\(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx\) [52]

Optimal result

Integrand size = 20, antiderivative size = 276 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) x}{e^7}+\frac {(b d-a e)^6 (B d-A e)}{2 e^8 (d+e x)^2}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{e^8 (d+e x)}-\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) (d+e x)^2}{2 e^8}+\frac {b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^3}{e^8}-\frac {b^5 (7 b B d-A b e-6 a B e) (d+e x)^4}{4 e^8}+\frac {b^6 B (d+e x)^5}{5 e^8}-\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) \log (d+e x)}{e^8} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=\frac {-20 b^2 e \left (-15 a^4 B e^4+12 a b^3 d^2 e (5 B d-3 A e)-5 b^4 d^3 (3 B d-2 A e)-20 a^3 b e^3 (-3 B d+A e)+45 a^2 b^2 d e^2 (-2 B d+A e)\right ) x+10 b^3 e^2 \left (20 a^3 B e^3+18 a b^2 d e (2 B d-A e)+15 a^2 b e^2 (-3 B d+A e)+2 b^3 d^2 (-5 B d+3 A e)\right ) x^2-20 b^4 e^3 \left (-5 a^2 B e^2-2 a b e (-3 B d+A e)+b^2 d (-2 B d+A e)\right ) x^3+5 b^5 e^4 (-3 b B d+A b e+6 a B e) x^4+4 b^6 B e^5 x^5+\frac {10 (b d-a e)^6 (B d-A e)}{(d+e x)^2}-\frac {20 (b d-a e)^5 (7 b B d-6 A b e-a B e)}{d+e x}-60 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e) \log (d+e x)}{20 e^8} \] Input:

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^3,x]
 

Output:

(-20*b^2*e*(-15*a^4*B*e^4 + 12*a*b^3*d^2*e*(5*B*d - 3*A*e) - 5*b^4*d^3*(3* 
B*d - 2*A*e) - 20*a^3*b*e^3*(-3*B*d + A*e) + 45*a^2*b^2*d*e^2*(-2*B*d + A* 
e))*x + 10*b^3*e^2*(20*a^3*B*e^3 + 18*a*b^2*d*e*(2*B*d - A*e) + 15*a^2*b*e 
^2*(-3*B*d + A*e) + 2*b^3*d^2*(-5*B*d + 3*A*e))*x^2 - 20*b^4*e^3*(-5*a^2*B 
*e^2 - 2*a*b*e*(-3*B*d + A*e) + b^2*d*(-2*B*d + A*e))*x^3 + 5*b^5*e^4*(-3* 
b*B*d + A*b*e + 6*a*B*e)*x^4 + 4*b^6*B*e^5*x^5 + (10*(b*d - a*e)^6*(B*d - 
A*e))/(d + e*x)^2 - (20*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(d + e* 
x) - 60*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*Log[d + e*x])/(20*e^ 
8)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.

\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^5 (d+e x)^4 (-6 a B e-A b e+7 b B d)}{4 e^8}+\frac {b^4 (d+e x)^3 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8}-\frac {5 b^3 (d+e x)^2 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8}+\frac {5 b^2 x (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^7}-\frac {(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8 (d+e x)}+\frac {(b d-a e)^6 (B d-A e)}{2 e^8 (d+e x)^2}-\frac {3 b (b d-a e)^4 \log (d+e x) (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac {b^6 B (d+e x)^5}{5 e^8}\)

Input:

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^3,x]
 

Output:

(5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*x)/e^7 + ((b*d - a*e)^6 
*(B*d - A*e))/(2*e^8*(d + e*x)^2) - ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a* 
B*e))/(e^8*(d + e*x)) - (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e) 
*(d + e*x)^2)/(2*e^8) + (b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d 
+ e*x)^3)/e^8 - (b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^4)/(4*e^8) + (b 
^6*B*(d + e*x)^5)/(5*e^8) - (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B* 
e)*Log[d + e*x])/e^8
 

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(788\) vs. \(2(268)=536\).

Time = 0.23 (sec) , antiderivative size = 789, normalized size of antiderivative = 2.86

method	result	size
norman	\(\frac {\frac {b^{2} \left (20 A \,a^{3} b \,e^{4}-30 A \,a^{2} b^{2} d \,e^{3}+20 A a \,b^{3} d^{2} e^{2}-5 A \,b^{4} d^{3} e +15 B \,a^{4} e^{4}-40 B \,a^{3} b d \,e^{3}+50 B \,a^{2} b^{2} d^{2} e^{2}-30 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) x^{3}}{e^{5}}-\frac {a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}-45 A \,a^{4} b^{2} d^{2} e^{5}+180 A \,a^{3} b^{3} d^{3} e^{4}-270 A \,a^{2} b^{4} d^{4} e^{3}+180 A a \,b^{5} d^{5} e^{2}-45 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}-18 B \,a^{5} b \,d^{2} e^{5}+135 B \,a^{4} b^{2} d^{3} e^{4}-360 B \,a^{3} b^{3} d^{4} e^{3}+450 B \,a^{2} b^{4} d^{5} e^{2}-270 B a \,b^{5} d^{6} e +63 b^{6} B \,d^{7}}{2 e^{8}}-\frac {\left (6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+120 A \,a^{3} b^{3} d^{2} e^{4}-180 A \,a^{2} b^{4} d^{3} e^{3}+120 A a \,b^{5} d^{4} e^{2}-30 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+90 B \,a^{4} b^{2} d^{2} e^{4}-240 B \,a^{3} b^{3} d^{3} e^{3}+300 B \,a^{2} b^{4} d^{4} e^{2}-180 B a \,b^{5} d^{5} e +42 b^{6} B \,d^{6}\right ) x}{e^{7}}+\frac {b^{3} \left (30 A \,a^{2} b \,e^{3}-20 A a \,b^{2} d \,e^{2}+5 A \,b^{3} d^{2} e +40 B \,a^{3} e^{3}-50 B \,a^{2} b d \,e^{2}+30 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right ) x^{4}}{4 e^{4}}+\frac {b^{4} \left (20 A a b \,e^{2}-5 A \,b^{2} d e +50 B \,a^{2} e^{2}-30 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{10 e^{3}}+\frac {b^{5} \left (5 A b e +30 B a e -7 B b d \right ) x^{6}}{20 e^{2}}+\frac {b^{6} B \,x^{7}}{5 e}}{\left (e x +d \right )^{2}}+\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(789\)
default	\(\frac {b^{2} \left (-\frac {3}{4} B \,b^{4} d \,e^{3} x^{4}+2 A a \,b^{3} e^{4} x^{3}-A \,b^{4} d \,e^{3} x^{3}+5 B \,a^{2} b^{2} e^{4} x^{3}+2 B \,b^{4} d^{2} e^{2} x^{3}+3 A \,b^{4} d^{2} e^{2} x^{2}+10 B \,a^{3} b \,e^{4} x^{2}-5 B \,b^{4} d^{3} e \,x^{2}+20 A \,a^{3} b \,e^{4} x -10 A \,b^{4} d^{3} e x -45 A \,a^{2} b^{2} d \,e^{3} x +36 A a \,b^{3} d^{2} e^{2} x -60 B \,a^{3} b d \,e^{3} x +90 B \,a^{2} b^{2} d^{2} e^{2} x -60 B a \,b^{3} d^{3} e x -\frac {45}{2} B \,a^{2} b^{2} d \,e^{3} x^{2}+18 B a \,b^{3} d^{2} e^{2} x^{2}+\frac {3}{2} B a \,b^{3} e^{4} x^{4}-6 B a \,b^{3} d \,e^{3} x^{3}-9 A a \,b^{3} d \,e^{3} x^{2}+\frac {1}{5} b^{4} B \,x^{5} e^{4}+\frac {1}{4} A \,b^{4} e^{4} x^{4}+15 B \,a^{4} e^{4} x +15 B \,b^{4} d^{4} x +\frac {15}{2} A \,a^{2} b^{2} e^{4} x^{2}\right )}{e^{7}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{e^{8} \left (e x +d \right )}-\frac {a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{2 e^{8} \left (e x +d \right )^{2}}+\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(846\)
risch	\(-\frac {60 b^{3} \ln \left (e x +d \right ) A \,a^{3} d}{e^{4}}+\frac {90 b^{4} \ln \left (e x +d \right ) A \,a^{2} d^{2}}{e^{5}}-\frac {60 b^{5} \ln \left (e x +d \right ) A a \,d^{3}}{e^{6}}-\frac {45 b^{2} \ln \left (e x +d \right ) B \,a^{4} d}{e^{4}}+\frac {120 b^{3} \ln \left (e x +d \right ) B \,a^{3} d^{2}}{e^{5}}-\frac {150 b^{4} \ln \left (e x +d \right ) B \,a^{2} d^{3}}{e^{6}}+\frac {90 b^{5} \ln \left (e x +d \right ) B a \,d^{4}}{e^{7}}-\frac {3 b^{6} B d \,x^{4}}{4 e^{4}}+\frac {2 b^{5} A a \,x^{3}}{e^{3}}+\frac {b^{6} B \,x^{5}}{5 e^{3}}+\frac {b^{6} A \,x^{4}}{4 e^{3}}-\frac {b^{6} A d \,x^{3}}{e^{4}}+\frac {5 b^{4} B \,a^{2} x^{3}}{e^{3}}+\frac {2 b^{6} B \,d^{2} x^{3}}{e^{5}}+\frac {3 b^{6} A \,d^{2} x^{2}}{e^{5}}+\frac {10 b^{3} B \,a^{3} x^{2}}{e^{3}}-\frac {5 b^{6} B \,d^{3} x^{2}}{e^{6}}+\frac {20 b^{3} A \,a^{3} x}{e^{3}}-\frac {10 b^{6} A \,d^{3} x}{e^{6}}+\frac {3 b^{5} B a \,x^{4}}{2 e^{3}}+\frac {15 b^{2} B \,a^{4} x}{e^{3}}+\frac {15 b^{6} B \,d^{4} x}{e^{7}}+\frac {15 b^{4} A \,a^{2} x^{2}}{2 e^{3}}+\frac {15 b^{2} \ln \left (e x +d \right ) A \,a^{4}}{e^{3}}+\frac {15 b^{6} \ln \left (e x +d \right ) A \,d^{4}}{e^{7}}+\frac {6 b \ln \left (e x +d \right ) B \,a^{5}}{e^{3}}-\frac {21 b^{6} \ln \left (e x +d \right ) B \,d^{5}}{e^{8}}+\frac {\left (-6 A \,a^{5} b \,e^{6}+30 A \,a^{4} b^{2} d \,e^{5}-60 A \,a^{3} b^{3} d^{2} e^{4}+60 A \,a^{2} b^{4} d^{3} e^{3}-30 A a \,b^{5} d^{4} e^{2}+6 A \,b^{6} d^{5} e -B \,a^{6} e^{6}+12 B \,a^{5} b d \,e^{5}-45 B \,a^{4} b^{2} d^{2} e^{4}+80 B \,a^{3} b^{3} d^{3} e^{3}-75 B \,a^{2} b^{4} d^{4} e^{2}+36 B a \,b^{5} d^{5} e -7 b^{6} B \,d^{6}\right ) x -\frac {a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}-45 A \,a^{4} b^{2} d^{2} e^{5}+100 A \,a^{3} b^{3} d^{3} e^{4}-105 A \,a^{2} b^{4} d^{4} e^{3}+54 A a \,b^{5} d^{5} e^{2}-11 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}-18 B \,a^{5} b \,d^{2} e^{5}+75 B \,a^{4} b^{2} d^{3} e^{4}-140 B \,a^{3} b^{3} d^{4} e^{3}+135 B \,a^{2} b^{4} d^{5} e^{2}-66 B a \,b^{5} d^{6} e +13 b^{6} B \,d^{7}}{2 e}}{e^{7} \left (e x +d \right )^{2}}-\frac {45 b^{4} A \,a^{2} d x}{e^{4}}+\frac {36 b^{5} A a \,d^{2} x}{e^{5}}-\frac {60 b^{3} B \,a^{3} d x}{e^{4}}+\frac {90 b^{4} B \,a^{2} d^{2} x}{e^{5}}-\frac {60 b^{5} B a \,d^{3} x}{e^{6}}-\frac {45 b^{4} B \,a^{2} d \,x^{2}}{2 e^{4}}+\frac {18 b^{5} B a \,d^{2} x^{2}}{e^{5}}-\frac {6 b^{5} B a d \,x^{3}}{e^{4}}-\frac {9 b^{5} A a d \,x^{2}}{e^{4}}\)	\(917\)
parallelrisch	\(\text {Expression too large to display}\)	\(1415\)

Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

(b^2*(20*A*a^3*b*e^4-30*A*a^2*b^2*d*e^3+20*A*a*b^3*d^2*e^2-5*A*b^4*d^3*e+1 
5*B*a^4*e^4-40*B*a^3*b*d*e^3+50*B*a^2*b^2*d^2*e^2-30*B*a*b^3*d^3*e+7*B*b^4 
*d^4)/e^5*x^3-1/2*(A*a^6*e^7+6*A*a^5*b*d*e^6-45*A*a^4*b^2*d^2*e^5+180*A*a^ 
3*b^3*d^3*e^4-270*A*a^2*b^4*d^4*e^3+180*A*a*b^5*d^5*e^2-45*A*b^6*d^6*e+B*a 
^6*d*e^6-18*B*a^5*b*d^2*e^5+135*B*a^4*b^2*d^3*e^4-360*B*a^3*b^3*d^4*e^3+45 
0*B*a^2*b^4*d^5*e^2-270*B*a*b^5*d^6*e+63*B*b^6*d^7)/e^8-(6*A*a^5*b*e^6-30* 
A*a^4*b^2*d*e^5+120*A*a^3*b^3*d^2*e^4-180*A*a^2*b^4*d^3*e^3+120*A*a*b^5*d^ 
4*e^2-30*A*b^6*d^5*e+B*a^6*e^6-12*B*a^5*b*d*e^5+90*B*a^4*b^2*d^2*e^4-240*B 
*a^3*b^3*d^3*e^3+300*B*a^2*b^4*d^4*e^2-180*B*a*b^5*d^5*e+42*B*b^6*d^6)/e^7 
*x+1/4*b^3*(30*A*a^2*b*e^3-20*A*a*b^2*d*e^2+5*A*b^3*d^2*e+40*B*a^3*e^3-50* 
B*a^2*b*d*e^2+30*B*a*b^2*d^2*e-7*B*b^3*d^3)/e^4*x^4+1/10*b^4*(20*A*a*b*e^2 
-5*A*b^2*d*e+50*B*a^2*e^2-30*B*a*b*d*e+7*B*b^2*d^2)/e^3*x^5+1/20*b^5*(5*A* 
b*e+30*B*a*e-7*B*b*d)/e^2*x^6+1/5*b^6*B/e*x^7)/(e*x+d)^2+3*b/e^8*(5*A*a^4* 
b*e^5-20*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d 
^4*e+2*B*a^5*e^5-15*B*a^4*b*d*e^4+40*B*a^3*b^2*d^2*e^3-50*B*a^2*b^3*d^3*e^ 
2+30*B*a*b^4*d^4*e-7*B*b^5*d^5)*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1177 vs. \(2 (268) = 536\).

Time = 0.08 (sec) , antiderivative size = 1177, normalized size of antiderivative = 4.26 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^3,x, algorithm="fricas")
 

Output:

1/20*(4*B*b^6*e^7*x^7 - 130*B*b^6*d^7 - 10*A*a^6*e^7 + 110*(6*B*a*b^5 + A* 
b^6)*d^6*e - 270*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 350*(4*B*a^3*b^3 + 3* 
A*a^2*b^4)*d^4*e^3 - 250*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 90*(2*B*a^5 
*b + 5*A*a^4*b^2)*d^2*e^5 - 10*(B*a^6 + 6*A*a^5*b)*d*e^6 - (7*B*b^6*d*e^6 
- 5*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 2*(7*B*b^6*d^2*e^5 - 5*(6*B*a*b^5 + A*b 
^6)*d*e^6 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 5*(7*B*b^6*d^3*e^4 - 5 
*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 10*(4* 
B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 20*(7*B*b^6*d^4*e^3 - 5*(6*B*a*b^5 + A 
*b^6)*d^3*e^4 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 10*(4*B*a^3*b^3 + 3 
*A*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 10*(50*B*b^6* 
d^5*e^2 - 34*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 63*(5*B*a^2*b^4 + 2*A*a*b^5)*d^ 
3*e^4 - 55*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 20*(3*B*a^4*b^2 + 4*A*a^3 
*b^3)*d*e^6)*x^2 + 20*(8*B*b^6*d^6*e - 4*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 3*( 
5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 - 
 10*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^ 
6 - (B*a^6 + 6*A*a^5*b)*e^7)*x - 60*(7*B*b^6*d^7 - 5*(6*B*a*b^5 + A*b^6)*d 
^6*e + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^ 
4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - (2*B*a^5*b + 5*A*a^4* 
b^2)*d^2*e^5 + (7*B*b^6*d^5*e^2 - 5*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 10*(5*B* 
a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 +...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 821 vs. \(2 (280) = 560\).

Time = 6.15 (sec) , antiderivative size = 821, normalized size of antiderivative = 2.97 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx=\frac {B b^{6} x^{5}}{5 e^{3}} + \frac {3 b \left (a e - b d\right )^{4} \cdot \left (5 A b e + 2 B a e - 7 B b d\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{4} \left (\frac {A b^{6}}{4 e^{3}} + \frac {3 B a b^{5}}{2 e^{3}} - \frac {3 B b^{6} d}{4 e^{4}}\right ) + x^{3} \cdot \left (\frac {2 A a b^{5}}{e^{3}} - \frac {A b^{6} d}{e^{4}} + \frac {5 B a^{2} b^{4}}{e^{3}} - \frac {6 B a b^{5} d}{e^{4}} + \frac {2 B b^{6} d^{2}}{e^{5}}\right ) + x^{2} \cdot \left (\frac {15 A a^{2} b^{4}}{2 e^{3}} - \frac {9 A a b^{5} d}{e^{4}} + \frac {3 A b^{6} d^{2}}{e^{5}} + \frac {10 B a^{3} b^{3}}{e^{3}} - \frac {45 B a^{2} b^{4} d}{2 e^{4}} + \frac {18 B a b^{5} d^{2}}{e^{5}} - \frac {5 B b^{6} d^{3}}{e^{6}}\right ) + x \left (\frac {20 A a^{3} b^{3}}{e^{3}} - \frac {45 A a^{2} b^{4} d}{e^{4}} + \frac {36 A a b^{5} d^{2}}{e^{5}} - \frac {10 A b^{6} d^{3}}{e^{6}} + \frac {15 B a^{4} b^{2}}{e^{3}} - \frac {60 B a^{3} b^{3} d}{e^{4}} + \frac {90 B a^{2} b^{4} d^{2}}{e^{5}} - \frac {60 B a b^{5} d^{3}}{e^{6}} + \frac {15 B b^{6} d^{4}}{e^{7}}\right ) + \frac {- A a^{6} e^{7} - 6 A a^{5} b d e^{6} + 45 A a^{4} b^{2} d^{2} e^{5} - 100 A a^{3} b^{3} d^{3} e^{4} + 105 A a^{2} b^{4} d^{4} e^{3} - 54 A a b^{5} d^{5} e^{2} + 11 A b^{6} d^{6} e - B a^{6} d e^{6} + 18 B a^{5} b d^{2} e^{5} - 75 B a^{4} b^{2} d^{3} e^{4} + 140 B a^{3} b^{3} d^{4} e^{3} - 135 B a^{2} b^{4} d^{5} e^{2} + 66 B a b^{5} d^{6} e - 13 B b^{6} d^{7} + x \left (- 12 A a^{5} b e^{7} + 60 A a^{4} b^{2} d e^{6} - 120 A a^{3} b^{3} d^{2} e^{5} + 120 A a^{2} b^{4} d^{3} e^{4} - 60 A a b^{5} d^{4} e^{3} + 12 A b^{6} d^{5} e^{2} - 2 B a^{6} e^{7} + 24 B a^{5} b d e^{6} - 90 B a^{4} b^{2} d^{2} e^{5} + 160 B a^{3} b^{3} d^{3} e^{4} - 150 B a^{2} b^{4} d^{4} e^{3} + 72 B a b^{5} d^{5} e^{2} - 14 B b^{6} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} \] Input:

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**3,x)
 

Output:

B*b**6*x**5/(5*e**3) + 3*b*(a*e - b*d)**4*(5*A*b*e + 2*B*a*e - 7*B*b*d)*lo 
g(d + e*x)/e**8 + x**4*(A*b**6/(4*e**3) + 3*B*a*b**5/(2*e**3) - 3*B*b**6*d 
/(4*e**4)) + x**3*(2*A*a*b**5/e**3 - A*b**6*d/e**4 + 5*B*a**2*b**4/e**3 - 
6*B*a*b**5*d/e**4 + 2*B*b**6*d**2/e**5) + x**2*(15*A*a**2*b**4/(2*e**3) - 
9*A*a*b**5*d/e**4 + 3*A*b**6*d**2/e**5 + 10*B*a**3*b**3/e**3 - 45*B*a**2*b 
**4*d/(2*e**4) + 18*B*a*b**5*d**2/e**5 - 5*B*b**6*d**3/e**6) + x*(20*A*a** 
3*b**3/e**3 - 45*A*a**2*b**4*d/e**4 + 36*A*a*b**5*d**2/e**5 - 10*A*b**6*d* 
*3/e**6 + 15*B*a**4*b**2/e**3 - 60*B*a**3*b**3*d/e**4 + 90*B*a**2*b**4*d** 
2/e**5 - 60*B*a*b**5*d**3/e**6 + 15*B*b**6*d**4/e**7) + (-A*a**6*e**7 - 6* 
A*a**5*b*d*e**6 + 45*A*a**4*b**2*d**2*e**5 - 100*A*a**3*b**3*d**3*e**4 + 1 
05*A*a**2*b**4*d**4*e**3 - 54*A*a*b**5*d**5*e**2 + 11*A*b**6*d**6*e - B*a* 
*6*d*e**6 + 18*B*a**5*b*d**2*e**5 - 75*B*a**4*b**2*d**3*e**4 + 140*B*a**3* 
b**3*d**4*e**3 - 135*B*a**2*b**4*d**5*e**2 + 66*B*a*b**5*d**6*e - 13*B*b** 
6*d**7 + x*(-12*A*a**5*b*e**7 + 60*A*a**4*b**2*d*e**6 - 120*A*a**3*b**3*d* 
*2*e**5 + 120*A*a**2*b**4*d**3*e**4 - 60*A*a*b**5*d**4*e**3 + 12*A*b**6*d* 
*5*e**2 - 2*B*a**6*e**7 + 24*B*a**5*b*d*e**6 - 90*B*a**4*b**2*d**2*e**5 + 
160*B*a**3*b**3*d**3*e**4 - 150*B*a**2*b**4*d**4*e**3 + 72*B*a*b**5*d**5*e 
**2 - 14*B*b**6*d**6*e))/(2*d**2*e**8 + 4*d*e**9*x + 2*e**10*x**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 779 vs. \(2 (268) = 536\).

Time = 0.05 (sec) , antiderivative size = 779, normalized size of antiderivative = 2.82 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^3,x, algorithm="maxima")
 

Output:

-1/2*(13*B*b^6*d^7 + A*a^6*e^7 - 11*(6*B*a*b^5 + A*b^6)*d^6*e + 27*(5*B*a^ 
2*b^4 + 2*A*a*b^5)*d^5*e^2 - 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 25*( 
3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 9*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + 
 (B*a^6 + 6*A*a^5*b)*d*e^6 + 2*(7*B*b^6*d^6*e - 6*(6*B*a*b^5 + A*b^6)*d^5* 
e^2 + 15*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4 
)*d^3*e^4 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 6*(2*B*a^5*b + 5*A*a^ 
4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^10*x^2 + 2*d*e^9*x + d^2*e^8 
) + 1/20*(4*B*b^6*e^4*x^5 - 5*(3*B*b^6*d*e^3 - (6*B*a*b^5 + A*b^6)*e^4)*x^ 
4 + 20*(2*B*b^6*d^2*e^2 - (6*B*a*b^5 + A*b^6)*d*e^3 + (5*B*a^2*b^4 + 2*A*a 
*b^5)*e^4)*x^3 - 10*(10*B*b^6*d^3*e - 6*(6*B*a*b^5 + A*b^6)*d^2*e^2 + 9*(5 
*B*a^2*b^4 + 2*A*a*b^5)*d*e^3 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^4)*x^2 + 2 
0*(15*B*b^6*d^4 - 10*(6*B*a*b^5 + A*b^6)*d^3*e + 18*(5*B*a^2*b^4 + 2*A*a*b 
^5)*d^2*e^2 - 15*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^3 + 5*(3*B*a^4*b^2 + 4*A* 
a^3*b^3)*e^4)*x)/e^7 - 3*(7*B*b^6*d^5 - 5*(6*B*a*b^5 + A*b^6)*d^4*e + 10*( 
5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 
+ 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*e^5)*log 
(e*x + d)/e^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 862 vs. \(2 (268) = 536\).

Time = 0.12 (sec) , antiderivative size = 862, normalized size of antiderivative = 3.12 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^3,x, algorithm="giac")
 

Output:

-3*(7*B*b^6*d^5 - 30*B*a*b^5*d^4*e - 5*A*b^6*d^4*e + 50*B*a^2*b^4*d^3*e^2 
+ 20*A*a*b^5*d^3*e^2 - 40*B*a^3*b^3*d^2*e^3 - 30*A*a^2*b^4*d^2*e^3 + 15*B* 
a^4*b^2*d*e^4 + 20*A*a^3*b^3*d*e^4 - 2*B*a^5*b*e^5 - 5*A*a^4*b^2*e^5)*log( 
abs(e*x + d))/e^8 - 1/2*(13*B*b^6*d^7 - 66*B*a*b^5*d^6*e - 11*A*b^6*d^6*e 
+ 135*B*a^2*b^4*d^5*e^2 + 54*A*a*b^5*d^5*e^2 - 140*B*a^3*b^3*d^4*e^3 - 105 
*A*a^2*b^4*d^4*e^3 + 75*B*a^4*b^2*d^3*e^4 + 100*A*a^3*b^3*d^3*e^4 - 18*B*a 
^5*b*d^2*e^5 - 45*A*a^4*b^2*d^2*e^5 + B*a^6*d*e^6 + 6*A*a^5*b*d*e^6 + A*a^ 
6*e^7 + 2*(7*B*b^6*d^6*e - 36*B*a*b^5*d^5*e^2 - 6*A*b^6*d^5*e^2 + 75*B*a^2 
*b^4*d^4*e^3 + 30*A*a*b^5*d^4*e^3 - 80*B*a^3*b^3*d^3*e^4 - 60*A*a^2*b^4*d^ 
3*e^4 + 45*B*a^4*b^2*d^2*e^5 + 60*A*a^3*b^3*d^2*e^5 - 12*B*a^5*b*d*e^6 - 3 
0*A*a^4*b^2*d*e^6 + B*a^6*e^7 + 6*A*a^5*b*e^7)*x)/((e*x + d)^2*e^8) + 1/20 
*(4*B*b^6*e^12*x^5 - 15*B*b^6*d*e^11*x^4 + 30*B*a*b^5*e^12*x^4 + 5*A*b^6*e 
^12*x^4 + 40*B*b^6*d^2*e^10*x^3 - 120*B*a*b^5*d*e^11*x^3 - 20*A*b^6*d*e^11 
*x^3 + 100*B*a^2*b^4*e^12*x^3 + 40*A*a*b^5*e^12*x^3 - 100*B*b^6*d^3*e^9*x^ 
2 + 360*B*a*b^5*d^2*e^10*x^2 + 60*A*b^6*d^2*e^10*x^2 - 450*B*a^2*b^4*d*e^1 
1*x^2 - 180*A*a*b^5*d*e^11*x^2 + 200*B*a^3*b^3*e^12*x^2 + 150*A*a^2*b^4*e^ 
12*x^2 + 300*B*b^6*d^4*e^8*x - 1200*B*a*b^5*d^3*e^9*x - 200*A*b^6*d^3*e^9* 
x + 1800*B*a^2*b^4*d^2*e^10*x + 720*A*a*b^5*d^2*e^10*x - 1200*B*a^3*b^3*d* 
e^11*x - 900*A*a^2*b^4*d*e^11*x + 300*B*a^4*b^2*e^12*x + 400*A*a^3*b^3*e^1 
2*x)/e^15
 

Mupad [B] (verification not implemented)

Time = 0.97 (sec) , antiderivative size = 1053, normalized size of antiderivative = 3.82 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^3,x)
 

Output:

x*((3*d*((3*d^2*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B*b^6*d)/e^4))/e^2 - (3*d*(( 
3*d*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B*b^6*d)/e^4))/e - (3*a*b^4*(2*A*b + 5*B 
*a))/e^3 + (3*B*b^6*d^2)/e^5))/e - (5*a^2*b^3*(3*A*b + 4*B*a))/e^3 + (B*b^ 
6*d^3)/e^6))/e - (d^3*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B*b^6*d)/e^4))/e^3 + ( 
3*d^2*((3*d*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B*b^6*d)/e^4))/e - (3*a*b^4*(2*A 
*b + 5*B*a))/e^3 + (3*B*b^6*d^2)/e^5))/e^2 + (5*a^3*b^2*(4*A*b + 3*B*a))/e 
^3) - x^3*((d*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B*b^6*d)/e^4))/e - (a*b^4*(2*A 
*b + 5*B*a))/e^3 + (B*b^6*d^2)/e^5) - x^2*((3*d^2*((A*b^6 + 6*B*a*b^5)/e^3 
 - (3*B*b^6*d)/e^4))/(2*e^2) - (3*d*((3*d*((A*b^6 + 6*B*a*b^5)/e^3 - (3*B* 
b^6*d)/e^4))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^3 + (3*B*b^6*d^2)/e^5))/(2*e) 
 - (5*a^2*b^3*(3*A*b + 4*B*a))/(2*e^3) + (B*b^6*d^3)/(2*e^6)) - ((A*a^6*e^ 
7 + 13*B*b^6*d^7 - 11*A*b^6*d^6*e + B*a^6*d*e^6 + 54*A*a*b^5*d^5*e^2 - 18* 
B*a^5*b*d^2*e^5 - 105*A*a^2*b^4*d^4*e^3 + 100*A*a^3*b^3*d^3*e^4 - 45*A*a^4 
*b^2*d^2*e^5 + 135*B*a^2*b^4*d^5*e^2 - 140*B*a^3*b^3*d^4*e^3 + 75*B*a^4*b^ 
2*d^3*e^4 + 6*A*a^5*b*d*e^6 - 66*B*a*b^5*d^6*e)/(2*e) + x*(B*a^6*e^6 + 7*B 
*b^6*d^6 + 6*A*a^5*b*e^6 - 6*A*b^6*d^5*e + 30*A*a*b^5*d^4*e^2 - 30*A*a^4*b 
^2*d*e^5 - 60*A*a^2*b^4*d^3*e^3 + 60*A*a^3*b^3*d^2*e^4 + 75*B*a^2*b^4*d^4* 
e^2 - 80*B*a^3*b^3*d^3*e^3 + 45*B*a^4*b^2*d^2*e^4 - 36*B*a*b^5*d^5*e - 12* 
B*a^5*b*d*e^5))/(d^2*e^7 + e^9*x^2 + 2*d*e^8*x) + x^4*((A*b^6 + 6*B*a*b^5) 
/(4*e^3) - (3*B*b^6*d)/(4*e^4)) + (log(d + e*x)*(6*B*a^5*b*e^5 - 21*B*b...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.89 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^3,x)
 

Output:

(420*log(d + e*x)*a**5*b**2*d**3*e**5 + 840*log(d + e*x)*a**5*b**2*d**2*e* 
*6*x + 420*log(d + e*x)*a**5*b**2*d*e**7*x**2 - 2100*log(d + e*x)*a**4*b** 
3*d**4*e**4 - 4200*log(d + e*x)*a**4*b**3*d**3*e**5*x - 2100*log(d + e*x)* 
a**4*b**3*d**2*e**6*x**2 + 4200*log(d + e*x)*a**3*b**4*d**5*e**3 + 8400*lo 
g(d + e*x)*a**3*b**4*d**4*e**4*x + 4200*log(d + e*x)*a**3*b**4*d**3*e**5*x 
**2 - 4200*log(d + e*x)*a**2*b**5*d**6*e**2 - 8400*log(d + e*x)*a**2*b**5* 
d**5*e**3*x - 4200*log(d + e*x)*a**2*b**5*d**4*e**4*x**2 + 2100*log(d + e* 
x)*a*b**6*d**7*e + 4200*log(d + e*x)*a*b**6*d**6*e**2*x + 2100*log(d + e*x 
)*a*b**6*d**5*e**3*x**2 - 420*log(d + e*x)*b**7*d**8 - 840*log(d + e*x)*b* 
*7*d**7*e*x - 420*log(d + e*x)*b**7*d**6*e**2*x**2 - 10*a**7*d*e**7 + 70*a 
**6*b*e**8*x**2 + 210*a**5*b**2*d**3*e**5 - 420*a**5*b**2*d*e**7*x**2 - 10 
50*a**4*b**3*d**4*e**4 + 2100*a**4*b**3*d**2*e**6*x**2 + 700*a**4*b**3*d*e 
**7*x**3 + 2100*a**3*b**4*d**5*e**3 - 4200*a**3*b**4*d**3*e**5*x**2 - 1400 
*a**3*b**4*d**2*e**6*x**3 + 350*a**3*b**4*d*e**7*x**4 - 2100*a**2*b**5*d** 
6*e**2 + 4200*a**2*b**5*d**4*e**4*x**2 + 1400*a**2*b**5*d**3*e**5*x**3 - 3 
50*a**2*b**5*d**2*e**6*x**4 + 140*a**2*b**5*d*e**7*x**5 + 1050*a*b**6*d**7 
*e - 2100*a*b**6*d**5*e**3*x**2 - 700*a*b**6*d**4*e**4*x**3 + 175*a*b**6*d 
**3*e**5*x**4 - 70*a*b**6*d**2*e**6*x**5 + 35*a*b**6*d*e**7*x**6 - 210*b** 
7*d**8 + 420*b**7*d**6*e**2*x**2 + 140*b**7*d**5*e**3*x**3 - 35*b**7*d**4* 
e**4*x**4 + 14*b**7*d**3*e**5*x**5 - 7*b**7*d**2*e**6*x**6 + 4*b**7*d*e...