3.11.56 \(\int (a+b x)^6 (A+B x) (d+e x)^3 \, dx\) [1056]

Optimal. Leaf size=159 \[ \frac {(A b-a B) (b d-a e)^3 (a+b x)^7}{7 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^8}{8 b^5}+\frac {e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^9}{3 b^5}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^{10}}{10 b^5}+\frac {B e^3 (a+b x)^{11}}{11 b^5} \]

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 159, 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^2 (a+b x)^{10} (-4 a B e+A b e+3 b B d)}{10 b^5}+\frac {e (a+b x)^9 (b d-a e) (-2 a B e+A b e+b B d)}{3 b^5}+\frac {(a+b x)^8 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{8 b^5}+\frac {(a+b x)^7 (A b-a B) (b d-a e)^3}{7 b^5}+\frac {B e^3 (a+b x)^{11}}{11 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(586\) vs. \(2(159)=318\).
time = 0.13, size = 586, normalized size = 3.69 \begin {gather*} a^6 A d^3 x+\frac {1}{2} a^5 d^2 (6 A b d+a B d+3 a A e) x^2+a^4 d \left (a B d (2 b d+a e)+A \left (5 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x^3+\frac {1}{4} a^3 \left (3 a B d \left (5 b^2 d^2+6 a b d e+a^2 e^2\right )+A \left (20 b^3 d^3+45 a b^2 d^2 e+18 a^2 b d e^2+a^3 e^3\right )\right ) x^4+\frac {1}{5} a^2 \left (a B \left (20 b^3 d^3+45 a b^2 d^2 e+18 a^2 b d e^2+a^3 e^3\right )+3 A b \left (5 b^3 d^3+20 a b^2 d^2 e+15 a^2 b d e^2+2 a^3 e^3\right )\right ) x^5+\frac {1}{2} a b \left (a B \left (5 b^3 d^3+20 a b^2 d^2 e+15 a^2 b d e^2+2 a^3 e^3\right )+A b \left (2 b^3 d^3+15 a b^2 d^2 e+20 a^2 b d e^2+5 a^3 e^3\right )\right ) x^6+\frac {1}{7} b^2 \left (3 a B \left (2 b^3 d^3+15 a b^2 d^2 e+20 a^2 b d e^2+5 a^3 e^3\right )+A b \left (b^3 d^3+18 a b^2 d^2 e+45 a^2 b d e^2+20 a^3 e^3\right )\right ) x^7+\frac {1}{8} b^3 \left (20 a^3 B e^3+18 a b^2 d e (B d+A e)+15 a^2 b e^2 (3 B d+A e)+b^3 d^2 (B d+3 A e)\right ) x^8+\frac {1}{3} b^4 e \left (5 a^2 B e^2+b^2 d (B d+A e)+2 a b e (3 B d+A e)\right ) x^9+\frac {1}{10} b^5 e^2 (3 b B d+A b e+6 a B e) x^{10}+\frac {1}{11} b^6 B e^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^6*A*d^3*x + (a^5*d^2*(6*A*b*d + a*B*d + 3*a*A*e)*x^2)/2 + a^4*d*(a*B*d*(2*b*d + a*e) + A*(5*b^2*d^2 + 6*a*b*
d*e + a^2*e^2))*x^3 + (a^3*(3*a*B*d*(5*b^2*d^2 + 6*a*b*d*e + a^2*e^2) + A*(20*b^3*d^3 + 45*a*b^2*d^2*e + 18*a^
2*b*d*e^2 + a^3*e^3))*x^4)/4 + (a^2*(a*B*(20*b^3*d^3 + 45*a*b^2*d^2*e + 18*a^2*b*d*e^2 + a^3*e^3) + 3*A*b*(5*b
^3*d^3 + 20*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 2*a^3*e^3))*x^5)/5 + (a*b*(a*B*(5*b^3*d^3 + 20*a*b^2*d^2*e + 15*a^2
*b*d*e^2 + 2*a^3*e^3) + A*b*(2*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d*e^2 + 5*a^3*e^3))*x^6)/2 + (b^2*(3*a*B*(2
*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d*e^2 + 5*a^3*e^3) + A*b*(b^3*d^3 + 18*a*b^2*d^2*e + 45*a^2*b*d*e^2 + 20*
a^3*e^3))*x^7)/7 + (b^3*(20*a^3*B*e^3 + 18*a*b^2*d*e*(B*d + A*e) + 15*a^2*b*e^2*(3*B*d + A*e) + b^3*d^2*(B*d +
3*A*e))*x^8)/8 + (b^4*e*(5*a^2*B*e^2 + b^2*d*(B*d + A*e) + 2*a*b*e*(3*B*d + A*e))*x^9)/3 + (b^5*e^2*(3*b*B*d
+ A*b*e + 6*a*B*e)*x^10)/10 + (b^6*B*e^3*x^11)/11

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(644\) vs. \(2(149)=298\).
time = 0.07, size = 645, normalized size = 4.06 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)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 657 vs. \(2 (157) = 314\).
time = 0.30, size = 657, normalized size = 4.13 \begin {gather*} \frac {1}{11} \, B b^{6} x^{11} e^{3} + A a^{6} d^{3} x + \frac {1}{10} \, {\left (3 \, B b^{6} d e^{2} + 6 \, B a b^{5} e^{3} + A b^{6} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (B b^{6} d^{2} e + 5 \, B a^{2} b^{4} e^{3} + 2 \, A a b^{5} e^{3} + {\left (6 \, B a b^{5} e^{2} + A b^{6} e^{2}\right )} d\right )} x^{9} + \frac {1}{8} \, {\left (B b^{6} d^{3} + 20 \, B a^{3} b^{3} e^{3} + 15 \, A a^{2} b^{4} e^{3} + 3 \, {\left (6 \, B a b^{5} e + A b^{6} e\right )} d^{2} + 9 \, {\left (5 \, B a^{2} b^{4} e^{2} + 2 \, A a b^{5} e^{2}\right )} d\right )} x^{8} + \frac {1}{7} \, {\left (15 \, B a^{4} b^{2} e^{3} + 20 \, A a^{3} b^{3} e^{3} + {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} + 9 \, {\left (5 \, B a^{2} b^{4} e + 2 \, A a b^{5} e\right )} d^{2} + 15 \, {\left (4 \, B a^{3} b^{3} e^{2} + 3 \, A a^{2} b^{4} e^{2}\right )} d\right )} x^{7} + \frac {1}{2} \, {\left (2 \, B a^{5} b e^{3} + 5 \, A a^{4} b^{2} e^{3} + {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} + 5 \, {\left (4 \, B a^{3} b^{3} e + 3 \, A a^{2} b^{4} e\right )} d^{2} + 5 \, {\left (3 \, B a^{4} b^{2} e^{2} + 4 \, A a^{3} b^{3} e^{2}\right )} d\right )} x^{6} + \frac {1}{5} \, {\left (B a^{6} e^{3} + 6 \, A a^{5} b e^{3} + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} + 15 \, {\left (3 \, B a^{4} b^{2} e + 4 \, A a^{3} b^{3} e\right )} d^{2} + 9 \, {\left (2 \, B a^{5} b e^{2} + 5 \, A a^{4} b^{2} e^{2}\right )} d\right )} x^{5} + \frac {1}{4} \, {\left (A a^{6} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} + 9 \, {\left (2 \, B a^{5} b e + 5 \, A a^{4} b^{2} e\right )} d^{2} + 3 \, {\left (B a^{6} e^{2} + 6 \, A a^{5} b e^{2}\right )} d\right )} x^{4} + {\left (A a^{6} d e^{2} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} + {\left (B a^{6} e + 6 \, A a^{5} b e\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, A a^{6} d^{2} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{3}\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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (157) = 314\).
time = 0.84, size = 651, normalized size = 4.09 \begin {gather*} \frac {1}{8} \, B b^{6} d^{3} x^{8} + A a^{6} d^{3} x + \frac {1}{7} \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} x^{7} + \frac {1}{2} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} x^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} x^{5} + \frac {5}{4} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} x^{4} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{3} x^{3} + \frac {1}{2} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{3} x^{2} + \frac {1}{9240} \, {\left (840 \, B b^{6} x^{11} + 2310 \, A a^{6} x^{4} + 924 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{10} + 3080 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{9} + 5775 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{8} + 6600 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{7} + 4620 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{6} + 1848 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{5}\right )} e^{3} + \frac {1}{840} \, {\left (252 \, B b^{6} d x^{10} + 840 \, A a^{6} d x^{3} + 280 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d x^{9} + 945 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d x^{8} + 1800 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d x^{7} + 2100 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d x^{6} + 1512 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d x^{5} + 630 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d x^{4}\right )} e^{2} + \frac {1}{168} \, {\left (56 \, B b^{6} d^{2} x^{9} + 252 \, A a^{6} d^{2} x^{2} + 63 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} x^{8} + 216 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} x^{7} + 420 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} x^{6} + 504 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} x^{5} + 378 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} x^{4} + 168 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2} 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)^3,x, algorithm="fricas")

[Out]

1/8*B*b^6*d^3*x^8 + A*a^6*d^3*x + 1/7*(6*B*a*b^5 + A*b^6)*d^3*x^7 + 1/2*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*x^6 + (4
*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*x^5 + 5/4*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*x^4 + (2*B*a^5*b + 5*A*a^4*b^2)*d^3*x^
3 + 1/2*(B*a^6 + 6*A*a^5*b)*d^3*x^2 + 1/9240*(840*B*b^6*x^11 + 2310*A*a^6*x^4 + 924*(6*B*a*b^5 + A*b^6)*x^10 +
3080*(5*B*a^2*b^4 + 2*A*a*b^5)*x^9 + 5775*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^8 + 6600*(3*B*a^4*b^2 + 4*A*a^3*b^3)*
x^7 + 4620*(2*B*a^5*b + 5*A*a^4*b^2)*x^6 + 1848*(B*a^6 + 6*A*a^5*b)*x^5)*e^3 + 1/840*(252*B*b^6*d*x^10 + 840*A
*a^6*d*x^3 + 280*(6*B*a*b^5 + A*b^6)*d*x^9 + 945*(5*B*a^2*b^4 + 2*A*a*b^5)*d*x^8 + 1800*(4*B*a^3*b^3 + 3*A*a^2
*b^4)*d*x^7 + 2100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*x^6 + 1512*(2*B*a^5*b + 5*A*a^4*b^2)*d*x^5 + 630*(B*a^6 + 6*A
*a^5*b)*d*x^4)*e^2 + 1/168*(56*B*b^6*d^2*x^9 + 252*A*a^6*d^2*x^2 + 63*(6*B*a*b^5 + A*b^6)*d^2*x^8 + 216*(5*B*a
^2*b^4 + 2*A*a*b^5)*d^2*x^7 + 420*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*x^6 + 504*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*x^
5 + 378*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*x^4 + 168*(B*a^6 + 6*A*a^5*b)*d^2*x^3)*e

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (155) = 310\).
time = 0.06, size = 802, normalized size = 5.04 \begin {gather*} A a^{6} d^{3} x + \frac {B b^{6} e^{3} x^{11}}{11} + x^{10} \left (\frac {A b^{6} e^{3}}{10} + \frac {3 B a b^{5} e^{3}}{5} + \frac {3 B b^{6} d e^{2}}{10}\right ) + x^{9} \cdot \left (\frac {2 A a b^{5} e^{3}}{3} + \frac {A b^{6} d e^{2}}{3} + \frac {5 B a^{2} b^{4} e^{3}}{3} + 2 B a b^{5} d e^{2} + \frac {B b^{6} d^{2} e}{3}\right ) + x^{8} \cdot \left (\frac {15 A a^{2} b^{4} e^{3}}{8} + \frac {9 A a b^{5} d e^{2}}{4} + \frac {3 A b^{6} d^{2} e}{8} + \frac {5 B a^{3} b^{3} e^{3}}{2} + \frac {45 B a^{2} b^{4} d e^{2}}{8} + \frac {9 B a b^{5} d^{2} e}{4} + \frac {B b^{6} d^{3}}{8}\right ) + x^{7} \cdot \left (\frac {20 A a^{3} b^{3} e^{3}}{7} + \frac {45 A a^{2} b^{4} d e^{2}}{7} + \frac {18 A a b^{5} d^{2} e}{7} + \frac {A b^{6} d^{3}}{7} + \frac {15 B a^{4} b^{2} e^{3}}{7} + \frac {60 B a^{3} b^{3} d e^{2}}{7} + \frac {45 B a^{2} b^{4} d^{2} e}{7} + \frac {6 B a b^{5} d^{3}}{7}\right ) + x^{6} \cdot \left (\frac {5 A a^{4} b^{2} e^{3}}{2} + 10 A a^{3} b^{3} d e^{2} + \frac {15 A a^{2} b^{4} d^{2} e}{2} + A a b^{5} d^{3} + B a^{5} b e^{3} + \frac {15 B a^{4} b^{2} d e^{2}}{2} + 10 B a^{3} b^{3} d^{2} e + \frac {5 B a^{2} b^{4} d^{3}}{2}\right ) + x^{5} \cdot \left (\frac {6 A a^{5} b e^{3}}{5} + 9 A a^{4} b^{2} d e^{2} + 12 A a^{3} b^{3} d^{2} e + 3 A a^{2} b^{4} d^{3} + \frac {B a^{6} e^{3}}{5} + \frac {18 B a^{5} b d e^{2}}{5} + 9 B a^{4} b^{2} d^{2} e + 4 B a^{3} b^{3} d^{3}\right ) + x^{4} \left (\frac {A a^{6} e^{3}}{4} + \frac {9 A a^{5} b d e^{2}}{2} + \frac {45 A a^{4} b^{2} d^{2} e}{4} + 5 A a^{3} b^{3} d^{3} + \frac {3 B a^{6} d e^{2}}{4} + \frac {9 B a^{5} b d^{2} e}{2} + \frac {15 B a^{4} b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{6} d e^{2} + 6 A a^{5} b d^{2} e + 5 A a^{4} b^{2} d^{3} + B a^{6} d^{2} e + 2 B a^{5} b d^{3}\right ) + x^{2} \cdot \left (\frac {3 A a^{6} d^{2} e}{2} + 3 A a^{5} b d^{3} + \frac {B a^{6} d^{3}}{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)**3,x)

[Out]

A*a**6*d**3*x + B*b**6*e**3*x**11/11 + x**10*(A*b**6*e**3/10 + 3*B*a*b**5*e**3/5 + 3*B*b**6*d*e**2/10) + x**9*
(2*A*a*b**5*e**3/3 + A*b**6*d*e**2/3 + 5*B*a**2*b**4*e**3/3 + 2*B*a*b**5*d*e**2 + B*b**6*d**2*e/3) + x**8*(15*
A*a**2*b**4*e**3/8 + 9*A*a*b**5*d*e**2/4 + 3*A*b**6*d**2*e/8 + 5*B*a**3*b**3*e**3/2 + 45*B*a**2*b**4*d*e**2/8
+ 9*B*a*b**5*d**2*e/4 + B*b**6*d**3/8) + x**7*(20*A*a**3*b**3*e**3/7 + 45*A*a**2*b**4*d*e**2/7 + 18*A*a*b**5*d
**2*e/7 + A*b**6*d**3/7 + 15*B*a**4*b**2*e**3/7 + 60*B*a**3*b**3*d*e**2/7 + 45*B*a**2*b**4*d**2*e/7 + 6*B*a*b*
*5*d**3/7) + x**6*(5*A*a**4*b**2*e**3/2 + 10*A*a**3*b**3*d*e**2 + 15*A*a**2*b**4*d**2*e/2 + A*a*b**5*d**3 + B*
a**5*b*e**3 + 15*B*a**4*b**2*d*e**2/2 + 10*B*a**3*b**3*d**2*e + 5*B*a**2*b**4*d**3/2) + x**5*(6*A*a**5*b*e**3/
5 + 9*A*a**4*b**2*d*e**2 + 12*A*a**3*b**3*d**2*e + 3*A*a**2*b**4*d**3 + B*a**6*e**3/5 + 18*B*a**5*b*d*e**2/5 +
9*B*a**4*b**2*d**2*e + 4*B*a**3*b**3*d**3) + x**4*(A*a**6*e**3/4 + 9*A*a**5*b*d*e**2/2 + 45*A*a**4*b**2*d**2*
e/4 + 5*A*a**3*b**3*d**3 + 3*B*a**6*d*e**2/4 + 9*B*a**5*b*d**2*e/2 + 15*B*a**4*b**2*d**3/4) + x**3*(A*a**6*d*e
**2 + 6*A*a**5*b*d**2*e + 5*A*a**4*b**2*d**3 + B*a**6*d**2*e + 2*B*a**5*b*d**3) + x**2*(3*A*a**6*d**2*e/2 + 3*
A*a**5*b*d**3 + B*a**6*d**3/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 768 vs. \(2 (157) = 314\).
time = 1.75, size = 768, normalized size = 4.83 \begin {gather*} \frac {1}{11} \, B b^{6} x^{11} e^{3} + \frac {3}{10} \, B b^{6} d x^{10} e^{2} + \frac {1}{3} \, B b^{6} d^{2} x^{9} e + \frac {1}{8} \, B b^{6} d^{3} x^{8} + \frac {3}{5} \, B a b^{5} x^{10} e^{3} + \frac {1}{10} \, A b^{6} x^{10} e^{3} + 2 \, B a b^{5} d x^{9} e^{2} + \frac {1}{3} \, A b^{6} d x^{9} e^{2} + \frac {9}{4} \, B a b^{5} d^{2} x^{8} e + \frac {3}{8} \, A b^{6} d^{2} x^{8} e + \frac {6}{7} \, B a b^{5} d^{3} x^{7} + \frac {1}{7} \, A b^{6} d^{3} x^{7} + \frac {5}{3} \, B a^{2} b^{4} x^{9} e^{3} + \frac {2}{3} \, A a b^{5} x^{9} e^{3} + \frac {45}{8} \, B a^{2} b^{4} d x^{8} e^{2} + \frac {9}{4} \, A a b^{5} d x^{8} e^{2} + \frac {45}{7} \, B a^{2} b^{4} d^{2} x^{7} e + \frac {18}{7} \, A a b^{5} d^{2} x^{7} e + \frac {5}{2} \, B a^{2} b^{4} d^{3} x^{6} + A a b^{5} d^{3} x^{6} + \frac {5}{2} \, B a^{3} b^{3} x^{8} e^{3} + \frac {15}{8} \, A a^{2} b^{4} x^{8} e^{3} + \frac {60}{7} \, B a^{3} b^{3} d x^{7} e^{2} + \frac {45}{7} \, A a^{2} b^{4} d x^{7} e^{2} + 10 \, B a^{3} b^{3} d^{2} x^{6} e + \frac {15}{2} \, A a^{2} b^{4} d^{2} x^{6} e + 4 \, B a^{3} b^{3} d^{3} x^{5} + 3 \, A a^{2} b^{4} d^{3} x^{5} + \frac {15}{7} \, B a^{4} b^{2} x^{7} e^{3} + \frac {20}{7} \, A a^{3} b^{3} x^{7} e^{3} + \frac {15}{2} \, B a^{4} b^{2} d x^{6} e^{2} + 10 \, A a^{3} b^{3} d x^{6} e^{2} + 9 \, B a^{4} b^{2} d^{2} x^{5} e + 12 \, A a^{3} b^{3} d^{2} x^{5} e + \frac {15}{4} \, B a^{4} b^{2} d^{3} x^{4} + 5 \, A a^{3} b^{3} d^{3} x^{4} + B a^{5} b x^{6} e^{3} + \frac {5}{2} \, A a^{4} b^{2} x^{6} e^{3} + \frac {18}{5} \, B a^{5} b d x^{5} e^{2} + 9 \, A a^{4} b^{2} d x^{5} e^{2} + \frac {9}{2} \, B a^{5} b d^{2} x^{4} e + \frac {45}{4} \, A a^{4} b^{2} d^{2} x^{4} e + 2 \, B a^{5} b d^{3} x^{3} + 5 \, A a^{4} b^{2} d^{3} x^{3} + \frac {1}{5} \, B a^{6} x^{5} e^{3} + \frac {6}{5} \, A a^{5} b x^{5} e^{3} + \frac {3}{4} \, B a^{6} d x^{4} e^{2} + \frac {9}{2} \, A a^{5} b d x^{4} e^{2} + B a^{6} d^{2} x^{3} e + 6 \, A a^{5} b d^{2} x^{3} e + \frac {1}{2} \, B a^{6} d^{3} x^{2} + 3 \, A a^{5} b d^{3} x^{2} + \frac {1}{4} \, A a^{6} x^{4} e^{3} + A a^{6} d x^{3} e^{2} + \frac {3}{2} \, A a^{6} d^{2} x^{2} e + A a^{6} d^{3} 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)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.24, size = 649, normalized size = 4.08 \begin {gather*} x^3\,\left (B\,a^6\,d^2\,e+A\,a^6\,d\,e^2+2\,B\,a^5\,b\,d^3+6\,A\,a^5\,b\,d^2\,e+5\,A\,a^4\,b^2\,d^3\right )+x^9\,\left (\frac {5\,B\,a^2\,b^4\,e^3}{3}+2\,B\,a\,b^5\,d\,e^2+\frac {2\,A\,a\,b^5\,e^3}{3}+\frac {B\,b^6\,d^2\,e}{3}+\frac {A\,b^6\,d\,e^2}{3}\right )+x^4\,\left (\frac {3\,B\,a^6\,d\,e^2}{4}+\frac {A\,a^6\,e^3}{4}+\frac {9\,B\,a^5\,b\,d^2\,e}{2}+\frac {9\,A\,a^5\,b\,d\,e^2}{2}+\frac {15\,B\,a^4\,b^2\,d^3}{4}+\frac {45\,A\,a^4\,b^2\,d^2\,e}{4}+5\,A\,a^3\,b^3\,d^3\right )+x^8\,\left (\frac {5\,B\,a^3\,b^3\,e^3}{2}+\frac {45\,B\,a^2\,b^4\,d\,e^2}{8}+\frac {15\,A\,a^2\,b^4\,e^3}{8}+\frac {9\,B\,a\,b^5\,d^2\,e}{4}+\frac {9\,A\,a\,b^5\,d\,e^2}{4}+\frac {B\,b^6\,d^3}{8}+\frac {3\,A\,b^6\,d^2\,e}{8}\right )+x^6\,\left (B\,a^5\,b\,e^3+\frac {15\,B\,a^4\,b^2\,d\,e^2}{2}+\frac {5\,A\,a^4\,b^2\,e^3}{2}+10\,B\,a^3\,b^3\,d^2\,e+10\,A\,a^3\,b^3\,d\,e^2+\frac {5\,B\,a^2\,b^4\,d^3}{2}+\frac {15\,A\,a^2\,b^4\,d^2\,e}{2}+A\,a\,b^5\,d^3\right )+x^5\,\left (\frac {B\,a^6\,e^3}{5}+\frac {18\,B\,a^5\,b\,d\,e^2}{5}+\frac {6\,A\,a^5\,b\,e^3}{5}+9\,B\,a^4\,b^2\,d^2\,e+9\,A\,a^4\,b^2\,d\,e^2+4\,B\,a^3\,b^3\,d^3+12\,A\,a^3\,b^3\,d^2\,e+3\,A\,a^2\,b^4\,d^3\right )+x^7\,\left (\frac {15\,B\,a^4\,b^2\,e^3}{7}+\frac {60\,B\,a^3\,b^3\,d\,e^2}{7}+\frac {20\,A\,a^3\,b^3\,e^3}{7}+\frac {45\,B\,a^2\,b^4\,d^2\,e}{7}+\frac {45\,A\,a^2\,b^4\,d\,e^2}{7}+\frac {6\,B\,a\,b^5\,d^3}{7}+\frac {18\,A\,a\,b^5\,d^2\,e}{7}+\frac {A\,b^6\,d^3}{7}\right )+\frac {a^5\,d^2\,x^2\,\left (3\,A\,a\,e+6\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^5\,e^2\,x^{10}\,\left (A\,b\,e+6\,B\,a\,e+3\,B\,b\,d\right )}{10}+A\,a^6\,d^3\,x+\frac {B\,b^6\,e^3\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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