\(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx\) [54]

Optimal result

Integrand size = 20, antiderivative size = 279 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx=\frac {3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) x}{e^7}+\frac {(b d-a e)^6 (B d-A e)}{4 e^8 (d+e x)^4}-\frac {(b d-a e)^5 (7 b B d-6 A b e-a B e)}{3 e^8 (d+e x)^3}+\frac {3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{2 e^8 (d+e x)^2}-\frac {5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{e^8 (d+e x)}-\frac {b^5 (7 b B d-A b e-6 a B e) (d+e x)^2}{2 e^8}+\frac {b^6 B (d+e x)^3}{3 e^8}-\frac {5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) \log (d+e x)}{e^8} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx=\frac {-12 b^4 e \left (-15 a^2 B e^2-6 a b e (-5 B d+A e)+5 b^2 d (-3 B d+A e)\right ) x+6 b^5 e^2 (-5 b B d+A b e+6 a B e) x^2+4 b^6 B e^3 x^3+\frac {3 (b d-a e)^6 (B d-A e)}{(d+e x)^4}-\frac {4 (b d-a e)^5 (7 b B d-6 A b e-a B e)}{(d+e x)^3}+\frac {18 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{(d+e x)^2}-\frac {60 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{d+e x}-60 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) \log (d+e x)}{12 e^8} \] Input:

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^5,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 279, 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.

\(\displaystyle \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx\)

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^5,x]
 

Output:

(3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*x)/e^7 + ((b*d - a*e)^6*( 
B*d - A*e))/(4*e^8*(d + e*x)^4) - ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B* 
e))/(3*e^8*(d + e*x)^3) + (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e) 
)/(2*e^8*(d + e*x)^2) - (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e) 
)/(e^8*(d + e*x)) - (b^5*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^2)/(2*e^8) 
+ (b^6*B*(d + e*x)^3)/(3*e^8) - (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 
4*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. \(797\) vs. \(2(269)=538\).

Time = 0.24 (sec) , antiderivative size = 798, normalized size of antiderivative = 2.86

method	result	size
norman	\(\frac {\frac {b^{4} \left (6 A a b \,e^{2}-3 A \,b^{2} d e +15 B \,a^{2} e^{2}-18 B a b d e +7 b^{2} B \,d^{2}\right ) x^{5}}{e^{3}}-\frac {3 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}+60 A \,a^{3} b^{3} d^{3} e^{4}-375 A \,a^{2} b^{4} d^{4} e^{3}+750 A a \,b^{5} d^{5} e^{2}-375 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}+45 B \,a^{4} b^{2} d^{3} e^{4}-500 B \,a^{3} b^{3} d^{4} e^{3}+1875 B \,a^{2} b^{4} d^{5} e^{2}-2250 B a \,b^{5} d^{6} e +875 b^{6} B \,d^{7}}{12 e^{8}}-\frac {\left (20 A \,a^{3} b^{3} e^{4}-60 A \,a^{2} b^{4} d \,e^{3}+120 A a \,b^{5} d^{2} e^{2}-60 A \,b^{6} d^{3} e +15 B \,a^{4} b^{2} e^{4}-80 B \,a^{3} b^{3} d \,e^{3}+300 B \,a^{2} b^{4} d^{2} e^{2}-360 B a \,b^{5} d^{3} e +140 b^{6} B \,d^{4}\right ) x^{3}}{e^{5}}-\frac {3 \left (5 A \,a^{4} b^{2} e^{5}+20 A \,a^{3} b^{3} d \,e^{4}-90 A \,a^{2} b^{4} d^{2} e^{3}+180 A a \,b^{5} d^{3} e^{2}-90 A \,b^{6} d^{4} e +2 B \,a^{5} b \,e^{5}+15 B \,a^{4} b^{2} d \,e^{4}-120 B \,a^{3} b^{3} d^{2} e^{3}+450 B \,a^{2} b^{4} d^{3} e^{2}-540 B a \,b^{5} d^{4} e +210 b^{6} B \,d^{5}\right ) x^{2}}{2 e^{6}}-\frac {\left (6 A \,a^{5} b \,e^{6}+15 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-330 A \,a^{2} b^{4} d^{3} e^{3}+660 A a \,b^{5} d^{4} e^{2}-330 A \,b^{6} d^{5} e +B \,a^{6} e^{6}+6 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-440 B \,a^{3} b^{3} d^{3} e^{3}+1650 B \,a^{2} b^{4} d^{4} e^{2}-1980 B a \,b^{5} d^{5} e +770 b^{6} B \,d^{6}\right ) x}{3 e^{7}}+\frac {b^{5} \left (3 A b e +18 B a e -7 B b d \right ) x^{6}}{6 e^{2}}+\frac {b^{6} B \,x^{7}}{3 e}}{\left (e x +d \right )^{4}}+\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right ) \ln \left (e x +d \right )}{e^{8}}\)	\(798\)
default	\(\frac {b^{4} \left (\frac {1}{3} b^{2} B \,x^{3} e^{2}+\frac {1}{2} A \,b^{2} e^{2} x^{2}+3 B a b \,e^{2} x^{2}-\frac {5}{2} B \,b^{2} d e \,x^{2}+6 A a b \,e^{2} x -5 A \,b^{2} d e x +15 B \,a^{2} e^{2} x -30 B a b d e x +15 b^{2} B \,d^{2} x \right )}{e^{7}}-\frac {5 b^{2} \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right )}{e^{8} \left (e x +d \right )}-\frac {3 b \left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right )}{2 e^{8} \left (e x +d \right )^{2}}-\frac {a^{6} A \,e^{7}-6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}-20 A \,a^{3} b^{3} d^{3} e^{4}+15 A \,a^{2} b^{4} d^{4} e^{3}-6 A a \,b^{5} d^{5} e^{2}+A \,b^{6} d^{6} e -B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}-15 B \,a^{4} b^{2} d^{3} e^{4}+20 B \,a^{3} b^{3} d^{4} e^{3}-15 B \,a^{2} b^{4} d^{5} e^{2}+6 B a \,b^{5} d^{6} e -b^{6} B \,d^{7}}{4 e^{8} \left (e x +d \right )^{4}}+\frac {5 b^{3} \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B \,a^{2} b d \,e^{2}+18 B a \,b^{2} d^{2} e -7 b^{3} B \,d^{3}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {6 A \,a^{5} b \,e^{6}-30 A \,a^{4} b^{2} d \,e^{5}+60 A \,a^{3} b^{3} d^{2} e^{4}-60 A \,a^{2} b^{4} d^{3} e^{3}+30 A a \,b^{5} d^{4} e^{2}-6 A \,b^{6} d^{5} e +B \,a^{6} e^{6}-12 B \,a^{5} b d \,e^{5}+45 B \,a^{4} b^{2} d^{2} e^{4}-80 B \,a^{3} b^{3} d^{3} e^{3}+75 B \,a^{2} b^{4} d^{4} e^{2}-36 B a \,b^{5} d^{5} e +7 b^{6} B \,d^{6}}{3 e^{8} \left (e x +d \right )^{3}}\)	\(810\)
risch	\(\frac {b^{6} B \,x^{3}}{3 e^{5}}+\frac {b^{6} A \,x^{2}}{2 e^{5}}+\frac {3 b^{5} B a \,x^{2}}{e^{5}}-\frac {5 b^{6} B d \,x^{2}}{2 e^{6}}+\frac {6 b^{5} A a x}{e^{5}}-\frac {5 b^{6} A d x}{e^{6}}+\frac {15 b^{4} B \,a^{2} x}{e^{5}}-\frac {30 b^{5} B a d x}{e^{6}}+\frac {15 b^{6} B \,d^{2} x}{e^{7}}+\frac {\left (-20 A \,a^{3} b^{3} e^{6}+60 A \,a^{2} b^{4} d \,e^{5}-60 A a \,b^{5} d^{2} e^{4}+20 A \,b^{6} d^{3} e^{3}-15 B \,a^{4} b^{2} e^{6}+80 B \,a^{3} b^{3} d \,e^{5}-150 B \,a^{2} b^{4} d^{2} e^{4}+120 B a \,b^{5} d^{3} e^{3}-35 b^{6} B \,d^{4} e^{2}\right ) x^{3}-\frac {3 e b \left (5 A \,a^{4} b \,e^{5}+20 A \,a^{3} b^{2} d \,e^{4}-90 A \,a^{2} b^{3} d^{2} e^{3}+100 A a \,b^{4} d^{3} e^{2}-35 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}+15 B \,a^{4} b d \,e^{4}-120 B \,a^{3} b^{2} d^{2} e^{3}+250 B \,a^{2} b^{3} d^{3} e^{2}-210 B a \,b^{4} d^{4} e +63 B \,b^{5} d^{5}\right ) x^{2}}{2}+\left (-2 A \,a^{5} b \,e^{6}-5 A \,a^{4} b^{2} d \,e^{5}-20 A \,a^{3} b^{3} d^{2} e^{4}+110 A \,a^{2} b^{4} d^{3} e^{3}-130 A a \,b^{5} d^{4} e^{2}+47 A \,b^{6} d^{5} e -\frac {1}{3} B \,a^{6} e^{6}-2 B \,a^{5} b d \,e^{5}-15 B \,a^{4} b^{2} d^{2} e^{4}+\frac {440}{3} B \,a^{3} b^{3} d^{3} e^{3}-325 B \,a^{2} b^{4} d^{4} e^{2}+282 B a \,b^{5} d^{5} e -\frac {259}{3} b^{6} B \,d^{6}\right ) x -\frac {3 a^{6} A \,e^{7}+6 A \,a^{5} b d \,e^{6}+15 A \,a^{4} b^{2} d^{2} e^{5}+60 A \,a^{3} b^{3} d^{3} e^{4}-375 A \,a^{2} b^{4} d^{4} e^{3}+462 A a \,b^{5} d^{5} e^{2}-171 A \,b^{6} d^{6} e +B \,a^{6} d \,e^{6}+6 B \,a^{5} b \,d^{2} e^{5}+45 B \,a^{4} b^{2} d^{3} e^{4}-500 B \,a^{3} b^{3} d^{4} e^{3}+1155 B \,a^{2} b^{4} d^{5} e^{2}-1026 B a \,b^{5} d^{6} e +319 b^{6} B \,d^{7}}{12 e}}{e^{7} \left (e x +d \right )^{4}}+\frac {15 b^{4} \ln \left (e x +d \right ) A \,a^{2}}{e^{5}}-\frac {30 b^{5} \ln \left (e x +d \right ) A a d}{e^{6}}+\frac {15 b^{6} \ln \left (e x +d \right ) A \,d^{2}}{e^{7}}+\frac {20 b^{3} \ln \left (e x +d \right ) B \,a^{3}}{e^{5}}-\frac {75 b^{4} \ln \left (e x +d \right ) B \,a^{2} d}{e^{6}}+\frac {90 b^{5} \ln \left (e x +d \right ) B a \,d^{2}}{e^{7}}-\frac {35 b^{6} \ln \left (e x +d \right ) B \,d^{3}}{e^{8}}\)	\(855\)
parallelrisch	\(\text {Expression too large to display}\)	\(1539\)

Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^5,x,method=_RETURNVERBOSE)
 

Output:

(b^4*(6*A*a*b*e^2-3*A*b^2*d*e+15*B*a^2*e^2-18*B*a*b*d*e+7*B*b^2*d^2)/e^3*x 
^5-1/12*(3*A*a^6*e^7+6*A*a^5*b*d*e^6+15*A*a^4*b^2*d^2*e^5+60*A*a^3*b^3*d^3 
*e^4-375*A*a^2*b^4*d^4*e^3+750*A*a*b^5*d^5*e^2-375*A*b^6*d^6*e+B*a^6*d*e^6 
+6*B*a^5*b*d^2*e^5+45*B*a^4*b^2*d^3*e^4-500*B*a^3*b^3*d^4*e^3+1875*B*a^2*b 
^4*d^5*e^2-2250*B*a*b^5*d^6*e+875*B*b^6*d^7)/e^8-(20*A*a^3*b^3*e^4-60*A*a^ 
2*b^4*d*e^3+120*A*a*b^5*d^2*e^2-60*A*b^6*d^3*e+15*B*a^4*b^2*e^4-80*B*a^3*b 
^3*d*e^3+300*B*a^2*b^4*d^2*e^2-360*B*a*b^5*d^3*e+140*B*b^6*d^4)/e^5*x^3-3/ 
2*(5*A*a^4*b^2*e^5+20*A*a^3*b^3*d*e^4-90*A*a^2*b^4*d^2*e^3+180*A*a*b^5*d^3 
*e^2-90*A*b^6*d^4*e+2*B*a^5*b*e^5+15*B*a^4*b^2*d*e^4-120*B*a^3*b^3*d^2*e^3 
+450*B*a^2*b^4*d^3*e^2-540*B*a*b^5*d^4*e+210*B*b^6*d^5)/e^6*x^2-1/3*(6*A*a 
^5*b*e^6+15*A*a^4*b^2*d*e^5+60*A*a^3*b^3*d^2*e^4-330*A*a^2*b^4*d^3*e^3+660 
*A*a*b^5*d^4*e^2-330*A*b^6*d^5*e+B*a^6*e^6+6*B*a^5*b*d*e^5+45*B*a^4*b^2*d^ 
2*e^4-440*B*a^3*b^3*d^3*e^3+1650*B*a^2*b^4*d^4*e^2-1980*B*a*b^5*d^5*e+770* 
B*b^6*d^6)/e^7*x+1/6*b^5*(3*A*b*e+18*B*a*e-7*B*b*d)/e^2*x^6+1/3*b^6*B/e*x^ 
7)/(e*x+d)^4+5/e^8*b^3*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+4*B*a^ 
3*e^3-15*B*a^2*b*d*e^2+18*B*a*b^2*d^2*e-7*B*b^3*d^3)*ln(e*x+d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1222 vs. \(2 (269) = 538\).

Time = 0.09 (sec) , antiderivative size = 1222, normalized size of antiderivative = 4.38 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^5,x, algorithm="fricas")
 

Output:

1/12*(4*B*b^6*e^7*x^7 - 319*B*b^6*d^7 - 3*A*a^6*e^7 + 171*(6*B*a*b^5 + A*b 
^6)*d^6*e - 231*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 125*(4*B*a^3*b^3 + 3*A 
*a^2*b^4)*d^4*e^3 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(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*e^6 - 3* 
(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 12*(7*B*b^6*d^2*e^5 - 3*(6*B*a*b^5 + A*b^6) 
*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 4*(139*B*b^6*d^3*e^4 - 51* 
(6*B*a*b^5 + A*b^6)*d^2*e^5 + 36*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6)*x^4 + 4* 
(136*B*b^6*d^4*e^3 - 24*(6*B*a*b^5 + A*b^6)*d^3*e^4 - 36*(5*B*a^2*b^4 + 2* 
A*a*b^5)*d^2*e^5 + 60*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 - 15*(3*B*a^4*b^2 
+ 4*A*a^3*b^3)*e^7)*x^3 - 6*(74*B*b^6*d^5*e^2 - 66*(6*B*a*b^5 + A*b^6)*d^4 
*e^3 + 126*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 90*(4*B*a^3*b^3 + 3*A*a^2*b 
^4)*d^2*e^5 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 3*(2*B*a^5*b + 5*A*a^ 
4*b^2)*e^7)*x^2 - 4*(214*B*b^6*d^6*e - 126*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 1 
86*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 110*(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 + 3*(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 - 3*(6*B*a*b^5 + A* 
b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - (4*B*a^3*b^3 + 3*A*a^2* 
b^4)*d^4*e^3 + (7*B*b^6*d^3*e^4 - 3*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*a 
^2*b^4 + 2*A*a*b^5)*d*e^6 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 4*(7*B* 
b^6*d^4*e^3 - 3*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**6*(B*x+A)/(e*x+d)**5,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 801 vs. \(2 (269) = 538\).

Time = 0.09 (sec) , antiderivative size = 801, normalized size of antiderivative = 2.87 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^5,x, algorithm="maxima")
 

Output:

-1/12*(319*B*b^6*d^7 + 3*A*a^6*e^7 - 171*(6*B*a*b^5 + A*b^6)*d^6*e + 231*( 
5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 125*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 
 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 3*(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 + 60*(7*B*b^6*d^4*e^3 - 4*(6*B*a*b^5 + A 
*b^6)*d^3*e^4 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 4*(4*B*a^3*b^3 + 3*A 
*a^2*b^4)*d*e^6 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 18*(63*B*b^6*d^5* 
e^2 - 35*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 50*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^ 
4 - 30*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3) 
*d*e^6 + (2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 4*(259*B*b^6*d^6*e - 141*(6* 
B*a*b^5 + A*b^6)*d^5*e^2 + 195*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 110*(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 
+ 3*(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^12*x^ 
4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8) + 1/6*(2*B*b^6* 
e^2*x^3 - 3*(5*B*b^6*d*e - (6*B*a*b^5 + A*b^6)*e^2)*x^2 + 6*(15*B*b^6*d^2 
- 5*(6*B*a*b^5 + A*b^6)*d*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*x)/e^7 - 5* 
(7*B*b^6*d^3 - 3*(6*B*a*b^5 + A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d 
*e^2 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^3)*log(e*x + d)/e^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (269) = 538\).

Time = 0.14 (sec) , antiderivative size = 1175, normalized size of antiderivative = 4.21 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^5,x, algorithm="giac")
 

Output:

1/6*(2*B*b^6 - 3*(7*B*b^6*d*e - 6*B*a*b^5*e^2 - A*b^6*e^2)/((e*x + d)*e) + 
 18*(7*B*b^6*d^2*e^2 - 12*B*a*b^5*d*e^3 - 2*A*b^6*d*e^3 + 5*B*a^2*b^4*e^4 
+ 2*A*a*b^5*e^4)/((e*x + d)^2*e^2))*(e*x + d)^3/e^8 + 5*(7*B*b^6*d^3 - 18* 
B*a*b^5*d^2*e - 3*A*b^6*d^2*e + 15*B*a^2*b^4*d*e^2 + 6*A*a*b^5*d*e^2 - 4*B 
*a^3*b^3*e^3 - 3*A*a^2*b^4*e^3)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^8 
 - 1/12*(420*B*b^6*d^4*e^36/(e*x + d) - 126*B*b^6*d^5*e^36/(e*x + d)^2 + 2 
8*B*b^6*d^6*e^36/(e*x + d)^3 - 3*B*b^6*d^7*e^36/(e*x + d)^4 - 1440*B*a*b^5 
*d^3*e^37/(e*x + d) - 240*A*b^6*d^3*e^37/(e*x + d) + 540*B*a*b^5*d^4*e^37/ 
(e*x + d)^2 + 90*A*b^6*d^4*e^37/(e*x + d)^2 - 144*B*a*b^5*d^5*e^37/(e*x + 
d)^3 - 24*A*b^6*d^5*e^37/(e*x + d)^3 + 18*B*a*b^5*d^6*e^37/(e*x + d)^4 + 3 
*A*b^6*d^6*e^37/(e*x + d)^4 + 1800*B*a^2*b^4*d^2*e^38/(e*x + d) + 720*A*a* 
b^5*d^2*e^38/(e*x + d) - 900*B*a^2*b^4*d^3*e^38/(e*x + d)^2 - 360*A*a*b^5* 
d^3*e^38/(e*x + d)^2 + 300*B*a^2*b^4*d^4*e^38/(e*x + d)^3 + 120*A*a*b^5*d^ 
4*e^38/(e*x + d)^3 - 45*B*a^2*b^4*d^5*e^38/(e*x + d)^4 - 18*A*a*b^5*d^5*e^ 
38/(e*x + d)^4 - 960*B*a^3*b^3*d*e^39/(e*x + d) - 720*A*a^2*b^4*d*e^39/(e* 
x + d) + 720*B*a^3*b^3*d^2*e^39/(e*x + d)^2 + 540*A*a^2*b^4*d^2*e^39/(e*x 
+ d)^2 - 320*B*a^3*b^3*d^3*e^39/(e*x + d)^3 - 240*A*a^2*b^4*d^3*e^39/(e*x 
+ d)^3 + 60*B*a^3*b^3*d^4*e^39/(e*x + d)^4 + 45*A*a^2*b^4*d^4*e^39/(e*x + 
d)^4 + 180*B*a^4*b^2*e^40/(e*x + d) + 240*A*a^3*b^3*e^40/(e*x + d) - 270*B 
*a^4*b^2*d*e^40/(e*x + d)^2 - 360*A*a^3*b^3*d*e^40/(e*x + d)^2 + 180*B*...
 

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 863, normalized size of antiderivative = 3.09 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx=x^2\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{2\,e^5}-\frac {5\,B\,b^6\,d}{2\,e^6}\right )-\frac {x^3\,\left (15\,B\,a^4\,b^2\,e^6-80\,B\,a^3\,b^3\,d\,e^5+20\,A\,a^3\,b^3\,e^6+150\,B\,a^2\,b^4\,d^2\,e^4-60\,A\,a^2\,b^4\,d\,e^5-120\,B\,a\,b^5\,d^3\,e^3+60\,A\,a\,b^5\,d^2\,e^4+35\,B\,b^6\,d^4\,e^2-20\,A\,b^6\,d^3\,e^3\right )+\frac {B\,a^6\,d\,e^6+3\,A\,a^6\,e^7+6\,B\,a^5\,b\,d^2\,e^5+6\,A\,a^5\,b\,d\,e^6+45\,B\,a^4\,b^2\,d^3\,e^4+15\,A\,a^4\,b^2\,d^2\,e^5-500\,B\,a^3\,b^3\,d^4\,e^3+60\,A\,a^3\,b^3\,d^3\,e^4+1155\,B\,a^2\,b^4\,d^5\,e^2-375\,A\,a^2\,b^4\,d^4\,e^3-1026\,B\,a\,b^5\,d^6\,e+462\,A\,a\,b^5\,d^5\,e^2+319\,B\,b^6\,d^7-171\,A\,b^6\,d^6\,e}{12\,e}+x\,\left (\frac {B\,a^6\,e^6}{3}+2\,B\,a^5\,b\,d\,e^5+2\,A\,a^5\,b\,e^6+15\,B\,a^4\,b^2\,d^2\,e^4+5\,A\,a^4\,b^2\,d\,e^5-\frac {440\,B\,a^3\,b^3\,d^3\,e^3}{3}+20\,A\,a^3\,b^3\,d^2\,e^4+325\,B\,a^2\,b^4\,d^4\,e^2-110\,A\,a^2\,b^4\,d^3\,e^3-282\,B\,a\,b^5\,d^5\,e+130\,A\,a\,b^5\,d^4\,e^2+\frac {259\,B\,b^6\,d^6}{3}-47\,A\,b^6\,d^5\,e\right )+x^2\,\left (3\,B\,a^5\,b\,e^6+\frac {45\,B\,a^4\,b^2\,d\,e^5}{2}+\frac {15\,A\,a^4\,b^2\,e^6}{2}-180\,B\,a^3\,b^3\,d^2\,e^4+30\,A\,a^3\,b^3\,d\,e^5+375\,B\,a^2\,b^4\,d^3\,e^3-135\,A\,a^2\,b^4\,d^2\,e^4-315\,B\,a\,b^5\,d^4\,e^2+150\,A\,a\,b^5\,d^3\,e^3+\frac {189\,B\,b^6\,d^5\,e}{2}-\frac {105\,A\,b^6\,d^4\,e^2}{2}\right )}{d^4\,e^7+4\,d^3\,e^8\,x+6\,d^2\,e^9\,x^2+4\,d\,e^{10}\,x^3+e^{11}\,x^4}-x\,\left (\frac {5\,d\,\left (\frac {A\,b^6+6\,B\,a\,b^5}{e^5}-\frac {5\,B\,b^6\,d}{e^6}\right )}{e}-\frac {3\,a\,b^4\,\left (2\,A\,b+5\,B\,a\right )}{e^5}+\frac {10\,B\,b^6\,d^2}{e^7}\right )+\frac {\ln \left (d+e\,x\right )\,\left (20\,B\,a^3\,b^3\,e^3-75\,B\,a^2\,b^4\,d\,e^2+15\,A\,a^2\,b^4\,e^3+90\,B\,a\,b^5\,d^2\,e-30\,A\,a\,b^5\,d\,e^2-35\,B\,b^6\,d^3+15\,A\,b^6\,d^2\,e\right )}{e^8}+\frac {B\,b^6\,x^3}{3\,e^5} \] Input:

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^5,x)
 

Output:

x^2*((A*b^6 + 6*B*a*b^5)/(2*e^5) - (5*B*b^6*d)/(2*e^6)) - (x^3*(20*A*a^3*b 
^3*e^6 + 15*B*a^4*b^2*e^6 - 20*A*b^6*d^3*e^3 + 35*B*b^6*d^4*e^2 + 60*A*a*b 
^5*d^2*e^4 - 60*A*a^2*b^4*d*e^5 - 120*B*a*b^5*d^3*e^3 - 80*B*a^3*b^3*d*e^5 
 + 150*B*a^2*b^4*d^2*e^4) + (3*A*a^6*e^7 + 319*B*b^6*d^7 - 171*A*b^6*d^6*e 
 + B*a^6*d*e^6 + 462*A*a*b^5*d^5*e^2 + 6*B*a^5*b*d^2*e^5 - 375*A*a^2*b^4*d 
^4*e^3 + 60*A*a^3*b^3*d^3*e^4 + 15*A*a^4*b^2*d^2*e^5 + 1155*B*a^2*b^4*d^5* 
e^2 - 500*B*a^3*b^3*d^4*e^3 + 45*B*a^4*b^2*d^3*e^4 + 6*A*a^5*b*d*e^6 - 102 
6*B*a*b^5*d^6*e)/(12*e) + x*((B*a^6*e^6)/3 + (259*B*b^6*d^6)/3 + 2*A*a^5*b 
*e^6 - 47*A*b^6*d^5*e + 130*A*a*b^5*d^4*e^2 + 5*A*a^4*b^2*d*e^5 - 110*A*a^ 
2*b^4*d^3*e^3 + 20*A*a^3*b^3*d^2*e^4 + 325*B*a^2*b^4*d^4*e^2 - (440*B*a^3* 
b^3*d^3*e^3)/3 + 15*B*a^4*b^2*d^2*e^4 - 282*B*a*b^5*d^5*e + 2*B*a^5*b*d*e^ 
5) + x^2*(3*B*a^5*b*e^6 + (189*B*b^6*d^5*e)/2 + (15*A*a^4*b^2*e^6)/2 - (10 
5*A*b^6*d^4*e^2)/2 + 150*A*a*b^5*d^3*e^3 + 30*A*a^3*b^3*d*e^5 - 315*B*a*b^ 
5*d^4*e^2 + (45*B*a^4*b^2*d*e^5)/2 - 135*A*a^2*b^4*d^2*e^4 + 375*B*a^2*b^4 
*d^3*e^3 - 180*B*a^3*b^3*d^2*e^4))/(d^4*e^7 + e^11*x^4 + 4*d^3*e^8*x + 4*d 
*e^10*x^3 + 6*d^2*e^9*x^2) - x*((5*d*((A*b^6 + 6*B*a*b^5)/e^5 - (5*B*b^6*d 
)/e^6))/e - (3*a*b^4*(2*A*b + 5*B*a))/e^5 + (10*B*b^6*d^2)/e^7) + (log(d + 
 e*x)*(15*A*b^6*d^2*e - 35*B*b^6*d^3 + 15*A*a^2*b^4*e^3 + 20*B*a^3*b^3*e^3 
 - 75*B*a^2*b^4*d*e^2 - 30*A*a*b^5*d*e^2 + 90*B*a*b^5*d^2*e))/e^8 + (B*b^6 
*x^3)/(3*e^5)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 851, normalized size of antiderivative = 3.05 \[ \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^6*(B*x+A)/(e*x+d)^5,x)
                                                                                    
                                                                                    
 

Output:

(420*log(d + e*x)*a**3*b**4*d**5*e**3 + 1680*log(d + e*x)*a**3*b**4*d**4*e 
**4*x + 2520*log(d + e*x)*a**3*b**4*d**3*e**5*x**2 + 1680*log(d + e*x)*a** 
3*b**4*d**2*e**6*x**3 + 420*log(d + e*x)*a**3*b**4*d*e**7*x**4 - 1260*log( 
d + e*x)*a**2*b**5*d**6*e**2 - 5040*log(d + e*x)*a**2*b**5*d**5*e**3*x - 7 
560*log(d + e*x)*a**2*b**5*d**4*e**4*x**2 - 5040*log(d + e*x)*a**2*b**5*d* 
*3*e**5*x**3 - 1260*log(d + e*x)*a**2*b**5*d**2*e**6*x**4 + 1260*log(d + e 
*x)*a*b**6*d**7*e + 5040*log(d + e*x)*a*b**6*d**6*e**2*x + 7560*log(d + e* 
x)*a*b**6*d**5*e**3*x**2 + 5040*log(d + e*x)*a*b**6*d**4*e**4*x**3 + 1260* 
log(d + e*x)*a*b**6*d**3*e**5*x**4 - 420*log(d + e*x)*b**7*d**8 - 1680*log 
(d + e*x)*b**7*d**7*e*x - 2520*log(d + e*x)*b**7*d**6*e**2*x**2 - 1680*log 
(d + e*x)*b**7*d**5*e**3*x**3 - 420*log(d + e*x)*b**7*d**4*e**4*x**4 - 3*a 
**7*d*e**7 - 7*a**6*b*d**2*e**6 - 28*a**6*b*d*e**7*x - 21*a**5*b**2*d**3*e 
**5 - 84*a**5*b**2*d**2*e**6*x - 126*a**5*b**2*d*e**7*x**2 + 105*a**4*b**3 
*e**8*x**4 + 455*a**3*b**4*d**5*e**3 + 1400*a**3*b**4*d**4*e**4*x + 1260*a 
**3*b**4*d**3*e**5*x**2 - 420*a**3*b**4*d*e**7*x**4 - 1365*a**2*b**5*d**6* 
e**2 - 4200*a**2*b**5*d**5*e**3*x - 3780*a**2*b**5*d**4*e**4*x**2 + 1260*a 
**2*b**5*d**2*e**6*x**4 + 252*a**2*b**5*d*e**7*x**5 + 1365*a*b**6*d**7*e + 
 4200*a*b**6*d**6*e**2*x + 3780*a*b**6*d**5*e**3*x**2 - 1260*a*b**6*d**3*e 
**5*x**4 - 252*a*b**6*d**2*e**6*x**5 + 42*a*b**6*d*e**7*x**6 - 455*b**7*d* 
*8 - 1400*b**7*d**7*e*x - 1260*b**7*d**6*e**2*x**2 + 420*b**7*d**4*e**4...