x}-\frac {-A \,b^{2} c \,e^{2}+2 A b \,c^{2} d e -A \,c^{3} d^{2}+b^{3} B \,e^{2}-2 B \,b^{2} c d e +B b \,c^{2} d^{2}}{2 b^{3} c \left (c x +b \right )^{2}}-\frac {\left (A \,b^{2} e^{2}-6 A b c d e +6 A \,c^{2} d^{2}+2 B \,b^{2} d e -3 B b c \,d^{2}\right ) \ln \left (c x +b \right )}{b^{5}}+\frac {A \,b^{2} e^{2}-4 A b c d e +3 A \,c^{2} d^{2}+2 B \,b^{2} d e -2 B b c \,d^{2}}{b^{4} \left (c x +b \right )}\)	\(259\)
norman	\(\frac {\frac {\left (A \,b^{2} c^{2} e^{2}-6 A b \,c^{3} d e +6 A \,c^{4} d^{2}+2 B \,b^{2} c^{2} d e -3 B b \,c^{3} d^{2}\right ) x^{3}}{b^{4} c}-\frac {A \,d^{2}}{2 b}-\frac {d \left (2 A b e -2 A c d +B b d \right ) x}{b^{2}}+\frac {\left (3 A \,b^{2} c^{2} e^{2}-18 A b \,c^{3} d e +18 A \,c^{4} d^{2}-B \,b^{3} c \,e^{2}+6 B \,b^{2} c^{2} d e -9 B b \,c^{3} d^{2}\right ) x^{2}}{2 c^{2} b^{3}}}{x^{2} \left (c x +b \right )^{2}}+\frac {\left (A \,b^{2} e^{2}-6 A b c d e +6 A \,c^{2} d^{2}+2 B \,b^{2} d e -3 B b c \,d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {\left (A \,b^{2} e^{2}-6 A b c d e +6 A \,c^{2} d^{2}+2 B \,b^{2} d e -3 B b c \,d^{2}\right ) \ln \left (c x +b \right )}{b^{5}}\)	\(278\)
risch	\(\frac {\frac {c \left (A \,b^{2} e^{2}-6 A b c d e +6 A \,c^{2} d^{2}+2 B \,b^{2} d e -3 B b c \,d^{2}\right ) x^{3}}{b^{4}}+\frac {\left (3 A \,b^{2} c \,e^{2}-18 A b \,c^{2} d e +18 A \,c^{3} d^{2}-b^{3} B \,e^{2}+6 B \,b^{2} c d e -9 B b \,c^{2} d^{2}\right ) x^{2}}{2 c \,b^{3}}-\frac {d \left (2 A b e -2 A c d +B b d \right ) x}{b^{2}}-\frac {A \,d^{2}}{2 b}}{x^{2} \left (c x +b \right )^{2}}-\frac {\ln \left (c x +b \right ) A \,e^{2}}{b^{3}}+\frac {6 \ln \left (c x +b \right ) A c d e}{b^{4}}-\frac {6 \ln \left (c x +b \right ) A \,c^{2} d^{2}}{b^{5}}-\frac {2 \ln \left (c x +b \right ) B d e}{b^{3}}+\frac {3 \ln \left (c x +b \right ) B c \,d^{2}}{b^{4}}+\frac {\ln \left (-x \right ) A \,e^{2}}{b^{3}}-\frac {6 \ln \left (-x \right ) A c d e}{b^{4}}+\frac {6 \ln \left (-x \right ) A \,c^{2} d^{2}}{b^{5}}+\frac {2 \ln \left (-x \right ) B d e}{b^{3}}-\frac {3 \ln \left (-x \right ) B c \,d^{2}}{b^{4}}\)	\(307\)
parallelrisch	\(\frac {2 A \ln \left (x \right ) x^{4} b^{2} c^{4} e^{2}-2 A \ln \left (c x +b \right ) x^{4} b^{2} c^{4} e^{2}-6 B \ln \left (x \right ) x^{4} b \,c^{5} d^{2}+6 B \ln \left (c x +b \right ) x^{4} b \,c^{5} d^{2}+4 A \ln \left (x \right ) x^{3} b^{3} c^{3} e^{2}+24 A \ln \left (x \right ) x^{3} b \,c^{5} d^{2}-4 A \ln \left (c x +b \right ) x^{3} b^{3} c^{3} e^{2}-24 A \ln \left (c x +b \right ) x^{3} b \,c^{5} d^{2}-12 B \ln \left (x \right ) x^{3} b^{2} c^{4} d^{2}+12 B \ln \left (c x +b \right ) x^{3} b^{2} c^{4} d^{2}+2 A \ln \left (x \right ) x^{2} b^{4} c^{2} e^{2}+12 A \ln \left (x \right ) x^{2} b^{2} c^{4} d^{2}-2 A \ln \left (c x +b \right ) x^{2} b^{4} c^{2} e^{2}-12 A \ln \left (c x +b \right ) x^{2} b^{2} c^{4} d^{2}-6 B \ln \left (x \right ) x^{2} b^{3} c^{3} d^{2}+6 B \ln \left (c x +b \right ) x^{2} b^{3} c^{3} d^{2}-12 A \,x^{3} b^{2} c^{4} d e +4 B \,x^{3} b^{3} c^{3} d e -18 A \,x^{2} b^{3} c^{3} d e +6 B \,x^{2} b^{4} c^{2} d e -4 A x \,b^{4} c^{2} d e -A \,b^{4} c^{2} d^{2}+18 A \,x^{2} b^{2} c^{4} d^{2}-B \,x^{2} b^{5} c \,e^{2}-9 B \,x^{2} b^{3} c^{3} d^{2}+4 A x \,b^{3} c^{3} d^{2}-2 B x \,b^{4} c^{2} d^{2}+12 A \ln \left (x \right ) x^{4} c^{6} d^{2}-12 A \ln \left (c x +b \right ) x^{4} c^{6} d^{2}+2 A \,x^{3} b^{3} c^{3} e^{2}+12 A \,x^{3} b \,c^{5} d^{2}-6 B \,x^{3} b^{2} c^{4} d^{2}+3 A \,x^{2} b^{4} c^{2} e^{2}-12 A \ln \left (x \right ) x^{4} b \,c^{5} d e +12 A \ln \left (c x +b \right ) x^{4} b \,c^{5} d e +4 B \ln \left (x \right ) x^{4} b^{2} c^{4} d e -4 B \ln \left (c x +b \right ) x^{4} b^{2} c^{4} d e -24 A \ln \left (x \right ) x^{3} b^{2} c^{4} d e +24 A \ln \left (c x +b \right ) x^{3} b^{2} c^{4} d e +8 B \ln \left (x \right ) x^{3} b^{3} c^{3} d e -8 B \ln \left (c x +b \right ) x^{3} b^{3} c^{3} d e -12 A \ln \left (x \right ) x^{2} b^{3} c^{3} d e +12 A \ln \left (c x +b \right ) x^{2} b^{3} c^{3} d e +4 B \ln \left (x \right ) x^{2} b^{4} c^{2} d e -4 B \ln \left (c x +b \right ) x^{2} b^{4} c^{2} d e}{2 c^{2} b^{5} x^{2} \left (c x +b \right )^{2}}\)	\(767\)

input

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

output

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

3.12.57.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (194) = 388\).

Time = 0.33 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.82 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=-\frac {A b^{4} c d^{2} - 2 \, {\left (A b^{3} c^{2} e^{2} - 3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 2 \, {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d e\right )} x^{3} + {\left (9 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} - 6 \, {\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e + {\left (B b^{5} - 3 \, A b^{4} c\right )} e^{2}\right )} x^{2} + 2 \, {\left (2 \, A b^{4} c d e + {\left (B b^{4} c - 2 \, A b^{3} c^{2}\right )} d^{2}\right )} x + 2 \, {\left ({\left (A b^{2} c^{3} e^{2} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} + 2 \, {\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d e\right )} x^{4} + 2 \, {\left (A b^{3} c^{2} e^{2} - 3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 2 \, {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d e\right )} x^{3} + {\left (A b^{4} c e^{2} - 3 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} + 2 \, {\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (A b^{2} c^{3} e^{2} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} + 2 \, {\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d e\right )} x^{4} + 2 \, {\left (A b^{3} c^{2} e^{2} - 3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} + 2 \, {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d e\right )} x^{3} + {\left (A b^{4} c e^{2} - 3 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} + 2 \, {\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{3} x^{4} + 2 \, b^{6} c^{2} x^{3} + b^{7} c x^{2}\right )}} \]

input

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

output

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

3.12.57.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (197) = 394\).

Time = 4.45 (sec) , antiderivative size = 660, normalized size of antiderivative = 3.33 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=\frac {- A b^{3} c d^{2} + x^{3} \cdot \left (2 A b^{2} c^{2} e^{2} - 12 A b c^{3} d e + 12 A c^{4} d^{2} + 4 B b^{2} c^{2} d e - 6 B b c^{3} d^{2}\right ) + x^{2} \cdot \left (3 A b^{3} c e^{2} - 18 A b^{2} c^{2} d e + 18 A b c^{3} d^{2} - B b^{4} e^{2} + 6 B b^{3} c d e - 9 B b^{2} c^{2} d^{2}\right ) + x \left (- 4 A b^{3} c d e + 4 A b^{2} c^{2} d^{2} - 2 B b^{3} c d^{2}\right )}{2 b^{6} c x^{2} + 4 b^{5} c^{2} x^{3} + 2 b^{4} c^{3} x^{4}} + \frac {\left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right ) \log {\left (x + \frac {A b^{3} e^{2} - 6 A b^{2} c d e + 6 A b c^{2} d^{2} + 2 B b^{3} d e - 3 B b^{2} c d^{2} - b \left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right )}{2 A b^{2} c e^{2} - 12 A b c^{2} d e + 12 A c^{3} d^{2} + 4 B b^{2} c d e - 6 B b c^{2} d^{2}} \right )}}{b^{5}} - \frac {\left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right ) \log {\left (x + \frac {A b^{3} e^{2} - 6 A b^{2} c d e + 6 A b c^{2} d^{2} + 2 B b^{3} d e - 3 B b^{2} c d^{2} + b \left (A b^{2} e^{2} - 6 A b c d e + 6 A c^{2} d^{2} + 2 B b^{2} d e - 3 B b c d^{2}\right )}{2 A b^{2} c e^{2} - 12 A b c^{2} d e + 12 A c^{3} d^{2} + 4 B b^{2} c d e - 6 B b c^{2} d^{2}} \right )}}{b^{5}} \]

input

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

output

(-A*b**3*c*d**2 + x**3*(2*A*b**2*c**2*e**2 - 12*A*b*c**3*d*e + 12*A*c**4*d 
**2 + 4*B*b**2*c**2*d*e - 6*B*b*c**3*d**2) + x**2*(3*A*b**3*c*e**2 - 18*A* 
b**2*c**2*d*e + 18*A*b*c**3*d**2 - B*b**4*e**2 + 6*B*b**3*c*d*e - 9*B*b**2 
*c**2*d**2) + x*(-4*A*b**3*c*d*e + 4*A*b**2*c**2*d**2 - 2*B*b**3*c*d**2))/ 
(2*b**6*c*x**2 + 4*b**5*c**2*x**3 + 2*b**4*c**3*x**4) + (A*b**2*e**2 - 6*A 
*b*c*d*e + 6*A*c**2*d**2 + 2*B*b**2*d*e - 3*B*b*c*d**2)*log(x + (A*b**3*e* 
*2 - 6*A*b**2*c*d*e + 6*A*b*c**2*d**2 + 2*B*b**3*d*e - 3*B*b**2*c*d**2 - b 
*(A*b**2*e**2 - 6*A*b*c*d*e + 6*A*c**2*d**2 + 2*B*b**2*d*e - 3*B*b*c*d**2) 
)/(2*A*b**2*c*e**2 - 12*A*b*c**2*d*e + 12*A*c**3*d**2 + 4*B*b**2*c*d*e - 6 
*B*b*c**2*d**2))/b**5 - (A*b**2*e**2 - 6*A*b*c*d*e + 6*A*c**2*d**2 + 2*B*b 
**2*d*e - 3*B*b*c*d**2)*log(x + (A*b**3*e**2 - 6*A*b**2*c*d*e + 6*A*b*c**2 
*d**2 + 2*B*b**3*d*e - 3*B*b**2*c*d**2 + b*(A*b**2*e**2 - 6*A*b*c*d*e + 6* 
A*c**2*d**2 + 2*B*b**2*d*e - 3*B*b*c*d**2))/(2*A*b**2*c*e**2 - 12*A*b*c**2 
*d*e + 12*A*c**3*d**2 + 4*B*b**2*c*d*e - 6*B*b*c**2*d**2))/b**5
 

3.12.57.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=-\frac {A b^{3} c d^{2} - 2 \, {\left (A b^{2} c^{2} e^{2} - 3 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} + 2 \, {\left (B b^{2} c^{2} - 3 \, A b c^{3}\right )} d e\right )} x^{3} + {\left (9 \, {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{2} - 6 \, {\left (B b^{3} c - 3 \, A b^{2} c^{2}\right )} d e + {\left (B b^{4} - 3 \, A b^{3} c\right )} e^{2}\right )} x^{2} + 2 \, {\left (2 \, A b^{3} c d e + {\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} d^{2}\right )} x}{2 \, {\left (b^{4} c^{3} x^{4} + 2 \, b^{5} c^{2} x^{3} + b^{6} c x^{2}\right )}} - \frac {{\left (A b^{2} e^{2} - 3 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} + 2 \, {\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \left (c x + b\right )}{b^{5}} + \frac {{\left (A b^{2} e^{2} - 3 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} + 2 \, {\left (B b^{2} - 3 \, A b c\right )} d e\right )} \log \left (x\right )}{b^{5}} \]

input

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

output

-1/2*(A*b^3*c*d^2 - 2*(A*b^2*c^2*e^2 - 3*(B*b*c^3 - 2*A*c^4)*d^2 + 2*(B*b^ 
2*c^2 - 3*A*b*c^3)*d*e)*x^3 + (9*(B*b^2*c^2 - 2*A*b*c^3)*d^2 - 6*(B*b^3*c 
- 3*A*b^2*c^2)*d*e + (B*b^4 - 3*A*b^3*c)*e^2)*x^2 + 2*(2*A*b^3*c*d*e + (B* 
b^3*c - 2*A*b^2*c^2)*d^2)*x)/(b^4*c^3*x^4 + 2*b^5*c^2*x^3 + b^6*c*x^2) - ( 
A*b^2*e^2 - 3*(B*b*c - 2*A*c^2)*d^2 + 2*(B*b^2 - 3*A*b*c)*d*e)*log(c*x + b 
)/b^5 + (A*b^2*e^2 - 3*(B*b*c - 2*A*c^2)*d^2 + 2*(B*b^2 - 3*A*b*c)*d*e)*lo 
g(x)/b^5
 

3.12.57.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.62 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=-\frac {{\left (3 \, B b c d^{2} - 6 \, A c^{2} d^{2} - 2 \, B b^{2} d e + 6 \, A b c d e - A b^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {{\left (3 \, B b c^{2} d^{2} - 6 \, A c^{3} d^{2} - 2 \, B b^{2} c d e + 6 \, A b c^{2} d e - A b^{2} c e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac {6 \, B b c^{3} d^{2} x^{3} - 12 \, A c^{4} d^{2} x^{3} - 4 \, B b^{2} c^{2} d e x^{3} + 12 \, A b c^{3} d e x^{3} - 2 \, A b^{2} c^{2} e^{2} x^{3} + 9 \, B b^{2} c^{2} d^{2} x^{2} - 18 \, A b c^{3} d^{2} x^{2} - 6 \, B b^{3} c d e x^{2} + 18 \, A b^{2} c^{2} d e x^{2} + B b^{4} e^{2} x^{2} - 3 \, A b^{3} c e^{2} x^{2} + 2 \, B b^{3} c d^{2} x - 4 \, A b^{2} c^{2} d^{2} x + 4 \, A b^{3} c d e x + A b^{3} c d^{2}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4} c} \]

input

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

output

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

3.12.57.9 Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.61 \[ \int \frac {(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {A\,d^2}{2\,b}-\frac {c\,x^3\,\left (2\,B\,b^2\,d\,e+A\,b^2\,e^2-3\,B\,b\,c\,d^2-6\,A\,b\,c\,d\,e+6\,A\,c^2\,d^2\right )}{b^4}+\frac {d\,x\,\left (2\,A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b^2}-\frac {x^2\,\left (-B\,b^3\,e^2+6\,B\,b^2\,c\,d\,e+3\,A\,b^2\,c\,e^2-9\,B\,b\,c^2\,d^2-18\,A\,b\,c^2\,d\,e+18\,A\,c^3\,d^2\right )}{2\,b^3\,c}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {2\,\mathrm {atanh}\left (\frac {\left (b+2\,c\,x\right )\,\left (b^2\,\left (A\,e^2+2\,B\,d\,e\right )-b\,\left (3\,B\,c\,d^2+6\,A\,c\,e\,d\right )+6\,A\,c^2\,d^2\right )}{b\,\left (2\,B\,b^2\,d\,e+A\,b^2\,e^2-3\,B\,b\,c\,d^2-6\,A\,b\,c\,d\,e+6\,A\,c^2\,d^2\right )}\right )\,\left (b^2\,\left (A\,e^2+2\,B\,d\,e\right )-b\,\left (3\,B\,c\,d^2+6\,A\,c\,e\,d\right )+6\,A\,c^2\,d^2\right )}{b^5} \]

input

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

output

- ((A*d^2)/(2*b) - (c*x^3*(A*b^2*e^2 + 6*A*c^2*d^2 - 3*B*b*c*d^2 + 2*B*b^2 
*d*e - 6*A*b*c*d*e))/b^4 + (d*x*(2*A*b*e - 2*A*c*d + B*b*d))/b^2 - (x^2*(1 
8*A*c^3*d^2 - B*b^3*e^2 + 3*A*b^2*c*e^2 - 9*B*b*c^2*d^2 - 18*A*b*c^2*d*e + 
 6*B*b^2*c*d*e))/(2*b^3*c))/(b^2*x^2 + c^2*x^4 + 2*b*c*x^3) - (2*atanh(((b 
 + 2*c*x)*(b^2*(A*e^2 + 2*B*d*e) - b*(3*B*c*d^2 + 6*A*c*d*e) + 6*A*c^2*d^2 
))/(b*(A*b^2*e^2 + 6*A*c^2*d^2 - 3*B*b*c*d^2 + 2*B*b^2*d*e - 6*A*b*c*d*e)) 
)*(b^2*(A*e^2 + 2*B*d*e) - b*(3*B*c*d^2 + 6*A*c*d*e) + 6*A*c^2*d^2))/b^5