\(\int (a+b x)^6 (A+B x) (d+e x)^4 \, dx\) [1055]

Optimal result

Integrand size = 20, antiderivative size = 204 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=\frac {(A b-a B) (b d-a e)^4 (a+b x)^7}{7 b^6}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^8}{8 b^6}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^9}{9 b^6}+\frac {e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^{10}}{5 b^6}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^{11}}{11 b^6}+\frac {B e^4 (a+b x)^{12}}{12 b^6} \]

[Out]

1/7*(A*b-B*a)*(-a*e+b*d)^4*(b*x+a)^7/b^6+1/8*(-a*e+b*d)^3*(4*A*b*e-5*B*a*e+B*b*d)*(b*x+a)^8/b^6+2/9*e*(-a*e+b*
d)^2*(3*A*b*e-5*B*a*e+2*B*b*d)*(b*x+a)^9/b^6+1/5*e^2*(-a*e+b*d)*(2*A*b*e-5*B*a*e+3*B*b*d)*(b*x+a)^10/b^6+1/11*
e^3*(A*b*e-5*B*a*e+4*B*b*d)*(b*x+a)^11/b^6+1/12*B*e^4*(b*x+a)^12/b^6

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=\frac {e^3 (a+b x)^{11} (-5 a B e+A b e+4 b B d)}{11 b^6}+\frac {e^2 (a+b x)^{10} (b d-a e) (-5 a B e+2 A b e+3 b B d)}{5 b^6}+\frac {2 e (a+b x)^9 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{9 b^6}+\frac {(a+b x)^8 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{8 b^6}+\frac {(a+b x)^7 (A b-a B) (b d-a e)^4}{7 b^6}+\frac {B e^4 (a+b x)^{12}}{12 b^6} \]

[In]

Int[(a + b*x)^6*(A + B*x)*(d + e*x)^4,x]

[Out]

((A*b - a*B)*(b*d - a*e)^4*(a + b*x)^7)/(7*b^6) + ((b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e)*(a + b*x)^8)/(8*b
^6) + (2*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*(a + b*x)^9)/(9*b^6) + (e^2*(b*d - a*e)*(3*b*B*d + 2*A*
b*e - 5*a*B*e)*(a + b*x)^10)/(5*b^6) + (e^3*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^11)/(11*b^6) + (B*e^4*(a + b
*x)^12)/(12*b^6)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (b d-a e)^4 (a+b x)^6}{b^5}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^7}{b^5}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^8}{b^5}+\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^9}{b^5}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^{10}}{b^5}+\frac {B e^4 (a+b x)^{11}}{b^5}\right ) \, dx \\ & = \frac {(A b-a B) (b d-a e)^4 (a+b x)^7}{7 b^6}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^8}{8 b^6}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^9}{9 b^6}+\frac {e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^{10}}{5 b^6}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^{11}}{11 b^6}+\frac {B e^4 (a+b x)^{12}}{12 b^6} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(762\) vs. \(2(204)=408\).

Time = 0.16 (sec) , antiderivative size = 762, normalized size of antiderivative = 3.74 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=a^6 A d^4 x+\frac {1}{2} a^5 d^3 (6 A b d+a B d+4 a A e) x^2+\frac {1}{3} a^4 d^2 \left (2 a B d (3 b d+2 a e)+3 A \left (5 b^2 d^2+8 a b d e+2 a^2 e^2\right )\right ) x^3+\frac {1}{4} a^3 d \left (3 a B d \left (5 b^2 d^2+8 a b d e+2 a^2 e^2\right )+4 A \left (5 b^3 d^3+15 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right )\right ) x^4+\frac {1}{5} a^2 \left (4 a B d \left (5 b^3 d^3+15 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right )+A \left (15 b^4 d^4+80 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+24 a^3 b d e^3+a^4 e^4\right )\right ) x^5+\frac {1}{6} a \left (6 A b \left (b^4 d^4+10 a b^3 d^3 e+20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4\right )+a B \left (15 b^4 d^4+80 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+24 a^3 b d e^3+a^4 e^4\right )\right ) x^6+\frac {1}{7} b \left (6 a B \left (b^4 d^4+10 a b^3 d^3 e+20 a^2 b^2 d^2 e^2+10 a^3 b d e^3+a^4 e^4\right )+A b \left (b^4 d^4+24 a b^3 d^3 e+90 a^2 b^2 d^2 e^2+80 a^3 b d e^3+15 a^4 e^4\right )\right ) x^7+\frac {1}{8} b^2 \left (15 a^4 B e^4+20 a^3 b e^3 (4 B d+A e)+30 a^2 b^2 d e^2 (3 B d+2 A e)+12 a b^3 d^2 e (2 B d+3 A e)+b^4 d^3 (B d+4 A e)\right ) x^8+\frac {1}{9} b^3 e \left (20 a^3 B e^3+15 a^2 b e^2 (4 B d+A e)+12 a b^2 d e (3 B d+2 A e)+2 b^3 d^2 (2 B d+3 A e)\right ) x^9+\frac {1}{10} b^4 e^2 \left (15 a^2 B e^2+6 a b e (4 B d+A e)+2 b^2 d (3 B d+2 A e)\right ) x^{10}+\frac {1}{11} b^5 e^3 (4 b B d+A b e+6 a B e) x^{11}+\frac {1}{12} b^6 B e^4 x^{12} \]

[In]

Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^4,x]

[Out]

a^6*A*d^4*x + (a^5*d^3*(6*A*b*d + a*B*d + 4*a*A*e)*x^2)/2 + (a^4*d^2*(2*a*B*d*(3*b*d + 2*a*e) + 3*A*(5*b^2*d^2
+ 8*a*b*d*e + 2*a^2*e^2))*x^3)/3 + (a^3*d*(3*a*B*d*(5*b^2*d^2 + 8*a*b*d*e + 2*a^2*e^2) + 4*A*(5*b^3*d^3 + 15*
a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3*e^3))*x^4)/4 + (a^2*(4*a*B*d*(5*b^3*d^3 + 15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a
^3*e^3) + A*(15*b^4*d^4 + 80*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 24*a^3*b*d*e^3 + a^4*e^4))*x^5)/5 + (a*(6*A*b*
(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4) + a*B*(15*b^4*d^4 + 80*a*b^3*d^3*e
+ 90*a^2*b^2*d^2*e^2 + 24*a^3*b*d*e^3 + a^4*e^4))*x^6)/6 + (b*(6*a*B*(b^4*d^4 + 10*a*b^3*d^3*e + 20*a^2*b^2*d^
2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4) + A*b*(b^4*d^4 + 24*a*b^3*d^3*e + 90*a^2*b^2*d^2*e^2 + 80*a^3*b*d*e^3 + 15*a
^4*e^4))*x^7)/7 + (b^2*(15*a^4*B*e^4 + 20*a^3*b*e^3*(4*B*d + A*e) + 30*a^2*b^2*d*e^2*(3*B*d + 2*A*e) + 12*a*b^
3*d^2*e*(2*B*d + 3*A*e) + b^4*d^3*(B*d + 4*A*e))*x^8)/8 + (b^3*e*(20*a^3*B*e^3 + 15*a^2*b*e^2*(4*B*d + A*e) +
12*a*b^2*d*e*(3*B*d + 2*A*e) + 2*b^3*d^2*(2*B*d + 3*A*e))*x^9)/9 + (b^4*e^2*(15*a^2*B*e^2 + 6*a*b*e*(4*B*d + A
*e) + 2*b^2*d*(3*B*d + 2*A*e))*x^10)/10 + (b^5*e^3*(4*b*B*d + A*b*e + 6*a*B*e)*x^11)/11 + (b^6*B*e^4*x^12)/12

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(820\) vs. \(2(192)=384\).

Time = 0.69 (sec) , antiderivative size = 821, normalized size of antiderivative = 4.02

method	result	size
default	\(\frac {b^{6} B \,e^{4} x^{12}}{12}+\frac {\left (\left (b^{6} A +6 a \,b^{5} B \right ) e^{4}+4 b^{6} B d \,e^{3}\right ) x^{11}}{11}+\frac {\left (\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) e^{4}+4 \left (b^{6} A +6 a \,b^{5} B \right ) d \,e^{3}+6 b^{6} B \,d^{2} e^{2}\right ) x^{10}}{10}+\frac {\left (\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) e^{4}+4 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d \,e^{3}+6 \left (b^{6} A +6 a \,b^{5} B \right ) d^{2} e^{2}+4 b^{6} B \,d^{3} e \right ) x^{9}}{9}+\frac {\left (\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) e^{4}+4 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d \,e^{3}+6 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{2} e^{2}+4 \left (b^{6} A +6 a \,b^{5} B \right ) d^{3} e +b^{6} B \,d^{4}\right ) x^{8}}{8}+\frac {\left (\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) e^{4}+4 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d \,e^{3}+6 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{2} e^{2}+4 \left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{3} e +\left (b^{6} A +6 a \,b^{5} B \right ) d^{4}\right ) x^{7}}{7}+\frac {\left (\left (6 A \,a^{5} b +B \,a^{6}\right ) e^{4}+4 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d \,e^{3}+6 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{2} e^{2}+4 \left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{3} e +\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d^{4}\right ) x^{6}}{6}+\frac {\left (A \,a^{6} e^{4}+4 \left (6 A \,a^{5} b +B \,a^{6}\right ) d \,e^{3}+6 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{2} e^{2}+4 \left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{3} e +\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d^{4}\right ) x^{5}}{5}+\frac {\left (4 A \,a^{6} d \,e^{3}+6 \left (6 A \,a^{5} b +B \,a^{6}\right ) d^{2} e^{2}+4 \left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{3} e +\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d^{4}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{6} d^{2} e^{2}+4 \left (6 A \,a^{5} b +B \,a^{6}\right ) d^{3} e +\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d^{4}\right ) x^{3}}{3}+\frac {\left (4 A \,a^{6} d^{3} e +\left (6 A \,a^{5} b +B \,a^{6}\right ) d^{4}\right ) x^{2}}{2}+A \,a^{6} d^{4} x\)	\(821\)
norman	\(\frac {b^{6} B \,e^{4} x^{12}}{12}+\left (\frac {1}{11} A \,b^{6} e^{4}+\frac {6}{11} B a \,b^{5} e^{4}+\frac {4}{11} b^{6} B d \,e^{3}\right ) x^{11}+\left (\frac {3}{5} A a \,b^{5} e^{4}+\frac {2}{5} A \,b^{6} d \,e^{3}+\frac {3}{2} B \,a^{2} b^{4} e^{4}+\frac {12}{5} B a \,b^{5} d \,e^{3}+\frac {3}{5} b^{6} B \,d^{2} e^{2}\right ) x^{10}+\left (\frac {5}{3} A \,a^{2} b^{4} e^{4}+\frac {8}{3} A a \,b^{5} d \,e^{3}+\frac {2}{3} A \,b^{6} d^{2} e^{2}+\frac {20}{9} B \,a^{3} b^{3} e^{4}+\frac {20}{3} B \,a^{2} b^{4} d \,e^{3}+4 B a \,b^{5} d^{2} e^{2}+\frac {4}{9} b^{6} B \,d^{3} e \right ) x^{9}+\left (\frac {5}{2} A \,a^{3} b^{3} e^{4}+\frac {15}{2} A \,a^{2} b^{4} d \,e^{3}+\frac {9}{2} A a \,b^{5} d^{2} e^{2}+\frac {1}{2} A \,b^{6} d^{3} e +\frac {15}{8} B \,a^{4} b^{2} e^{4}+10 B \,a^{3} b^{3} d \,e^{3}+\frac {45}{4} B \,a^{2} b^{4} d^{2} e^{2}+3 B a \,b^{5} d^{3} e +\frac {1}{8} b^{6} B \,d^{4}\right ) x^{8}+\left (\frac {15}{7} A \,a^{4} b^{2} e^{4}+\frac {80}{7} A \,a^{3} b^{3} d \,e^{3}+\frac {90}{7} A \,a^{2} b^{4} d^{2} e^{2}+\frac {24}{7} A a \,b^{5} d^{3} e +\frac {1}{7} A \,b^{6} d^{4}+\frac {6}{7} B \,a^{5} b \,e^{4}+\frac {60}{7} B \,a^{4} b^{2} d \,e^{3}+\frac {120}{7} B \,a^{3} b^{3} d^{2} e^{2}+\frac {60}{7} B \,a^{2} b^{4} d^{3} e +\frac {6}{7} B a \,b^{5} d^{4}\right ) x^{7}+\left (A \,a^{5} b \,e^{4}+10 A \,a^{4} b^{2} d \,e^{3}+20 A \,a^{3} b^{3} d^{2} e^{2}+10 A \,a^{2} b^{4} d^{3} e +A a \,b^{5} d^{4}+\frac {1}{6} B \,a^{6} e^{4}+4 B \,a^{5} b d \,e^{3}+15 B \,a^{4} b^{2} d^{2} e^{2}+\frac {40}{3} B \,a^{3} b^{3} d^{3} e +\frac {5}{2} B \,a^{2} b^{4} d^{4}\right ) x^{6}+\left (\frac {1}{5} A \,a^{6} e^{4}+\frac {24}{5} A \,a^{5} b d \,e^{3}+18 A \,a^{4} b^{2} d^{2} e^{2}+16 A \,a^{3} b^{3} d^{3} e +3 A \,a^{2} b^{4} d^{4}+\frac {4}{5} B \,a^{6} d \,e^{3}+\frac {36}{5} B \,a^{5} b \,d^{2} e^{2}+12 B \,a^{4} b^{2} d^{3} e +4 B \,a^{3} b^{3} d^{4}\right ) x^{5}+\left (A \,a^{6} d \,e^{3}+9 A \,a^{5} b \,d^{2} e^{2}+15 A \,a^{4} b^{2} d^{3} e +5 A \,a^{3} b^{3} d^{4}+\frac {3}{2} B \,a^{6} d^{2} e^{2}+6 B \,a^{5} b \,d^{3} e +\frac {15}{4} B \,a^{4} b^{2} d^{4}\right ) x^{4}+\left (2 A \,a^{6} d^{2} e^{2}+8 A \,a^{5} b \,d^{3} e +5 A \,a^{4} b^{2} d^{4}+\frac {4}{3} B \,a^{6} d^{3} e +2 B \,a^{5} b \,d^{4}\right ) x^{3}+\left (2 A \,a^{6} d^{3} e +3 A \,a^{5} b \,d^{4}+\frac {1}{2} B \,a^{6} d^{4}\right ) x^{2}+A \,a^{6} d^{4} x\)	\(862\)
gosper	\(\text {Expression too large to display}\)	\(1016\)
risch	\(\text {Expression too large to display}\)	\(1016\)
parallelrisch	\(\text {Expression too large to display}\)	\(1016\)

[In]

int((b*x+a)^6*(B*x+A)*(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/12*b^6*B*e^4*x^12+1/11*((A*b^6+6*B*a*b^5)*e^4+4*b^6*B*d*e^3)*x^11+1/10*((6*A*a*b^5+15*B*a^2*b^4)*e^4+4*(A*b^
6+6*B*a*b^5)*d*e^3+6*b^6*B*d^2*e^2)*x^10+1/9*((15*A*a^2*b^4+20*B*a^3*b^3)*e^4+4*(6*A*a*b^5+15*B*a^2*b^4)*d*e^3
+6*(A*b^6+6*B*a*b^5)*d^2*e^2+4*b^6*B*d^3*e)*x^9+1/8*((20*A*a^3*b^3+15*B*a^4*b^2)*e^4+4*(15*A*a^2*b^4+20*B*a^3*
b^3)*d*e^3+6*(6*A*a*b^5+15*B*a^2*b^4)*d^2*e^2+4*(A*b^6+6*B*a*b^5)*d^3*e+b^6*B*d^4)*x^8+1/7*((15*A*a^4*b^2+6*B*
a^5*b)*e^4+4*(20*A*a^3*b^3+15*B*a^4*b^2)*d*e^3+6*(15*A*a^2*b^4+20*B*a^3*b^3)*d^2*e^2+4*(6*A*a*b^5+15*B*a^2*b^4
)*d^3*e+(A*b^6+6*B*a*b^5)*d^4)*x^7+1/6*((6*A*a^5*b+B*a^6)*e^4+4*(15*A*a^4*b^2+6*B*a^5*b)*d*e^3+6*(20*A*a^3*b^3
+15*B*a^4*b^2)*d^2*e^2+4*(15*A*a^2*b^4+20*B*a^3*b^3)*d^3*e+(6*A*a*b^5+15*B*a^2*b^4)*d^4)*x^6+1/5*(A*a^6*e^4+4*
(6*A*a^5*b+B*a^6)*d*e^3+6*(15*A*a^4*b^2+6*B*a^5*b)*d^2*e^2+4*(20*A*a^3*b^3+15*B*a^4*b^2)*d^3*e+(15*A*a^2*b^4+2
0*B*a^3*b^3)*d^4)*x^5+1/4*(4*A*a^6*d*e^3+6*(6*A*a^5*b+B*a^6)*d^2*e^2+4*(15*A*a^4*b^2+6*B*a^5*b)*d^3*e+(20*A*a^
3*b^3+15*B*a^4*b^2)*d^4)*x^4+1/3*(6*A*a^6*d^2*e^2+4*(6*A*a^5*b+B*a^6)*d^3*e+(15*A*a^4*b^2+6*B*a^5*b)*d^4)*x^3+
1/2*(4*A*a^6*d^3*e+(6*A*a^5*b+B*a^6)*d^4)*x^2+A*a^6*d^4*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (192) = 384\).

Time = 0.22 (sec) , antiderivative size = 828, normalized size of antiderivative = 4.06 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=\frac {1}{12} \, B b^{6} e^{4} x^{12} + A a^{6} d^{4} x + \frac {1}{11} \, {\left (4 \, B b^{6} d e^{3} + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B b^{6} d^{2} e^{2} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{3} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B b^{6} d^{3} e + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{2} + 12 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{3} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{4} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e + 18 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{2} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} + 12 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e + 30 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{2} + 20 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{3} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e + 30 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{2} + 12 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{3} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{6} e^{4} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} + 20 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e + 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{6} d e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{4} + 12 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} e + 6 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{6} d^{2} e^{2} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{4} + 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{6} d^{3} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{4}\right )} x^{2} \]

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^4,x, algorithm="fricas")

[Out]

1/12*B*b^6*e^4*x^12 + A*a^6*d^4*x + 1/11*(4*B*b^6*d*e^3 + (6*B*a*b^5 + A*b^6)*e^4)*x^11 + 1/10*(6*B*b^6*d^2*e^
2 + 4*(6*B*a*b^5 + A*b^6)*d*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^4)*x^10 + 1/9*(4*B*b^6*d^3*e + 6*(6*B*a*b^5 +
A*b^6)*d^2*e^2 + 12*(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^9 + 1/8*(B*b^6*d^4
+ 4*(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 + 20*(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^8 + 1/7*((6*B*a*b^5 + A*b^6)*d^4 + 12*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e + 3
0*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^2 + 20*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^3 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^4
)*x^7 + 1/6*(3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e + 30*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*d^2*e^2 + 12*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^3 + (B*a^6 + 6*A*a^5*b)*e^4)*x^6 + 1/5*(A*a^6*e^4 + 5*(4*B*a
^3*b^3 + 3*A*a^2*b^4)*d^4 + 20*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e + 18*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^2 + 4*(B
*a^6 + 6*A*a^5*b)*d*e^3)*x^5 + 1/4*(4*A*a^6*d*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4 + 12*(2*B*a^5*b + 5*A*a^
4*b^2)*d^3*e + 6*(B*a^6 + 6*A*a^5*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^6*d^2*e^2 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^4 + 4
*(B*a^6 + 6*A*a^5*b)*d^3*e)*x^3 + 1/2*(4*A*a^6*d^3*e + (B*a^6 + 6*A*a^5*b)*d^4)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1035 vs. \(2 (204) = 408\).

Time = 0.08 (sec) , antiderivative size = 1035, normalized size of antiderivative = 5.07 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=A a^{6} d^{4} x + \frac {B b^{6} e^{4} x^{12}}{12} + x^{11} \left (\frac {A b^{6} e^{4}}{11} + \frac {6 B a b^{5} e^{4}}{11} + \frac {4 B b^{6} d e^{3}}{11}\right ) + x^{10} \cdot \left (\frac {3 A a b^{5} e^{4}}{5} + \frac {2 A b^{6} d e^{3}}{5} + \frac {3 B a^{2} b^{4} e^{4}}{2} + \frac {12 B a b^{5} d e^{3}}{5} + \frac {3 B b^{6} d^{2} e^{2}}{5}\right ) + x^{9} \cdot \left (\frac {5 A a^{2} b^{4} e^{4}}{3} + \frac {8 A a b^{5} d e^{3}}{3} + \frac {2 A b^{6} d^{2} e^{2}}{3} + \frac {20 B a^{3} b^{3} e^{4}}{9} + \frac {20 B a^{2} b^{4} d e^{3}}{3} + 4 B a b^{5} d^{2} e^{2} + \frac {4 B b^{6} d^{3} e}{9}\right ) + x^{8} \cdot \left (\frac {5 A a^{3} b^{3} e^{4}}{2} + \frac {15 A a^{2} b^{4} d e^{3}}{2} + \frac {9 A a b^{5} d^{2} e^{2}}{2} + \frac {A b^{6} d^{3} e}{2} + \frac {15 B a^{4} b^{2} e^{4}}{8} + 10 B a^{3} b^{3} d e^{3} + \frac {45 B a^{2} b^{4} d^{2} e^{2}}{4} + 3 B a b^{5} d^{3} e + \frac {B b^{6} d^{4}}{8}\right ) + x^{7} \cdot \left (\frac {15 A a^{4} b^{2} e^{4}}{7} + \frac {80 A a^{3} b^{3} d e^{3}}{7} + \frac {90 A a^{2} b^{4} d^{2} e^{2}}{7} + \frac {24 A a b^{5} d^{3} e}{7} + \frac {A b^{6} d^{4}}{7} + \frac {6 B a^{5} b e^{4}}{7} + \frac {60 B a^{4} b^{2} d e^{3}}{7} + \frac {120 B a^{3} b^{3} d^{2} e^{2}}{7} + \frac {60 B a^{2} b^{4} d^{3} e}{7} + \frac {6 B a b^{5} d^{4}}{7}\right ) + x^{6} \left (A a^{5} b e^{4} + 10 A a^{4} b^{2} d e^{3} + 20 A a^{3} b^{3} d^{2} e^{2} + 10 A a^{2} b^{4} d^{3} e + A a b^{5} d^{4} + \frac {B a^{6} e^{4}}{6} + 4 B a^{5} b d e^{3} + 15 B a^{4} b^{2} d^{2} e^{2} + \frac {40 B a^{3} b^{3} d^{3} e}{3} + \frac {5 B a^{2} b^{4} d^{4}}{2}\right ) + x^{5} \left (\frac {A a^{6} e^{4}}{5} + \frac {24 A a^{5} b d e^{3}}{5} + 18 A a^{4} b^{2} d^{2} e^{2} + 16 A a^{3} b^{3} d^{3} e + 3 A a^{2} b^{4} d^{4} + \frac {4 B a^{6} d e^{3}}{5} + \frac {36 B a^{5} b d^{2} e^{2}}{5} + 12 B a^{4} b^{2} d^{3} e + 4 B a^{3} b^{3} d^{4}\right ) + x^{4} \left (A a^{6} d e^{3} + 9 A a^{5} b d^{2} e^{2} + 15 A a^{4} b^{2} d^{3} e + 5 A a^{3} b^{3} d^{4} + \frac {3 B a^{6} d^{2} e^{2}}{2} + 6 B a^{5} b d^{3} e + \frac {15 B a^{4} b^{2} d^{4}}{4}\right ) + x^{3} \cdot \left (2 A a^{6} d^{2} e^{2} + 8 A a^{5} b d^{3} e + 5 A a^{4} b^{2} d^{4} + \frac {4 B a^{6} d^{3} e}{3} + 2 B a^{5} b d^{4}\right ) + x^{2} \cdot \left (2 A a^{6} d^{3} e + 3 A a^{5} b d^{4} + \frac {B a^{6} d^{4}}{2}\right ) \]

[In]

integrate((b*x+a)**6*(B*x+A)*(e*x+d)**4,x)

[Out]

A*a**6*d**4*x + B*b**6*e**4*x**12/12 + x**11*(A*b**6*e**4/11 + 6*B*a*b**5*e**4/11 + 4*B*b**6*d*e**3/11) + x**1
0*(3*A*a*b**5*e**4/5 + 2*A*b**6*d*e**3/5 + 3*B*a**2*b**4*e**4/2 + 12*B*a*b**5*d*e**3/5 + 3*B*b**6*d**2*e**2/5)
+ x**9*(5*A*a**2*b**4*e**4/3 + 8*A*a*b**5*d*e**3/3 + 2*A*b**6*d**2*e**2/3 + 20*B*a**3*b**3*e**4/9 + 20*B*a**2
*b**4*d*e**3/3 + 4*B*a*b**5*d**2*e**2 + 4*B*b**6*d**3*e/9) + x**8*(5*A*a**3*b**3*e**4/2 + 15*A*a**2*b**4*d*e**
3/2 + 9*A*a*b**5*d**2*e**2/2 + A*b**6*d**3*e/2 + 15*B*a**4*b**2*e**4/8 + 10*B*a**3*b**3*d*e**3 + 45*B*a**2*b**
4*d**2*e**2/4 + 3*B*a*b**5*d**3*e + B*b**6*d**4/8) + x**7*(15*A*a**4*b**2*e**4/7 + 80*A*a**3*b**3*d*e**3/7 + 9
0*A*a**2*b**4*d**2*e**2/7 + 24*A*a*b**5*d**3*e/7 + A*b**6*d**4/7 + 6*B*a**5*b*e**4/7 + 60*B*a**4*b**2*d*e**3/7
+ 120*B*a**3*b**3*d**2*e**2/7 + 60*B*a**2*b**4*d**3*e/7 + 6*B*a*b**5*d**4/7) + x**6*(A*a**5*b*e**4 + 10*A*a**
4*b**2*d*e**3 + 20*A*a**3*b**3*d**2*e**2 + 10*A*a**2*b**4*d**3*e + A*a*b**5*d**4 + B*a**6*e**4/6 + 4*B*a**5*b*
d*e**3 + 15*B*a**4*b**2*d**2*e**2 + 40*B*a**3*b**3*d**3*e/3 + 5*B*a**2*b**4*d**4/2) + x**5*(A*a**6*e**4/5 + 24
*A*a**5*b*d*e**3/5 + 18*A*a**4*b**2*d**2*e**2 + 16*A*a**3*b**3*d**3*e + 3*A*a**2*b**4*d**4 + 4*B*a**6*d*e**3/5
+ 36*B*a**5*b*d**2*e**2/5 + 12*B*a**4*b**2*d**3*e + 4*B*a**3*b**3*d**4) + x**4*(A*a**6*d*e**3 + 9*A*a**5*b*d*
*2*e**2 + 15*A*a**4*b**2*d**3*e + 5*A*a**3*b**3*d**4 + 3*B*a**6*d**2*e**2/2 + 6*B*a**5*b*d**3*e + 15*B*a**4*b*
*2*d**4/4) + x**3*(2*A*a**6*d**2*e**2 + 8*A*a**5*b*d**3*e + 5*A*a**4*b**2*d**4 + 4*B*a**6*d**3*e/3 + 2*B*a**5*
b*d**4) + x**2*(2*A*a**6*d**3*e + 3*A*a**5*b*d**4 + B*a**6*d**4/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 828 vs. \(2 (192) = 384\).

Time = 0.22 (sec) , antiderivative size = 828, normalized size of antiderivative = 4.06 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=\frac {1}{12} \, B b^{6} e^{4} x^{12} + A a^{6} d^{4} x + \frac {1}{11} \, {\left (4 \, B b^{6} d e^{3} + {\left (6 \, B a b^{5} + A b^{6}\right )} e^{4}\right )} x^{11} + \frac {1}{10} \, {\left (6 \, B b^{6} d^{2} e^{2} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{3} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{4}\right )} x^{10} + \frac {1}{9} \, {\left (4 \, B b^{6} d^{3} e + 6 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{2} + 12 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{3} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{4} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e + 18 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{2} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} + 12 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e + 30 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{2} + 20 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{3} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e + 30 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{2} + 12 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{3} + {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{6} e^{4} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} + 20 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e + 18 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{6} d e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{4} + 12 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} e + 6 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{6} d^{2} e^{2} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{4} + 4 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{6} d^{3} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{4}\right )} x^{2} \]

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^4,x, algorithm="maxima")

[Out]

1/12*B*b^6*e^4*x^12 + A*a^6*d^4*x + 1/11*(4*B*b^6*d*e^3 + (6*B*a*b^5 + A*b^6)*e^4)*x^11 + 1/10*(6*B*b^6*d^2*e^
2 + 4*(6*B*a*b^5 + A*b^6)*d*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^4)*x^10 + 1/9*(4*B*b^6*d^3*e + 6*(6*B*a*b^5 +
A*b^6)*d^2*e^2 + 12*(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^9 + 1/8*(B*b^6*d^4
+ 4*(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 + 20*(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^8 + 1/7*((6*B*a*b^5 + A*b^6)*d^4 + 12*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e + 3
0*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^2 + 20*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^3 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^4
)*x^7 + 1/6*(3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e + 30*(3*B*a^4*b^2 + 4*A*a^
3*b^3)*d^2*e^2 + 12*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^3 + (B*a^6 + 6*A*a^5*b)*e^4)*x^6 + 1/5*(A*a^6*e^4 + 5*(4*B*a
^3*b^3 + 3*A*a^2*b^4)*d^4 + 20*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e + 18*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^2 + 4*(B
*a^6 + 6*A*a^5*b)*d*e^3)*x^5 + 1/4*(4*A*a^6*d*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4 + 12*(2*B*a^5*b + 5*A*a^
4*b^2)*d^3*e + 6*(B*a^6 + 6*A*a^5*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^6*d^2*e^2 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d^4 + 4
*(B*a^6 + 6*A*a^5*b)*d^3*e)*x^3 + 1/2*(4*A*a^6*d^3*e + (B*a^6 + 6*A*a^5*b)*d^4)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (192) = 384\).

Time = 0.28 (sec) , antiderivative size = 1015, normalized size of antiderivative = 4.98 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=\frac {1}{12} \, B b^{6} e^{4} x^{12} + \frac {4}{11} \, B b^{6} d e^{3} x^{11} + \frac {6}{11} \, B a b^{5} e^{4} x^{11} + \frac {1}{11} \, A b^{6} e^{4} x^{11} + \frac {3}{5} \, B b^{6} d^{2} e^{2} x^{10} + \frac {12}{5} \, B a b^{5} d e^{3} x^{10} + \frac {2}{5} \, A b^{6} d e^{3} x^{10} + \frac {3}{2} \, B a^{2} b^{4} e^{4} x^{10} + \frac {3}{5} \, A a b^{5} e^{4} x^{10} + \frac {4}{9} \, B b^{6} d^{3} e x^{9} + 4 \, B a b^{5} d^{2} e^{2} x^{9} + \frac {2}{3} \, A b^{6} d^{2} e^{2} x^{9} + \frac {20}{3} \, B a^{2} b^{4} d e^{3} x^{9} + \frac {8}{3} \, A a b^{5} d e^{3} x^{9} + \frac {20}{9} \, B a^{3} b^{3} e^{4} x^{9} + \frac {5}{3} \, A a^{2} b^{4} e^{4} x^{9} + \frac {1}{8} \, B b^{6} d^{4} x^{8} + 3 \, B a b^{5} d^{3} e x^{8} + \frac {1}{2} \, A b^{6} d^{3} e x^{8} + \frac {45}{4} \, B a^{2} b^{4} d^{2} e^{2} x^{8} + \frac {9}{2} \, A a b^{5} d^{2} e^{2} x^{8} + 10 \, B a^{3} b^{3} d e^{3} x^{8} + \frac {15}{2} \, A a^{2} b^{4} d e^{3} x^{8} + \frac {15}{8} \, B a^{4} b^{2} e^{4} x^{8} + \frac {5}{2} \, A a^{3} b^{3} e^{4} x^{8} + \frac {6}{7} \, B a b^{5} d^{4} x^{7} + \frac {1}{7} \, A b^{6} d^{4} x^{7} + \frac {60}{7} \, B a^{2} b^{4} d^{3} e x^{7} + \frac {24}{7} \, A a b^{5} d^{3} e x^{7} + \frac {120}{7} \, B a^{3} b^{3} d^{2} e^{2} x^{7} + \frac {90}{7} \, A a^{2} b^{4} d^{2} e^{2} x^{7} + \frac {60}{7} \, B a^{4} b^{2} d e^{3} x^{7} + \frac {80}{7} \, A a^{3} b^{3} d e^{3} x^{7} + \frac {6}{7} \, B a^{5} b e^{4} x^{7} + \frac {15}{7} \, A a^{4} b^{2} e^{4} x^{7} + \frac {5}{2} \, B a^{2} b^{4} d^{4} x^{6} + A a b^{5} d^{4} x^{6} + \frac {40}{3} \, B a^{3} b^{3} d^{3} e x^{6} + 10 \, A a^{2} b^{4} d^{3} e x^{6} + 15 \, B a^{4} b^{2} d^{2} e^{2} x^{6} + 20 \, A a^{3} b^{3} d^{2} e^{2} x^{6} + 4 \, B a^{5} b d e^{3} x^{6} + 10 \, A a^{4} b^{2} d e^{3} x^{6} + \frac {1}{6} \, B a^{6} e^{4} x^{6} + A a^{5} b e^{4} x^{6} + 4 \, B a^{3} b^{3} d^{4} x^{5} + 3 \, A a^{2} b^{4} d^{4} x^{5} + 12 \, B a^{4} b^{2} d^{3} e x^{5} + 16 \, A a^{3} b^{3} d^{3} e x^{5} + \frac {36}{5} \, B a^{5} b d^{2} e^{2} x^{5} + 18 \, A a^{4} b^{2} d^{2} e^{2} x^{5} + \frac {4}{5} \, B a^{6} d e^{3} x^{5} + \frac {24}{5} \, A a^{5} b d e^{3} x^{5} + \frac {1}{5} \, A a^{6} e^{4} x^{5} + \frac {15}{4} \, B a^{4} b^{2} d^{4} x^{4} + 5 \, A a^{3} b^{3} d^{4} x^{4} + 6 \, B a^{5} b d^{3} e x^{4} + 15 \, A a^{4} b^{2} d^{3} e x^{4} + \frac {3}{2} \, B a^{6} d^{2} e^{2} x^{4} + 9 \, A a^{5} b d^{2} e^{2} x^{4} + A a^{6} d e^{3} x^{4} + 2 \, B a^{5} b d^{4} x^{3} + 5 \, A a^{4} b^{2} d^{4} x^{3} + \frac {4}{3} \, B a^{6} d^{3} e x^{3} + 8 \, A a^{5} b d^{3} e x^{3} + 2 \, A a^{6} d^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{6} d^{4} x^{2} + 3 \, A a^{5} b d^{4} x^{2} + 2 \, A a^{6} d^{3} e x^{2} + A a^{6} d^{4} x \]

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d)^4,x, algorithm="giac")

[Out]

1/12*B*b^6*e^4*x^12 + 4/11*B*b^6*d*e^3*x^11 + 6/11*B*a*b^5*e^4*x^11 + 1/11*A*b^6*e^4*x^11 + 3/5*B*b^6*d^2*e^2*
x^10 + 12/5*B*a*b^5*d*e^3*x^10 + 2/5*A*b^6*d*e^3*x^10 + 3/2*B*a^2*b^4*e^4*x^10 + 3/5*A*a*b^5*e^4*x^10 + 4/9*B*
b^6*d^3*e*x^9 + 4*B*a*b^5*d^2*e^2*x^9 + 2/3*A*b^6*d^2*e^2*x^9 + 20/3*B*a^2*b^4*d*e^3*x^9 + 8/3*A*a*b^5*d*e^3*x
^9 + 20/9*B*a^3*b^3*e^4*x^9 + 5/3*A*a^2*b^4*e^4*x^9 + 1/8*B*b^6*d^4*x^8 + 3*B*a*b^5*d^3*e*x^8 + 1/2*A*b^6*d^3*
e*x^8 + 45/4*B*a^2*b^4*d^2*e^2*x^8 + 9/2*A*a*b^5*d^2*e^2*x^8 + 10*B*a^3*b^3*d*e^3*x^8 + 15/2*A*a^2*b^4*d*e^3*x
^8 + 15/8*B*a^4*b^2*e^4*x^8 + 5/2*A*a^3*b^3*e^4*x^8 + 6/7*B*a*b^5*d^4*x^7 + 1/7*A*b^6*d^4*x^7 + 60/7*B*a^2*b^4
*d^3*e*x^7 + 24/7*A*a*b^5*d^3*e*x^7 + 120/7*B*a^3*b^3*d^2*e^2*x^7 + 90/7*A*a^2*b^4*d^2*e^2*x^7 + 60/7*B*a^4*b^
2*d*e^3*x^7 + 80/7*A*a^3*b^3*d*e^3*x^7 + 6/7*B*a^5*b*e^4*x^7 + 15/7*A*a^4*b^2*e^4*x^7 + 5/2*B*a^2*b^4*d^4*x^6
+ A*a*b^5*d^4*x^6 + 40/3*B*a^3*b^3*d^3*e*x^6 + 10*A*a^2*b^4*d^3*e*x^6 + 15*B*a^4*b^2*d^2*e^2*x^6 + 20*A*a^3*b^
3*d^2*e^2*x^6 + 4*B*a^5*b*d*e^3*x^6 + 10*A*a^4*b^2*d*e^3*x^6 + 1/6*B*a^6*e^4*x^6 + A*a^5*b*e^4*x^6 + 4*B*a^3*b
^3*d^4*x^5 + 3*A*a^2*b^4*d^4*x^5 + 12*B*a^4*b^2*d^3*e*x^5 + 16*A*a^3*b^3*d^3*e*x^5 + 36/5*B*a^5*b*d^2*e^2*x^5
+ 18*A*a^4*b^2*d^2*e^2*x^5 + 4/5*B*a^6*d*e^3*x^5 + 24/5*A*a^5*b*d*e^3*x^5 + 1/5*A*a^6*e^4*x^5 + 15/4*B*a^4*b^2
*d^4*x^4 + 5*A*a^3*b^3*d^4*x^4 + 6*B*a^5*b*d^3*e*x^4 + 15*A*a^4*b^2*d^3*e*x^4 + 3/2*B*a^6*d^2*e^2*x^4 + 9*A*a^
5*b*d^2*e^2*x^4 + A*a^6*d*e^3*x^4 + 2*B*a^5*b*d^4*x^3 + 5*A*a^4*b^2*d^4*x^3 + 4/3*B*a^6*d^3*e*x^3 + 8*A*a^5*b*
d^3*e*x^3 + 2*A*a^6*d^2*e^2*x^3 + 1/2*B*a^6*d^4*x^2 + 3*A*a^5*b*d^4*x^2 + 2*A*a^6*d^3*e*x^2 + A*a^6*d^4*x

Mupad [B] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 845, normalized size of antiderivative = 4.14 \[ \int (a+b x)^6 (A+B x) (d+e x)^4 \, dx=x^4\,\left (\frac {3\,B\,a^6\,d^2\,e^2}{2}+A\,a^6\,d\,e^3+6\,B\,a^5\,b\,d^3\,e+9\,A\,a^5\,b\,d^2\,e^2+\frac {15\,B\,a^4\,b^2\,d^4}{4}+15\,A\,a^4\,b^2\,d^3\,e+5\,A\,a^3\,b^3\,d^4\right )+x^9\,\left (\frac {20\,B\,a^3\,b^3\,e^4}{9}+\frac {20\,B\,a^2\,b^4\,d\,e^3}{3}+\frac {5\,A\,a^2\,b^4\,e^4}{3}+4\,B\,a\,b^5\,d^2\,e^2+\frac {8\,A\,a\,b^5\,d\,e^3}{3}+\frac {4\,B\,b^6\,d^3\,e}{9}+\frac {2\,A\,b^6\,d^2\,e^2}{3}\right )+x^3\,\left (\frac {4\,B\,a^6\,d^3\,e}{3}+2\,A\,a^6\,d^2\,e^2+2\,B\,a^5\,b\,d^4+8\,A\,a^5\,b\,d^3\,e+5\,A\,a^4\,b^2\,d^4\right )+x^{10}\,\left (\frac {3\,B\,a^2\,b^4\,e^4}{2}+\frac {12\,B\,a\,b^5\,d\,e^3}{5}+\frac {3\,A\,a\,b^5\,e^4}{5}+\frac {3\,B\,b^6\,d^2\,e^2}{5}+\frac {2\,A\,b^6\,d\,e^3}{5}\right )+x^5\,\left (\frac {4\,B\,a^6\,d\,e^3}{5}+\frac {A\,a^6\,e^4}{5}+\frac {36\,B\,a^5\,b\,d^2\,e^2}{5}+\frac {24\,A\,a^5\,b\,d\,e^3}{5}+12\,B\,a^4\,b^2\,d^3\,e+18\,A\,a^4\,b^2\,d^2\,e^2+4\,B\,a^3\,b^3\,d^4+16\,A\,a^3\,b^3\,d^3\,e+3\,A\,a^2\,b^4\,d^4\right )+x^8\,\left (\frac {15\,B\,a^4\,b^2\,e^4}{8}+10\,B\,a^3\,b^3\,d\,e^3+\frac {5\,A\,a^3\,b^3\,e^4}{2}+\frac {45\,B\,a^2\,b^4\,d^2\,e^2}{4}+\frac {15\,A\,a^2\,b^4\,d\,e^3}{2}+3\,B\,a\,b^5\,d^3\,e+\frac {9\,A\,a\,b^5\,d^2\,e^2}{2}+\frac {B\,b^6\,d^4}{8}+\frac {A\,b^6\,d^3\,e}{2}\right )+x^6\,\left (\frac {B\,a^6\,e^4}{6}+4\,B\,a^5\,b\,d\,e^3+A\,a^5\,b\,e^4+15\,B\,a^4\,b^2\,d^2\,e^2+10\,A\,a^4\,b^2\,d\,e^3+\frac {40\,B\,a^3\,b^3\,d^3\,e}{3}+20\,A\,a^3\,b^3\,d^2\,e^2+\frac {5\,B\,a^2\,b^4\,d^4}{2}+10\,A\,a^2\,b^4\,d^3\,e+A\,a\,b^5\,d^4\right )+x^7\,\left (\frac {6\,B\,a^5\,b\,e^4}{7}+\frac {60\,B\,a^4\,b^2\,d\,e^3}{7}+\frac {15\,A\,a^4\,b^2\,e^4}{7}+\frac {120\,B\,a^3\,b^3\,d^2\,e^2}{7}+\frac {80\,A\,a^3\,b^3\,d\,e^3}{7}+\frac {60\,B\,a^2\,b^4\,d^3\,e}{7}+\frac {90\,A\,a^2\,b^4\,d^2\,e^2}{7}+\frac {6\,B\,a\,b^5\,d^4}{7}+\frac {24\,A\,a\,b^5\,d^3\,e}{7}+\frac {A\,b^6\,d^4}{7}\right )+\frac {a^5\,d^3\,x^2\,\left (4\,A\,a\,e+6\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^5\,e^3\,x^{11}\,\left (A\,b\,e+6\,B\,a\,e+4\,B\,b\,d\right )}{11}+A\,a^6\,d^4\,x+\frac {B\,b^6\,e^4\,x^{12}}{12} \]

[In]

int((A + B*x)*(a + b*x)^6*(d + e*x)^4,x)

[Out]

x^4*(A*a^6*d*e^3 + 5*A*a^3*b^3*d^4 + (15*B*a^4*b^2*d^4)/4 + (3*B*a^6*d^2*e^2)/2 + 15*A*a^4*b^2*d^3*e + 9*A*a^5
*b*d^2*e^2 + 6*B*a^5*b*d^3*e) + x^9*((4*B*b^6*d^3*e)/9 + (5*A*a^2*b^4*e^4)/3 + (20*B*a^3*b^3*e^4)/9 + (2*A*b^6
*d^2*e^2)/3 + 4*B*a*b^5*d^2*e^2 + (20*B*a^2*b^4*d*e^3)/3 + (8*A*a*b^5*d*e^3)/3) + x^3*(2*B*a^5*b*d^4 + (4*B*a^
6*d^3*e)/3 + 5*A*a^4*b^2*d^4 + 2*A*a^6*d^2*e^2 + 8*A*a^5*b*d^3*e) + x^10*((3*A*a*b^5*e^4)/5 + (2*A*b^6*d*e^3)/
5 + (3*B*a^2*b^4*e^4)/2 + (3*B*b^6*d^2*e^2)/5 + (12*B*a*b^5*d*e^3)/5) + x^5*((A*a^6*e^4)/5 + (4*B*a^6*d*e^3)/5
+ 3*A*a^2*b^4*d^4 + 4*B*a^3*b^3*d^4 + 16*A*a^3*b^3*d^3*e + 12*B*a^4*b^2*d^3*e + (36*B*a^5*b*d^2*e^2)/5 + 18*A
*a^4*b^2*d^2*e^2 + (24*A*a^5*b*d*e^3)/5) + x^8*((B*b^6*d^4)/8 + (A*b^6*d^3*e)/2 + (5*A*a^3*b^3*e^4)/2 + (15*B*
a^4*b^2*e^4)/8 + (9*A*a*b^5*d^2*e^2)/2 + (15*A*a^2*b^4*d*e^3)/2 + 10*B*a^3*b^3*d*e^3 + (45*B*a^2*b^4*d^2*e^2)/
4 + 3*B*a*b^5*d^3*e) + x^6*((B*a^6*e^4)/6 + A*a*b^5*d^4 + A*a^5*b*e^4 + (5*B*a^2*b^4*d^4)/2 + 10*A*a^2*b^4*d^3
*e + 10*A*a^4*b^2*d*e^3 + (40*B*a^3*b^3*d^3*e)/3 + 20*A*a^3*b^3*d^2*e^2 + 15*B*a^4*b^2*d^2*e^2 + 4*B*a^5*b*d*e
^3) + x^7*((A*b^6*d^4)/7 + (6*B*a*b^5*d^4)/7 + (6*B*a^5*b*e^4)/7 + (15*A*a^4*b^2*e^4)/7 + (80*A*a^3*b^3*d*e^3)
/7 + (60*B*a^2*b^4*d^3*e)/7 + (60*B*a^4*b^2*d*e^3)/7 + (90*A*a^2*b^4*d^2*e^2)/7 + (120*B*a^3*b^3*d^2*e^2)/7 +
(24*A*a*b^5*d^3*e)/7) + (a^5*d^3*x^2*(4*A*a*e + 6*A*b*d + B*a*d))/2 + (b^5*e^3*x^11*(A*b*e + 6*B*a*e + 4*B*b*d
))/11 + A*a^6*d^4*x + (B*b^6*e^4*x^12)/12