3.11.54 \(\int (a+b x)^6 (A+B x) (d+e x)^5 \, dx\) [1054]

Optimal. Leaf size=240 \[ \frac {(A b-a B) (b d-a e)^5 (a+b x)^7}{7 b^7}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) (a+b x)^8}{8 b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) (a+b x)^9}{9 b^7}+\frac {e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)^{10}}{b^7}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^{11}}{11 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^{12}}{12 b^7}+\frac {B e^5 (a+b x)^{13}}{13 b^7} \]

[Out]

1/7*(A*b-B*a)*(-a*e+b*d)^5*(b*x+a)^7/b^7+1/8*(-a*e+b*d)^4*(5*A*b*e-6*B*a*e+B*b*d)*(b*x+a)^8/b^7+5/9*e*(-a*e+b*
d)^3*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^9/b^7+e^2*(-a*e+b*d)^2*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^10/b^7+5/11*e^3*(-a*
e+b*d)*(A*b*e-3*B*a*e+2*B*b*d)*(b*x+a)^11/b^7+1/12*e^4*(A*b*e-6*B*a*e+5*B*b*d)*(b*x+a)^12/b^7+1/13*B*e^5*(b*x+
a)^13/b^7

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Rubi [A]
time = 0.53, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} \frac {e^4 (a+b x)^{12} (-6 a B e+A b e+5 b B d)}{12 b^7}+\frac {5 e^3 (a+b x)^{11} (b d-a e) (-3 a B e+A b e+2 b B d)}{11 b^7}+\frac {e^2 (a+b x)^{10} (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7}+\frac {5 e (a+b x)^9 (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{9 b^7}+\frac {(a+b x)^8 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{8 b^7}+\frac {(a+b x)^7 (A b-a B) (b d-a e)^5}{7 b^7}+\frac {B e^5 (a+b x)^{13}}{13 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*(b*d - a*e)^5*(a + b*x)^7)/(7*b^7) + ((b*d - a*e)^4*(b*B*d + 5*A*b*e - 6*a*B*e)*(a + b*x)^8)/(8*b
^7) + (5*e*(b*d - a*e)^3*(b*B*d + 2*A*b*e - 3*a*B*e)*(a + b*x)^9)/(9*b^7) + (e^2*(b*d - a*e)^2*(b*B*d + A*b*e
- 2*a*B*e)*(a + b*x)^10)/b^7 + (5*e^3*(b*d - a*e)*(2*b*B*d + A*b*e - 3*a*B*e)*(a + b*x)^11)/(11*b^7) + (e^4*(5
*b*B*d + A*b*e - 6*a*B*e)*(a + b*x)^12)/(12*b^7) + (B*e^5*(a + b*x)^13)/(13*b^7)

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*} \int (a+b x)^6 (A+B x) (d+e x)^5 \, dx &=\int \left (\frac {(A b-a B) (b d-a e)^5 (a+b x)^6}{b^6}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) (a+b x)^7}{b^6}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) (a+b x)^8}{b^6}+\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)^9}{b^6}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^{10}}{b^6}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^{11}}{b^6}+\frac {B e^5 (a+b x)^{12}}{b^6}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e)^5 (a+b x)^7}{7 b^7}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) (a+b x)^8}{8 b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) (a+b x)^9}{9 b^7}+\frac {e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)^{10}}{b^7}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^{11}}{11 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^{12}}{12 b^7}+\frac {B e^5 (a+b x)^{13}}{13 b^7}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(907\) vs. \(2(240)=480\).
time = 0.23, size = 907, normalized size = 3.78 \begin {gather*} a^6 A d^5 x+\frac {1}{2} a^5 d^4 (6 A b d+a B d+5 a A e) x^2+\frac {1}{3} a^4 d^3 \left (a B d (6 b d+5 a e)+5 A \left (3 b^2 d^2+6 a b d e+2 a^2 e^2\right )\right ) x^3+\frac {5}{4} a^3 d^2 \left (a B d \left (3 b^2 d^2+6 a b d e+2 a^2 e^2\right )+A \left (4 b^3 d^3+15 a b^2 d^2 e+12 a^2 b d e^2+2 a^3 e^3\right )\right ) x^4+a^2 d \left (a B d \left (4 b^3 d^3+15 a b^2 d^2 e+12 a^2 b d e^2+2 a^3 e^3\right )+A \left (3 b^4 d^4+20 a b^3 d^3 e+30 a^2 b^2 d^2 e^2+12 a^3 b d e^3+a^4 e^4\right )\right ) x^5+\frac {1}{6} a \left (5 a B d \left (3 b^4 d^4+20 a b^3 d^3 e+30 a^2 b^2 d^2 e^2+12 a^3 b d e^3+a^4 e^4\right )+A \left (6 b^5 d^5+75 a b^4 d^4 e+200 a^2 b^3 d^3 e^2+150 a^3 b^2 d^2 e^3+30 a^4 b d e^4+a^5 e^5\right )\right ) x^6+\frac {1}{7} \left (a B \left (6 b^5 d^5+75 a b^4 d^4 e+200 a^2 b^3 d^3 e^2+150 a^3 b^2 d^2 e^3+30 a^4 b d e^4+a^5 e^5\right )+A b \left (b^5 d^5+30 a b^4 d^4 e+150 a^2 b^3 d^3 e^2+200 a^3 b^2 d^2 e^3+75 a^4 b d e^4+6 a^5 e^5\right )\right ) x^7+\frac {1}{8} b \left (6 a^5 B e^5+150 a^2 b^3 d^2 e^2 (B d+A e)+100 a^3 b^2 d e^3 (2 B d+A e)+15 a^4 b e^4 (5 B d+A e)+30 a b^4 d^3 e (B d+2 A e)+b^5 d^4 (B d+5 A e)\right ) x^8+\frac {5}{9} b^2 e \left (3 a^4 B e^4+12 a b^3 d^2 e (B d+A e)+15 a^2 b^2 d e^2 (2 B d+A e)+4 a^3 b e^3 (5 B d+A e)+b^4 d^3 (B d+2 A e)\right ) x^9+\frac {1}{2} b^3 e^2 \left (4 a^3 B e^3+2 b^3 d^2 (B d+A e)+6 a b^2 d e (2 B d+A e)+3 a^2 b e^2 (5 B d+A e)\right ) x^{10}+\frac {1}{11} b^4 e^3 \left (15 a^2 B e^2+5 b^2 d (2 B d+A e)+6 a b e (5 B d+A e)\right ) x^{11}+\frac {1}{12} b^5 e^4 (5 b B d+A b e+6 a B e) x^{12}+\frac {1}{13} b^6 B e^5 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^6*A*d^5*x + (a^5*d^4*(6*A*b*d + a*B*d + 5*a*A*e)*x^2)/2 + (a^4*d^3*(a*B*d*(6*b*d + 5*a*e) + 5*A*(3*b^2*d^2 +
6*a*b*d*e + 2*a^2*e^2))*x^3)/3 + (5*a^3*d^2*(a*B*d*(3*b^2*d^2 + 6*a*b*d*e + 2*a^2*e^2) + A*(4*b^3*d^3 + 15*a*
b^2*d^2*e + 12*a^2*b*d*e^2 + 2*a^3*e^3))*x^4)/4 + a^2*d*(a*B*d*(4*b^3*d^3 + 15*a*b^2*d^2*e + 12*a^2*b*d*e^2 +
2*a^3*e^3) + A*(3*b^4*d^4 + 20*a*b^3*d^3*e + 30*a^2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 + a^4*e^4))*x^5 + (a*(5*a*B*d
*(3*b^4*d^4 + 20*a*b^3*d^3*e + 30*a^2*b^2*d^2*e^2 + 12*a^3*b*d*e^3 + a^4*e^4) + A*(6*b^5*d^5 + 75*a*b^4*d^4*e
+ 200*a^2*b^3*d^3*e^2 + 150*a^3*b^2*d^2*e^3 + 30*a^4*b*d*e^4 + a^5*e^5))*x^6)/6 + ((a*B*(6*b^5*d^5 + 75*a*b^4*
d^4*e + 200*a^2*b^3*d^3*e^2 + 150*a^3*b^2*d^2*e^3 + 30*a^4*b*d*e^4 + a^5*e^5) + A*b*(b^5*d^5 + 30*a*b^4*d^4*e
+ 150*a^2*b^3*d^3*e^2 + 200*a^3*b^2*d^2*e^3 + 75*a^4*b*d*e^4 + 6*a^5*e^5))*x^7)/7 + (b*(6*a^5*B*e^5 + 150*a^2*
b^3*d^2*e^2*(B*d + A*e) + 100*a^3*b^2*d*e^3*(2*B*d + A*e) + 15*a^4*b*e^4*(5*B*d + A*e) + 30*a*b^4*d^3*e*(B*d +
2*A*e) + b^5*d^4*(B*d + 5*A*e))*x^8)/8 + (5*b^2*e*(3*a^4*B*e^4 + 12*a*b^3*d^2*e*(B*d + A*e) + 15*a^2*b^2*d*e^
2*(2*B*d + A*e) + 4*a^3*b*e^3*(5*B*d + A*e) + b^4*d^3*(B*d + 2*A*e))*x^9)/9 + (b^3*e^2*(4*a^3*B*e^3 + 2*b^3*d^
2*(B*d + A*e) + 6*a*b^2*d*e*(2*B*d + A*e) + 3*a^2*b*e^2*(5*B*d + A*e))*x^10)/2 + (b^4*e^3*(15*a^2*B*e^2 + 5*b^
2*d*(2*B*d + A*e) + 6*a*b*e*(5*B*d + A*e))*x^11)/11 + (b^5*e^4*(5*b*B*d + A*b*e + 6*a*B*e)*x^12)/12 + (b^6*B*e
^5*x^13)/13

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(996\) vs. \(2(228)=456\).
time = 0.07, size = 997, normalized size = 4.15 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b^6*B*e^5*x^13+1/12*((A*b^6+6*B*a*b^5)*e^5+5*b^6*B*d*e^4)*x^12+1/11*((6*A*a*b^5+15*B*a^2*b^4)*e^5+5*(A*b^
6+6*B*a*b^5)*d*e^4+10*b^6*B*d^2*e^3)*x^11+1/10*((15*A*a^2*b^4+20*B*a^3*b^3)*e^5+5*(6*A*a*b^5+15*B*a^2*b^4)*d*e
^4+10*(A*b^6+6*B*a*b^5)*d^2*e^3+10*b^6*B*d^3*e^2)*x^10+1/9*((20*A*a^3*b^3+15*B*a^4*b^2)*e^5+5*(15*A*a^2*b^4+20
*B*a^3*b^3)*d*e^4+10*(6*A*a*b^5+15*B*a^2*b^4)*d^2*e^3+10*(A*b^6+6*B*a*b^5)*d^3*e^2+5*b^6*B*d^4*e)*x^9+1/8*((15
*A*a^4*b^2+6*B*a^5*b)*e^5+5*(20*A*a^3*b^3+15*B*a^4*b^2)*d*e^4+10*(15*A*a^2*b^4+20*B*a^3*b^3)*d^2*e^3+10*(6*A*a
*b^5+15*B*a^2*b^4)*d^3*e^2+5*(A*b^6+6*B*a*b^5)*d^4*e+b^6*B*d^5)*x^8+1/7*((6*A*a^5*b+B*a^6)*e^5+5*(15*A*a^4*b^2
+6*B*a^5*b)*d*e^4+10*(20*A*a^3*b^3+15*B*a^4*b^2)*d^2*e^3+10*(15*A*a^2*b^4+20*B*a^3*b^3)*d^3*e^2+5*(6*A*a*b^5+1
5*B*a^2*b^4)*d^4*e+(A*b^6+6*B*a*b^5)*d^5)*x^7+1/6*(a^6*A*e^5+5*(6*A*a^5*b+B*a^6)*d*e^4+10*(15*A*a^4*b^2+6*B*a^
5*b)*d^2*e^3+10*(20*A*a^3*b^3+15*B*a^4*b^2)*d^3*e^2+5*(15*A*a^2*b^4+20*B*a^3*b^3)*d^4*e+(6*A*a*b^5+15*B*a^2*b^
4)*d^5)*x^6+1/5*(5*a^6*A*d*e^4+10*(6*A*a^5*b+B*a^6)*d^2*e^3+10*(15*A*a^4*b^2+6*B*a^5*b)*d^3*e^2+5*(20*A*a^3*b^
3+15*B*a^4*b^2)*d^4*e+(15*A*a^2*b^4+20*B*a^3*b^3)*d^5)*x^5+1/4*(10*a^6*A*d^2*e^3+10*(6*A*a^5*b+B*a^6)*d^3*e^2+
5*(15*A*a^4*b^2+6*B*a^5*b)*d^4*e+(20*A*a^3*b^3+15*B*a^4*b^2)*d^5)*x^4+1/3*(10*a^6*A*d^3*e^2+5*(6*A*a^5*b+B*a^6
)*d^4*e+(15*A*a^4*b^2+6*B*a^5*b)*d^5)*x^3+1/2*(5*a^6*A*d^4*e+(6*A*a^5*b+B*a^6)*d^5)*x^2+a^6*A*d^5*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1019 vs. \(2 (240) = 480\).
time = 0.28, size = 1019, normalized size = 4.25 \begin {gather*} \frac {1}{13} \, B b^{6} x^{13} e^{5} + A a^{6} d^{5} x + \frac {1}{12} \, {\left (5 \, B b^{6} d e^{4} + 6 \, B a b^{5} e^{5} + A b^{6} e^{5}\right )} x^{12} + \frac {1}{11} \, {\left (10 \, B b^{6} d^{2} e^{3} + 15 \, B a^{2} b^{4} e^{5} + 6 \, A a b^{5} e^{5} + 5 \, {\left (6 \, B a b^{5} e^{4} + A b^{6} e^{4}\right )} d\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B b^{6} d^{3} e^{2} + 4 \, B a^{3} b^{3} e^{5} + 3 \, A a^{2} b^{4} e^{5} + 2 \, {\left (6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} d^{2} + 3 \, {\left (5 \, B a^{2} b^{4} e^{4} + 2 \, A a b^{5} e^{4}\right )} d\right )} x^{10} + \frac {5}{9} \, {\left (B b^{6} d^{4} e + 3 \, B a^{4} b^{2} e^{5} + 4 \, A a^{3} b^{3} e^{5} + 2 \, {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d^{3} + 6 \, {\left (5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3}\right )} d^{2} + 5 \, {\left (4 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )} d\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{5} + 6 \, B a^{5} b e^{5} + 15 \, A a^{4} b^{2} e^{5} + 5 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{4} + 30 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d^{3} + 50 \, {\left (4 \, B a^{3} b^{3} e^{3} + 3 \, A a^{2} b^{4} e^{3}\right )} d^{2} + 25 \, {\left (3 \, B a^{4} b^{2} e^{4} + 4 \, A a^{3} b^{3} e^{4}\right )} d\right )} x^{8} + \frac {1}{7} \, {\left (B a^{6} e^{5} + 6 \, A a^{5} b e^{5} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} + 15 \, {\left (5 \, B a^{2} b^{4} e + 2 \, A a b^{5} e\right )} d^{4} + 50 \, {\left (4 \, B a^{3} b^{3} e^{2} + 3 \, A a^{2} b^{4} e^{2}\right )} d^{3} + 50 \, {\left (3 \, B a^{4} b^{2} e^{3} + 4 \, A a^{3} b^{3} e^{3}\right )} d^{2} + 15 \, {\left (2 \, B a^{5} b e^{4} + 5 \, A a^{4} b^{2} e^{4}\right )} d\right )} x^{7} + \frac {1}{6} \, {\left (A a^{6} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} + 25 \, {\left (4 \, B a^{3} b^{3} e + 3 \, A a^{2} b^{4} e\right )} d^{4} + 50 \, {\left (3 \, B a^{4} b^{2} e^{2} + 4 \, A a^{3} b^{3} e^{2}\right )} d^{3} + 30 \, {\left (2 \, B a^{5} b e^{3} + 5 \, A a^{4} b^{2} e^{3}\right )} d^{2} + 5 \, {\left (B a^{6} e^{4} + 6 \, A a^{5} b e^{4}\right )} d\right )} x^{6} + {\left (A a^{6} d e^{4} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{5} + 5 \, {\left (3 \, B a^{4} b^{2} e + 4 \, A a^{3} b^{3} e\right )} d^{4} + 6 \, {\left (2 \, B a^{5} b e^{2} + 5 \, A a^{4} b^{2} e^{2}\right )} d^{3} + 2 \, {\left (B a^{6} e^{3} + 6 \, A a^{5} b e^{3}\right )} d^{2}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, A a^{6} d^{2} e^{3} + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{5} + 3 \, {\left (2 \, B a^{5} b e + 5 \, A a^{4} b^{2} e\right )} d^{4} + 2 \, {\left (B a^{6} e^{2} + 6 \, A a^{5} b e^{2}\right )} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, A a^{6} d^{3} e^{2} + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{5} + 5 \, {\left (B a^{6} e + 6 \, A a^{5} b e\right )} d^{4}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{6} d^{4} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{5}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*B*b^6*x^13*e^5 + A*a^6*d^5*x + 1/12*(5*B*b^6*d*e^4 + 6*B*a*b^5*e^5 + A*b^6*e^5)*x^12 + 1/11*(10*B*b^6*d^2
*e^3 + 15*B*a^2*b^4*e^5 + 6*A*a*b^5*e^5 + 5*(6*B*a*b^5*e^4 + A*b^6*e^4)*d)*x^11 + 1/2*(2*B*b^6*d^3*e^2 + 4*B*a
^3*b^3*e^5 + 3*A*a^2*b^4*e^5 + 2*(6*B*a*b^5*e^3 + A*b^6*e^3)*d^2 + 3*(5*B*a^2*b^4*e^4 + 2*A*a*b^5*e^4)*d)*x^10
+ 5/9*(B*b^6*d^4*e + 3*B*a^4*b^2*e^5 + 4*A*a^3*b^3*e^5 + 2*(6*B*a*b^5*e^2 + A*b^6*e^2)*d^3 + 6*(5*B*a^2*b^4*e
^3 + 2*A*a*b^5*e^3)*d^2 + 5*(4*B*a^3*b^3*e^4 + 3*A*a^2*b^4*e^4)*d)*x^9 + 1/8*(B*b^6*d^5 + 6*B*a^5*b*e^5 + 15*A
*a^4*b^2*e^5 + 5*(6*B*a*b^5*e + A*b^6*e)*d^4 + 30*(5*B*a^2*b^4*e^2 + 2*A*a*b^5*e^2)*d^3 + 50*(4*B*a^3*b^3*e^3
+ 3*A*a^2*b^4*e^3)*d^2 + 25*(3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*d)*x^8 + 1/7*(B*a^6*e^5 + 6*A*a^5*b*e^5 + (6*B
*a*b^5 + A*b^6)*d^5 + 15*(5*B*a^2*b^4*e + 2*A*a*b^5*e)*d^4 + 50*(4*B*a^3*b^3*e^2 + 3*A*a^2*b^4*e^2)*d^3 + 50*(
3*B*a^4*b^2*e^3 + 4*A*a^3*b^3*e^3)*d^2 + 15*(2*B*a^5*b*e^4 + 5*A*a^4*b^2*e^4)*d)*x^7 + 1/6*(A*a^6*e^5 + 3*(5*B
*a^2*b^4 + 2*A*a*b^5)*d^5 + 25*(4*B*a^3*b^3*e + 3*A*a^2*b^4*e)*d^4 + 50*(3*B*a^4*b^2*e^2 + 4*A*a^3*b^3*e^2)*d^
3 + 30*(2*B*a^5*b*e^3 + 5*A*a^4*b^2*e^3)*d^2 + 5*(B*a^6*e^4 + 6*A*a^5*b*e^4)*d)*x^6 + (A*a^6*d*e^4 + (4*B*a^3*
b^3 + 3*A*a^2*b^4)*d^5 + 5*(3*B*a^4*b^2*e + 4*A*a^3*b^3*e)*d^4 + 6*(2*B*a^5*b*e^2 + 5*A*a^4*b^2*e^2)*d^3 + 2*(
B*a^6*e^3 + 6*A*a^5*b*e^3)*d^2)*x^5 + 5/4*(2*A*a^6*d^2*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5 + 3*(2*B*a^5*b*e
+ 5*A*a^4*b^2*e)*d^4 + 2*(B*a^6*e^2 + 6*A*a^5*b*e^2)*d^3)*x^4 + 1/3*(10*A*a^6*d^3*e^2 + 3*(2*B*a^5*b + 5*A*a^4
*b^2)*d^5 + 5*(B*a^6*e + 6*A*a^5*b*e)*d^4)*x^3 + 1/2*(5*A*a^6*d^4*e + (B*a^6 + 6*A*a^5*b)*d^5)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (240) = 480\).
time = 0.95, size = 1001, normalized size = 4.17 \begin {gather*} \frac {1}{8} \, B b^{6} d^{5} x^{8} + A a^{6} d^{5} x + \frac {1}{7} \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} x^{7} + \frac {1}{2} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} x^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{5} x^{5} + \frac {5}{4} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{5} x^{4} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{5} x^{3} + \frac {1}{2} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{5} x^{2} + \frac {1}{72072} \, {\left (5544 \, B b^{6} x^{13} + 12012 \, A a^{6} x^{6} + 6006 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{12} + 19656 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{11} + 36036 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{10} + 40040 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{9} + 27027 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{8} + 10296 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{7}\right )} e^{5} + \frac {1}{5544} \, {\left (2310 \, B b^{6} d x^{12} + 5544 \, A a^{6} d x^{5} + 2520 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{11} + 8316 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{10} + 15400 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{9} + 17325 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{8} + 11880 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x^{7} + 4620 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d x^{6}\right )} e^{4} + \frac {1}{924} \, {\left (840 \, B b^{6} d^{2} x^{11} + 2310 \, A a^{6} d^{2} x^{4} + 924 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{10} + 3080 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{9} + 5775 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{8} + 6600 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x^{7} + 4620 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} x^{6} + 1848 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} x^{5}\right )} e^{3} + \frac {1}{252} \, {\left (252 \, B b^{6} d^{3} x^{10} + 840 \, A a^{6} d^{3} x^{3} + 280 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{9} + 945 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{8} + 1800 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x^{7} + 2100 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} x^{6} + 1512 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} x^{5} + 630 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{3} x^{4}\right )} e^{2} + \frac {5}{504} \, {\left (56 \, B b^{6} d^{4} x^{9} + 252 \, A a^{6} d^{4} x^{2} + 63 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} x^{8} + 216 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} x^{7} + 420 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} x^{6} + 504 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{4} x^{5} + 378 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{4} x^{4} + 168 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{4} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*B*b^6*d^5*x^8 + A*a^6*d^5*x + 1/7*(6*B*a*b^5 + A*b^6)*d^5*x^7 + 1/2*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*x^6 + (4
*B*a^3*b^3 + 3*A*a^2*b^4)*d^5*x^5 + 5/4*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^5*x^4 + (2*B*a^5*b + 5*A*a^4*b^2)*d^5*x^
3 + 1/2*(B*a^6 + 6*A*a^5*b)*d^5*x^2 + 1/72072*(5544*B*b^6*x^13 + 12012*A*a^6*x^6 + 6006*(6*B*a*b^5 + A*b^6)*x^
12 + 19656*(5*B*a^2*b^4 + 2*A*a*b^5)*x^11 + 36036*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^10 + 40040*(3*B*a^4*b^2 + 4*A*
a^3*b^3)*x^9 + 27027*(2*B*a^5*b + 5*A*a^4*b^2)*x^8 + 10296*(B*a^6 + 6*A*a^5*b)*x^7)*e^5 + 1/5544*(2310*B*b^6*d
*x^12 + 5544*A*a^6*d*x^5 + 2520*(6*B*a*b^5 + A*b^6)*d*x^11 + 8316*(5*B*a^2*b^4 + 2*A*a*b^5)*d*x^10 + 15400*(4*
B*a^3*b^3 + 3*A*a^2*b^4)*d*x^9 + 17325*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*x^8 + 11880*(2*B*a^5*b + 5*A*a^4*b^2)*d*x
^7 + 4620*(B*a^6 + 6*A*a^5*b)*d*x^6)*e^4 + 1/924*(840*B*b^6*d^2*x^11 + 2310*A*a^6*d^2*x^4 + 924*(6*B*a*b^5 + A
*b^6)*d^2*x^10 + 3080*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*x^9 + 5775*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*x^8 + 6600*(3*B
*a^4*b^2 + 4*A*a^3*b^3)*d^2*x^7 + 4620*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*x^6 + 1848*(B*a^6 + 6*A*a^5*b)*d^2*x^5)*e
^3 + 1/252*(252*B*b^6*d^3*x^10 + 840*A*a^6*d^3*x^3 + 280*(6*B*a*b^5 + A*b^6)*d^3*x^9 + 945*(5*B*a^2*b^4 + 2*A*
a*b^5)*d^3*x^8 + 1800*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*x^7 + 2100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*x^6 + 1512*(2
*B*a^5*b + 5*A*a^4*b^2)*d^3*x^5 + 630*(B*a^6 + 6*A*a^5*b)*d^3*x^4)*e^2 + 5/504*(56*B*b^6*d^4*x^9 + 252*A*a^6*d
^4*x^2 + 63*(6*B*a*b^5 + A*b^6)*d^4*x^8 + 216*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*x^7 + 420*(4*B*a^3*b^3 + 3*A*a^2*b
^4)*d^4*x^6 + 504*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^4*x^5 + 378*(2*B*a^5*b + 5*A*a^4*b^2)*d^4*x^4 + 168*(B*a^6 + 6
*A*a^5*b)*d^4*x^3)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1278 vs. \(2 (241) = 482\).
time = 0.08, size = 1278, normalized size = 5.32 \begin {gather*} A a^{6} d^{5} x + \frac {B b^{6} e^{5} x^{13}}{13} + x^{12} \left (\frac {A b^{6} e^{5}}{12} + \frac {B a b^{5} e^{5}}{2} + \frac {5 B b^{6} d e^{4}}{12}\right ) + x^{11} \cdot \left (\frac {6 A a b^{5} e^{5}}{11} + \frac {5 A b^{6} d e^{4}}{11} + \frac {15 B a^{2} b^{4} e^{5}}{11} + \frac {30 B a b^{5} d e^{4}}{11} + \frac {10 B b^{6} d^{2} e^{3}}{11}\right ) + x^{10} \cdot \left (\frac {3 A a^{2} b^{4} e^{5}}{2} + 3 A a b^{5} d e^{4} + A b^{6} d^{2} e^{3} + 2 B a^{3} b^{3} e^{5} + \frac {15 B a^{2} b^{4} d e^{4}}{2} + 6 B a b^{5} d^{2} e^{3} + B b^{6} d^{3} e^{2}\right ) + x^{9} \cdot \left (\frac {20 A a^{3} b^{3} e^{5}}{9} + \frac {25 A a^{2} b^{4} d e^{4}}{3} + \frac {20 A a b^{5} d^{2} e^{3}}{3} + \frac {10 A b^{6} d^{3} e^{2}}{9} + \frac {5 B a^{4} b^{2} e^{5}}{3} + \frac {100 B a^{3} b^{3} d e^{4}}{9} + \frac {50 B a^{2} b^{4} d^{2} e^{3}}{3} + \frac {20 B a b^{5} d^{3} e^{2}}{3} + \frac {5 B b^{6} d^{4} e}{9}\right ) + x^{8} \cdot \left (\frac {15 A a^{4} b^{2} e^{5}}{8} + \frac {25 A a^{3} b^{3} d e^{4}}{2} + \frac {75 A a^{2} b^{4} d^{2} e^{3}}{4} + \frac {15 A a b^{5} d^{3} e^{2}}{2} + \frac {5 A b^{6} d^{4} e}{8} + \frac {3 B a^{5} b e^{5}}{4} + \frac {75 B a^{4} b^{2} d e^{4}}{8} + 25 B a^{3} b^{3} d^{2} e^{3} + \frac {75 B a^{2} b^{4} d^{3} e^{2}}{4} + \frac {15 B a b^{5} d^{4} e}{4} + \frac {B b^{6} d^{5}}{8}\right ) + x^{7} \cdot \left (\frac {6 A a^{5} b e^{5}}{7} + \frac {75 A a^{4} b^{2} d e^{4}}{7} + \frac {200 A a^{3} b^{3} d^{2} e^{3}}{7} + \frac {150 A a^{2} b^{4} d^{3} e^{2}}{7} + \frac {30 A a b^{5} d^{4} e}{7} + \frac {A b^{6} d^{5}}{7} + \frac {B a^{6} e^{5}}{7} + \frac {30 B a^{5} b d e^{4}}{7} + \frac {150 B a^{4} b^{2} d^{2} e^{3}}{7} + \frac {200 B a^{3} b^{3} d^{3} e^{2}}{7} + \frac {75 B a^{2} b^{4} d^{4} e}{7} + \frac {6 B a b^{5} d^{5}}{7}\right ) + x^{6} \left (\frac {A a^{6} e^{5}}{6} + 5 A a^{5} b d e^{4} + 25 A a^{4} b^{2} d^{2} e^{3} + \frac {100 A a^{3} b^{3} d^{3} e^{2}}{3} + \frac {25 A a^{2} b^{4} d^{4} e}{2} + A a b^{5} d^{5} + \frac {5 B a^{6} d e^{4}}{6} + 10 B a^{5} b d^{2} e^{3} + 25 B a^{4} b^{2} d^{3} e^{2} + \frac {50 B a^{3} b^{3} d^{4} e}{3} + \frac {5 B a^{2} b^{4} d^{5}}{2}\right ) + x^{5} \left (A a^{6} d e^{4} + 12 A a^{5} b d^{2} e^{3} + 30 A a^{4} b^{2} d^{3} e^{2} + 20 A a^{3} b^{3} d^{4} e + 3 A a^{2} b^{4} d^{5} + 2 B a^{6} d^{2} e^{3} + 12 B a^{5} b d^{3} e^{2} + 15 B a^{4} b^{2} d^{4} e + 4 B a^{3} b^{3} d^{5}\right ) + x^{4} \cdot \left (\frac {5 A a^{6} d^{2} e^{3}}{2} + 15 A a^{5} b d^{3} e^{2} + \frac {75 A a^{4} b^{2} d^{4} e}{4} + 5 A a^{3} b^{3} d^{5} + \frac {5 B a^{6} d^{3} e^{2}}{2} + \frac {15 B a^{5} b d^{4} e}{2} + \frac {15 B a^{4} b^{2} d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {10 A a^{6} d^{3} e^{2}}{3} + 10 A a^{5} b d^{4} e + 5 A a^{4} b^{2} d^{5} + \frac {5 B a^{6} d^{4} e}{3} + 2 B a^{5} b d^{5}\right ) + x^{2} \cdot \left (\frac {5 A a^{6} d^{4} e}{2} + 3 A a^{5} b d^{5} + \frac {B a^{6} d^{5}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**6*d**5*x + B*b**6*e**5*x**13/13 + x**12*(A*b**6*e**5/12 + B*a*b**5*e**5/2 + 5*B*b**6*d*e**4/12) + x**11*(
6*A*a*b**5*e**5/11 + 5*A*b**6*d*e**4/11 + 15*B*a**2*b**4*e**5/11 + 30*B*a*b**5*d*e**4/11 + 10*B*b**6*d**2*e**3
/11) + x**10*(3*A*a**2*b**4*e**5/2 + 3*A*a*b**5*d*e**4 + A*b**6*d**2*e**3 + 2*B*a**3*b**3*e**5 + 15*B*a**2*b**
4*d*e**4/2 + 6*B*a*b**5*d**2*e**3 + B*b**6*d**3*e**2) + x**9*(20*A*a**3*b**3*e**5/9 + 25*A*a**2*b**4*d*e**4/3
+ 20*A*a*b**5*d**2*e**3/3 + 10*A*b**6*d**3*e**2/9 + 5*B*a**4*b**2*e**5/3 + 100*B*a**3*b**3*d*e**4/9 + 50*B*a**
2*b**4*d**2*e**3/3 + 20*B*a*b**5*d**3*e**2/3 + 5*B*b**6*d**4*e/9) + x**8*(15*A*a**4*b**2*e**5/8 + 25*A*a**3*b*
*3*d*e**4/2 + 75*A*a**2*b**4*d**2*e**3/4 + 15*A*a*b**5*d**3*e**2/2 + 5*A*b**6*d**4*e/8 + 3*B*a**5*b*e**5/4 + 7
5*B*a**4*b**2*d*e**4/8 + 25*B*a**3*b**3*d**2*e**3 + 75*B*a**2*b**4*d**3*e**2/4 + 15*B*a*b**5*d**4*e/4 + B*b**6
*d**5/8) + x**7*(6*A*a**5*b*e**5/7 + 75*A*a**4*b**2*d*e**4/7 + 200*A*a**3*b**3*d**2*e**3/7 + 150*A*a**2*b**4*d
**3*e**2/7 + 30*A*a*b**5*d**4*e/7 + A*b**6*d**5/7 + B*a**6*e**5/7 + 30*B*a**5*b*d*e**4/7 + 150*B*a**4*b**2*d**
2*e**3/7 + 200*B*a**3*b**3*d**3*e**2/7 + 75*B*a**2*b**4*d**4*e/7 + 6*B*a*b**5*d**5/7) + x**6*(A*a**6*e**5/6 +
5*A*a**5*b*d*e**4 + 25*A*a**4*b**2*d**2*e**3 + 100*A*a**3*b**3*d**3*e**2/3 + 25*A*a**2*b**4*d**4*e/2 + A*a*b**
5*d**5 + 5*B*a**6*d*e**4/6 + 10*B*a**5*b*d**2*e**3 + 25*B*a**4*b**2*d**3*e**2 + 50*B*a**3*b**3*d**4*e/3 + 5*B*
a**2*b**4*d**5/2) + x**5*(A*a**6*d*e**4 + 12*A*a**5*b*d**2*e**3 + 30*A*a**4*b**2*d**3*e**2 + 20*A*a**3*b**3*d*
*4*e + 3*A*a**2*b**4*d**5 + 2*B*a**6*d**2*e**3 + 12*B*a**5*b*d**3*e**2 + 15*B*a**4*b**2*d**4*e + 4*B*a**3*b**3
*d**5) + x**4*(5*A*a**6*d**2*e**3/2 + 15*A*a**5*b*d**3*e**2 + 75*A*a**4*b**2*d**4*e/4 + 5*A*a**3*b**3*d**5 + 5
*B*a**6*d**3*e**2/2 + 15*B*a**5*b*d**4*e/2 + 15*B*a**4*b**2*d**5/4) + x**3*(10*A*a**6*d**3*e**2/3 + 10*A*a**5*
b*d**4*e + 5*A*a**4*b**2*d**5 + 5*B*a**6*d**4*e/3 + 2*B*a**5*b*d**5) + x**2*(5*A*a**6*d**4*e/2 + 3*A*a**5*b*d*
*5 + B*a**6*d**5/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1204 vs. \(2 (240) = 480\).
time = 1.66, size = 1204, normalized size = 5.02 \begin {gather*} \frac {1}{13} \, B b^{6} x^{13} e^{5} + \frac {5}{12} \, B b^{6} d x^{12} e^{4} + \frac {10}{11} \, B b^{6} d^{2} x^{11} e^{3} + B b^{6} d^{3} x^{10} e^{2} + \frac {5}{9} \, B b^{6} d^{4} x^{9} e + \frac {1}{8} \, B b^{6} d^{5} x^{8} + \frac {1}{2} \, B a b^{5} x^{12} e^{5} + \frac {1}{12} \, A b^{6} x^{12} e^{5} + \frac {30}{11} \, B a b^{5} d x^{11} e^{4} + \frac {5}{11} \, A b^{6} d x^{11} e^{4} + 6 \, B a b^{5} d^{2} x^{10} e^{3} + A b^{6} d^{2} x^{10} e^{3} + \frac {20}{3} \, B a b^{5} d^{3} x^{9} e^{2} + \frac {10}{9} \, A b^{6} d^{3} x^{9} e^{2} + \frac {15}{4} \, B a b^{5} d^{4} x^{8} e + \frac {5}{8} \, A b^{6} d^{4} x^{8} e + \frac {6}{7} \, B a b^{5} d^{5} x^{7} + \frac {1}{7} \, A b^{6} d^{5} x^{7} + \frac {15}{11} \, B a^{2} b^{4} x^{11} e^{5} + \frac {6}{11} \, A a b^{5} x^{11} e^{5} + \frac {15}{2} \, B a^{2} b^{4} d x^{10} e^{4} + 3 \, A a b^{5} d x^{10} e^{4} + \frac {50}{3} \, B a^{2} b^{4} d^{2} x^{9} e^{3} + \frac {20}{3} \, A a b^{5} d^{2} x^{9} e^{3} + \frac {75}{4} \, B a^{2} b^{4} d^{3} x^{8} e^{2} + \frac {15}{2} \, A a b^{5} d^{3} x^{8} e^{2} + \frac {75}{7} \, B a^{2} b^{4} d^{4} x^{7} e + \frac {30}{7} \, A a b^{5} d^{4} x^{7} e + \frac {5}{2} \, B a^{2} b^{4} d^{5} x^{6} + A a b^{5} d^{5} x^{6} + 2 \, B a^{3} b^{3} x^{10} e^{5} + \frac {3}{2} \, A a^{2} b^{4} x^{10} e^{5} + \frac {100}{9} \, B a^{3} b^{3} d x^{9} e^{4} + \frac {25}{3} \, A a^{2} b^{4} d x^{9} e^{4} + 25 \, B a^{3} b^{3} d^{2} x^{8} e^{3} + \frac {75}{4} \, A a^{2} b^{4} d^{2} x^{8} e^{3} + \frac {200}{7} \, B a^{3} b^{3} d^{3} x^{7} e^{2} + \frac {150}{7} \, A a^{2} b^{4} d^{3} x^{7} e^{2} + \frac {50}{3} \, B a^{3} b^{3} d^{4} x^{6} e + \frac {25}{2} \, A a^{2} b^{4} d^{4} x^{6} e + 4 \, B a^{3} b^{3} d^{5} x^{5} + 3 \, A a^{2} b^{4} d^{5} x^{5} + \frac {5}{3} \, B a^{4} b^{2} x^{9} e^{5} + \frac {20}{9} \, A a^{3} b^{3} x^{9} e^{5} + \frac {75}{8} \, B a^{4} b^{2} d x^{8} e^{4} + \frac {25}{2} \, A a^{3} b^{3} d x^{8} e^{4} + \frac {150}{7} \, B a^{4} b^{2} d^{2} x^{7} e^{3} + \frac {200}{7} \, A a^{3} b^{3} d^{2} x^{7} e^{3} + 25 \, B a^{4} b^{2} d^{3} x^{6} e^{2} + \frac {100}{3} \, A a^{3} b^{3} d^{3} x^{6} e^{2} + 15 \, B a^{4} b^{2} d^{4} x^{5} e + 20 \, A a^{3} b^{3} d^{4} x^{5} e + \frac {15}{4} \, B a^{4} b^{2} d^{5} x^{4} + 5 \, A a^{3} b^{3} d^{5} x^{4} + \frac {3}{4} \, B a^{5} b x^{8} e^{5} + \frac {15}{8} \, A a^{4} b^{2} x^{8} e^{5} + \frac {30}{7} \, B a^{5} b d x^{7} e^{4} + \frac {75}{7} \, A a^{4} b^{2} d x^{7} e^{4} + 10 \, B a^{5} b d^{2} x^{6} e^{3} + 25 \, A a^{4} b^{2} d^{2} x^{6} e^{3} + 12 \, B a^{5} b d^{3} x^{5} e^{2} + 30 \, A a^{4} b^{2} d^{3} x^{5} e^{2} + \frac {15}{2} \, B a^{5} b d^{4} x^{4} e + \frac {75}{4} \, A a^{4} b^{2} d^{4} x^{4} e + 2 \, B a^{5} b d^{5} x^{3} + 5 \, A a^{4} b^{2} d^{5} x^{3} + \frac {1}{7} \, B a^{6} x^{7} e^{5} + \frac {6}{7} \, A a^{5} b x^{7} e^{5} + \frac {5}{6} \, B a^{6} d x^{6} e^{4} + 5 \, A a^{5} b d x^{6} e^{4} + 2 \, B a^{6} d^{2} x^{5} e^{3} + 12 \, A a^{5} b d^{2} x^{5} e^{3} + \frac {5}{2} \, B a^{6} d^{3} x^{4} e^{2} + 15 \, A a^{5} b d^{3} x^{4} e^{2} + \frac {5}{3} \, B a^{6} d^{4} x^{3} e + 10 \, A a^{5} b d^{4} x^{3} e + \frac {1}{2} \, B a^{6} d^{5} x^{2} + 3 \, A a^{5} b d^{5} x^{2} + \frac {1}{6} \, A a^{6} x^{6} e^{5} + A a^{6} d x^{5} e^{4} + \frac {5}{2} \, A a^{6} d^{2} x^{4} e^{3} + \frac {10}{3} \, A a^{6} d^{3} x^{3} e^{2} + \frac {5}{2} \, A a^{6} d^{4} x^{2} e + A a^{6} d^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*B*b^6*x^13*e^5 + 5/12*B*b^6*d*x^12*e^4 + 10/11*B*b^6*d^2*x^11*e^3 + B*b^6*d^3*x^10*e^2 + 5/9*B*b^6*d^4*x^
9*e + 1/8*B*b^6*d^5*x^8 + 1/2*B*a*b^5*x^12*e^5 + 1/12*A*b^6*x^12*e^5 + 30/11*B*a*b^5*d*x^11*e^4 + 5/11*A*b^6*d
*x^11*e^4 + 6*B*a*b^5*d^2*x^10*e^3 + A*b^6*d^2*x^10*e^3 + 20/3*B*a*b^5*d^3*x^9*e^2 + 10/9*A*b^6*d^3*x^9*e^2 +
15/4*B*a*b^5*d^4*x^8*e + 5/8*A*b^6*d^4*x^8*e + 6/7*B*a*b^5*d^5*x^7 + 1/7*A*b^6*d^5*x^7 + 15/11*B*a^2*b^4*x^11*
e^5 + 6/11*A*a*b^5*x^11*e^5 + 15/2*B*a^2*b^4*d*x^10*e^4 + 3*A*a*b^5*d*x^10*e^4 + 50/3*B*a^2*b^4*d^2*x^9*e^3 +
20/3*A*a*b^5*d^2*x^9*e^3 + 75/4*B*a^2*b^4*d^3*x^8*e^2 + 15/2*A*a*b^5*d^3*x^8*e^2 + 75/7*B*a^2*b^4*d^4*x^7*e +
30/7*A*a*b^5*d^4*x^7*e + 5/2*B*a^2*b^4*d^5*x^6 + A*a*b^5*d^5*x^6 + 2*B*a^3*b^3*x^10*e^5 + 3/2*A*a^2*b^4*x^10*e
^5 + 100/9*B*a^3*b^3*d*x^9*e^4 + 25/3*A*a^2*b^4*d*x^9*e^4 + 25*B*a^3*b^3*d^2*x^8*e^3 + 75/4*A*a^2*b^4*d^2*x^8*
e^3 + 200/7*B*a^3*b^3*d^3*x^7*e^2 + 150/7*A*a^2*b^4*d^3*x^7*e^2 + 50/3*B*a^3*b^3*d^4*x^6*e + 25/2*A*a^2*b^4*d^
4*x^6*e + 4*B*a^3*b^3*d^5*x^5 + 3*A*a^2*b^4*d^5*x^5 + 5/3*B*a^4*b^2*x^9*e^5 + 20/9*A*a^3*b^3*x^9*e^5 + 75/8*B*
a^4*b^2*d*x^8*e^4 + 25/2*A*a^3*b^3*d*x^8*e^4 + 150/7*B*a^4*b^2*d^2*x^7*e^3 + 200/7*A*a^3*b^3*d^2*x^7*e^3 + 25*
B*a^4*b^2*d^3*x^6*e^2 + 100/3*A*a^3*b^3*d^3*x^6*e^2 + 15*B*a^4*b^2*d^4*x^5*e + 20*A*a^3*b^3*d^4*x^5*e + 15/4*B
*a^4*b^2*d^5*x^4 + 5*A*a^3*b^3*d^5*x^4 + 3/4*B*a^5*b*x^8*e^5 + 15/8*A*a^4*b^2*x^8*e^5 + 30/7*B*a^5*b*d*x^7*e^4
+ 75/7*A*a^4*b^2*d*x^7*e^4 + 10*B*a^5*b*d^2*x^6*e^3 + 25*A*a^4*b^2*d^2*x^6*e^3 + 12*B*a^5*b*d^3*x^5*e^2 + 30*
A*a^4*b^2*d^3*x^5*e^2 + 15/2*B*a^5*b*d^4*x^4*e + 75/4*A*a^4*b^2*d^4*x^4*e + 2*B*a^5*b*d^5*x^3 + 5*A*a^4*b^2*d^
5*x^3 + 1/7*B*a^6*x^7*e^5 + 6/7*A*a^5*b*x^7*e^5 + 5/6*B*a^6*d*x^6*e^4 + 5*A*a^5*b*d*x^6*e^4 + 2*B*a^6*d^2*x^5*
e^3 + 12*A*a^5*b*d^2*x^5*e^3 + 5/2*B*a^6*d^3*x^4*e^2 + 15*A*a^5*b*d^3*x^4*e^2 + 5/3*B*a^6*d^4*x^3*e + 10*A*a^5
*b*d^4*x^3*e + 1/2*B*a^6*d^5*x^2 + 3*A*a^5*b*d^5*x^2 + 1/6*A*a^6*x^6*e^5 + A*a^6*d*x^5*e^4 + 5/2*A*a^6*d^2*x^4
*e^3 + 10/3*A*a^6*d^3*x^3*e^2 + 5/2*A*a^6*d^4*x^2*e + A*a^6*d^5*x

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Mupad [B]
time = 1.35, size = 1039, normalized size = 4.33 \begin {gather*} x^7\,\left (\frac {B\,a^6\,e^5}{7}+\frac {30\,B\,a^5\,b\,d\,e^4}{7}+\frac {6\,A\,a^5\,b\,e^5}{7}+\frac {150\,B\,a^4\,b^2\,d^2\,e^3}{7}+\frac {75\,A\,a^4\,b^2\,d\,e^4}{7}+\frac {200\,B\,a^3\,b^3\,d^3\,e^2}{7}+\frac {200\,A\,a^3\,b^3\,d^2\,e^3}{7}+\frac {75\,B\,a^2\,b^4\,d^4\,e}{7}+\frac {150\,A\,a^2\,b^4\,d^3\,e^2}{7}+\frac {6\,B\,a\,b^5\,d^5}{7}+\frac {30\,A\,a\,b^5\,d^4\,e}{7}+\frac {A\,b^6\,d^5}{7}\right )+x^3\,\left (\frac {5\,B\,a^6\,d^4\,e}{3}+\frac {10\,A\,a^6\,d^3\,e^2}{3}+2\,B\,a^5\,b\,d^5+10\,A\,a^5\,b\,d^4\,e+5\,A\,a^4\,b^2\,d^5\right )+x^{11}\,\left (\frac {15\,B\,a^2\,b^4\,e^5}{11}+\frac {30\,B\,a\,b^5\,d\,e^4}{11}+\frac {6\,A\,a\,b^5\,e^5}{11}+\frac {10\,B\,b^6\,d^2\,e^3}{11}+\frac {5\,A\,b^6\,d\,e^4}{11}\right )+x^6\,\left (\frac {5\,B\,a^6\,d\,e^4}{6}+\frac {A\,a^6\,e^5}{6}+10\,B\,a^5\,b\,d^2\,e^3+5\,A\,a^5\,b\,d\,e^4+25\,B\,a^4\,b^2\,d^3\,e^2+25\,A\,a^4\,b^2\,d^2\,e^3+\frac {50\,B\,a^3\,b^3\,d^4\,e}{3}+\frac {100\,A\,a^3\,b^3\,d^3\,e^2}{3}+\frac {5\,B\,a^2\,b^4\,d^5}{2}+\frac {25\,A\,a^2\,b^4\,d^4\,e}{2}+A\,a\,b^5\,d^5\right )+x^8\,\left (\frac {3\,B\,a^5\,b\,e^5}{4}+\frac {75\,B\,a^4\,b^2\,d\,e^4}{8}+\frac {15\,A\,a^4\,b^2\,e^5}{8}+25\,B\,a^3\,b^3\,d^2\,e^3+\frac {25\,A\,a^3\,b^3\,d\,e^4}{2}+\frac {75\,B\,a^2\,b^4\,d^3\,e^2}{4}+\frac {75\,A\,a^2\,b^4\,d^2\,e^3}{4}+\frac {15\,B\,a\,b^5\,d^4\,e}{4}+\frac {15\,A\,a\,b^5\,d^3\,e^2}{2}+\frac {B\,b^6\,d^5}{8}+\frac {5\,A\,b^6\,d^4\,e}{8}\right )+x^5\,\left (2\,B\,a^6\,d^2\,e^3+A\,a^6\,d\,e^4+12\,B\,a^5\,b\,d^3\,e^2+12\,A\,a^5\,b\,d^2\,e^3+15\,B\,a^4\,b^2\,d^4\,e+30\,A\,a^4\,b^2\,d^3\,e^2+4\,B\,a^3\,b^3\,d^5+20\,A\,a^3\,b^3\,d^4\,e+3\,A\,a^2\,b^4\,d^5\right )+x^9\,\left (\frac {5\,B\,a^4\,b^2\,e^5}{3}+\frac {100\,B\,a^3\,b^3\,d\,e^4}{9}+\frac {20\,A\,a^3\,b^3\,e^5}{9}+\frac {50\,B\,a^2\,b^4\,d^2\,e^3}{3}+\frac {25\,A\,a^2\,b^4\,d\,e^4}{3}+\frac {20\,B\,a\,b^5\,d^3\,e^2}{3}+\frac {20\,A\,a\,b^5\,d^2\,e^3}{3}+\frac {5\,B\,b^6\,d^4\,e}{9}+\frac {10\,A\,b^6\,d^3\,e^2}{9}\right )+x^4\,\left (\frac {5\,B\,a^6\,d^3\,e^2}{2}+\frac {5\,A\,a^6\,d^2\,e^3}{2}+\frac {15\,B\,a^5\,b\,d^4\,e}{2}+15\,A\,a^5\,b\,d^3\,e^2+\frac {15\,B\,a^4\,b^2\,d^5}{4}+\frac {75\,A\,a^4\,b^2\,d^4\,e}{4}+5\,A\,a^3\,b^3\,d^5\right )+x^{10}\,\left (2\,B\,a^3\,b^3\,e^5+\frac {15\,B\,a^2\,b^4\,d\,e^4}{2}+\frac {3\,A\,a^2\,b^4\,e^5}{2}+6\,B\,a\,b^5\,d^2\,e^3+3\,A\,a\,b^5\,d\,e^4+B\,b^6\,d^3\,e^2+A\,b^6\,d^2\,e^3\right )+\frac {a^5\,d^4\,x^2\,\left (5\,A\,a\,e+6\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^5\,e^4\,x^{12}\,\left (A\,b\,e+6\,B\,a\,e+5\,B\,b\,d\right )}{12}+A\,a^6\,d^5\,x+\frac {B\,b^6\,e^5\,x^{13}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^7*((A*b^6*d^5)/7 + (B*a^6*e^5)/7 + (6*A*a^5*b*e^5)/7 + (6*B*a*b^5*d^5)/7 + (75*A*a^4*b^2*d*e^4)/7 + (75*B*a^
2*b^4*d^4*e)/7 + (150*A*a^2*b^4*d^3*e^2)/7 + (200*A*a^3*b^3*d^2*e^3)/7 + (200*B*a^3*b^3*d^3*e^2)/7 + (150*B*a^
4*b^2*d^2*e^3)/7 + (30*A*a*b^5*d^4*e)/7 + (30*B*a^5*b*d*e^4)/7) + x^3*(2*B*a^5*b*d^5 + (5*B*a^6*d^4*e)/3 + 5*A
*a^4*b^2*d^5 + (10*A*a^6*d^3*e^2)/3 + 10*A*a^5*b*d^4*e) + x^11*((6*A*a*b^5*e^5)/11 + (5*A*b^6*d*e^4)/11 + (15*
B*a^2*b^4*e^5)/11 + (10*B*b^6*d^2*e^3)/11 + (30*B*a*b^5*d*e^4)/11) + x^6*((A*a^6*e^5)/6 + A*a*b^5*d^5 + (5*B*a
^6*d*e^4)/6 + (5*B*a^2*b^4*d^5)/2 + (25*A*a^2*b^4*d^4*e)/2 + (50*B*a^3*b^3*d^4*e)/3 + 10*B*a^5*b*d^2*e^3 + (10
0*A*a^3*b^3*d^3*e^2)/3 + 25*A*a^4*b^2*d^2*e^3 + 25*B*a^4*b^2*d^3*e^2 + 5*A*a^5*b*d*e^4) + x^8*((B*b^6*d^5)/8 +
(3*B*a^5*b*e^5)/4 + (5*A*b^6*d^4*e)/8 + (15*A*a^4*b^2*e^5)/8 + (15*A*a*b^5*d^3*e^2)/2 + (25*A*a^3*b^3*d*e^4)/
2 + (75*B*a^4*b^2*d*e^4)/8 + (75*A*a^2*b^4*d^2*e^3)/4 + (75*B*a^2*b^4*d^3*e^2)/4 + 25*B*a^3*b^3*d^2*e^3 + (15*
B*a*b^5*d^4*e)/4) + x^5*(A*a^6*d*e^4 + 3*A*a^2*b^4*d^5 + 4*B*a^3*b^3*d^5 + 2*B*a^6*d^2*e^3 + 20*A*a^3*b^3*d^4*
e + 12*A*a^5*b*d^2*e^3 + 15*B*a^4*b^2*d^4*e + 12*B*a^5*b*d^3*e^2 + 30*A*a^4*b^2*d^3*e^2) + x^9*((5*B*b^6*d^4*e
)/9 + (20*A*a^3*b^3*e^5)/9 + (5*B*a^4*b^2*e^5)/3 + (10*A*b^6*d^3*e^2)/9 + (20*A*a*b^5*d^2*e^3)/3 + (25*A*a^2*b
^4*d*e^4)/3 + (20*B*a*b^5*d^3*e^2)/3 + (100*B*a^3*b^3*d*e^4)/9 + (50*B*a^2*b^4*d^2*e^3)/3) + x^4*(5*A*a^3*b^3*
d^5 + (15*B*a^4*b^2*d^5)/4 + (5*A*a^6*d^2*e^3)/2 + (5*B*a^6*d^3*e^2)/2 + (75*A*a^4*b^2*d^4*e)/4 + 15*A*a^5*b*d
^3*e^2 + (15*B*a^5*b*d^4*e)/2) + x^10*((3*A*a^2*b^4*e^5)/2 + 2*B*a^3*b^3*e^5 + A*b^6*d^2*e^3 + B*b^6*d^3*e^2 +
6*B*a*b^5*d^2*e^3 + (15*B*a^2*b^4*d*e^4)/2 + 3*A*a*b^5*d*e^4) + (a^5*d^4*x^2*(5*A*a*e + 6*A*b*d + B*a*d))/2 +
(b^5*e^4*x^12*(A*b*e + 6*B*a*e + 5*B*b*d))/12 + A*a^6*d^5*x + (B*b^6*e^5*x^13)/13

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