3.9.96 \(\int (a+b x)^3 (A+B x) (d+e x)^5 \, dx\)

Optimal. Leaf size=163 \[ -\frac {b^2 (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac {3 b (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5}-\frac {(d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5}+\frac {(d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5}+\frac {b^3 B (d+e x)^{10}}{10 e^5} \]

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Rubi [A]  time = 0.41, antiderivative size = 163, 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 = {77} \begin {gather*} -\frac {b^2 (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac {3 b (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5}-\frac {(d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5}+\frac {(d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5}+\frac {b^3 B (d+e x)^{10}}{10 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(A + B*x)*(d + e*x)^5,x]

[Out]

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

Rule 77

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

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Mathematica [B]  time = 0.17, size = 471, normalized size = 2.89 \begin {gather*} a^3 A d^5 x+\frac {1}{3} a d^3 x^3 \left (A \left (10 a^2 e^2+15 a b d e+3 b^2 d^2\right )+a B d (5 a e+3 b d)\right )+\frac {1}{8} b e^3 x^8 \left (3 a^2 B e^2+3 a b e (A e+5 B d)+5 b^2 d (A e+2 B d)\right )+\frac {1}{2} a^2 d^4 x^2 (5 a A e+a B d+3 A b d)+\frac {1}{7} e^2 x^7 \left (a^3 B e^3+3 a^2 b e^2 (A e+5 B d)+15 a b^2 d e (A e+2 B d)+10 b^3 d^2 (A e+B d)\right )+\frac {1}{6} e x^6 \left (a^3 e^3 (A e+5 B d)+15 a^2 b d e^2 (A e+2 B d)+30 a b^2 d^2 e (A e+B d)+5 b^3 d^3 (2 A e+B d)\right )+\frac {1}{5} d x^5 \left (5 a^3 e^3 (A e+2 B d)+30 a^2 b d e^2 (A e+B d)+15 a b^2 d^2 e (2 A e+B d)+b^3 d^3 (5 A e+B d)\right )+\frac {1}{4} d^2 x^4 \left (a B d \left (10 a^2 e^2+15 a b d e+3 b^2 d^2\right )+A \left (10 a^3 e^3+30 a^2 b d e^2+15 a b^2 d^2 e+b^3 d^3\right )\right )+\frac {1}{9} b^2 e^4 x^9 (3 a B e+A b e+5 b B d)+\frac {1}{10} b^3 B e^5 x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^5,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^3*(A + B*x)*(d + e*x)^5,x]

[Out]

IntegrateAlgebraic[(a + b*x)^3*(A + B*x)*(d + e*x)^5, x]

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fricas [B]  time = 0.86, size = 658, normalized size = 4.04 \begin {gather*} \frac {1}{10} x^{10} e^{5} b^{3} B + \frac {5}{9} x^{9} e^{4} d b^{3} B + \frac {1}{3} x^{9} e^{5} b^{2} a B + \frac {1}{9} x^{9} e^{5} b^{3} A + \frac {5}{4} x^{8} e^{3} d^{2} b^{3} B + \frac {15}{8} x^{8} e^{4} d b^{2} a B + \frac {3}{8} x^{8} e^{5} b a^{2} B + \frac {5}{8} x^{8} e^{4} d b^{3} A + \frac {3}{8} x^{8} e^{5} b^{2} a A + \frac {10}{7} x^{7} e^{2} d^{3} b^{3} B + \frac {30}{7} x^{7} e^{3} d^{2} b^{2} a B + \frac {15}{7} x^{7} e^{4} d b a^{2} B + \frac {1}{7} x^{7} e^{5} a^{3} B + \frac {10}{7} x^{7} e^{3} d^{2} b^{3} A + \frac {15}{7} x^{7} e^{4} d b^{2} a A + \frac {3}{7} x^{7} e^{5} b a^{2} A + \frac {5}{6} x^{6} e d^{4} b^{3} B + 5 x^{6} e^{2} d^{3} b^{2} a B + 5 x^{6} e^{3} d^{2} b a^{2} B + \frac {5}{6} x^{6} e^{4} d a^{3} B + \frac {5}{3} x^{6} e^{2} d^{3} b^{3} A + 5 x^{6} e^{3} d^{2} b^{2} a A + \frac {5}{2} x^{6} e^{4} d b a^{2} A + \frac {1}{6} x^{6} e^{5} a^{3} A + \frac {1}{5} x^{5} d^{5} b^{3} B + 3 x^{5} e d^{4} b^{2} a B + 6 x^{5} e^{2} d^{3} b a^{2} B + 2 x^{5} e^{3} d^{2} a^{3} B + x^{5} e d^{4} b^{3} A + 6 x^{5} e^{2} d^{3} b^{2} a A + 6 x^{5} e^{3} d^{2} b a^{2} A + x^{5} e^{4} d a^{3} A + \frac {3}{4} x^{4} d^{5} b^{2} a B + \frac {15}{4} x^{4} e d^{4} b a^{2} B + \frac {5}{2} x^{4} e^{2} d^{3} a^{3} B + \frac {1}{4} x^{4} d^{5} b^{3} A + \frac {15}{4} x^{4} e d^{4} b^{2} a A + \frac {15}{2} x^{4} e^{2} d^{3} b a^{2} A + \frac {5}{2} x^{4} e^{3} d^{2} a^{3} A + x^{3} d^{5} b a^{2} B + \frac {5}{3} x^{3} e d^{4} a^{3} B + x^{3} d^{5} b^{2} a A + 5 x^{3} e d^{4} b a^{2} A + \frac {10}{3} x^{3} e^{2} d^{3} a^{3} A + \frac {1}{2} x^{2} d^{5} a^{3} B + \frac {3}{2} x^{2} d^{5} b a^{2} A + \frac {5}{2} x^{2} e d^{4} a^{3} A + x d^{5} a^{3} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^5,x, algorithm="fricas")

[Out]

1/10*x^10*e^5*b^3*B + 5/9*x^9*e^4*d*b^3*B + 1/3*x^9*e^5*b^2*a*B + 1/9*x^9*e^5*b^3*A + 5/4*x^8*e^3*d^2*b^3*B +
15/8*x^8*e^4*d*b^2*a*B + 3/8*x^8*e^5*b*a^2*B + 5/8*x^8*e^4*d*b^3*A + 3/8*x^8*e^5*b^2*a*A + 10/7*x^7*e^2*d^3*b^
3*B + 30/7*x^7*e^3*d^2*b^2*a*B + 15/7*x^7*e^4*d*b*a^2*B + 1/7*x^7*e^5*a^3*B + 10/7*x^7*e^3*d^2*b^3*A + 15/7*x^
7*e^4*d*b^2*a*A + 3/7*x^7*e^5*b*a^2*A + 5/6*x^6*e*d^4*b^3*B + 5*x^6*e^2*d^3*b^2*a*B + 5*x^6*e^3*d^2*b*a^2*B +
5/6*x^6*e^4*d*a^3*B + 5/3*x^6*e^2*d^3*b^3*A + 5*x^6*e^3*d^2*b^2*a*A + 5/2*x^6*e^4*d*b*a^2*A + 1/6*x^6*e^5*a^3*
A + 1/5*x^5*d^5*b^3*B + 3*x^5*e*d^4*b^2*a*B + 6*x^5*e^2*d^3*b*a^2*B + 2*x^5*e^3*d^2*a^3*B + x^5*e*d^4*b^3*A +
6*x^5*e^2*d^3*b^2*a*A + 6*x^5*e^3*d^2*b*a^2*A + x^5*e^4*d*a^3*A + 3/4*x^4*d^5*b^2*a*B + 15/4*x^4*e*d^4*b*a^2*B
+ 5/2*x^4*e^2*d^3*a^3*B + 1/4*x^4*d^5*b^3*A + 15/4*x^4*e*d^4*b^2*a*A + 15/2*x^4*e^2*d^3*b*a^2*A + 5/2*x^4*e^3
*d^2*a^3*A + x^3*d^5*b*a^2*B + 5/3*x^3*e*d^4*a^3*B + x^3*d^5*b^2*a*A + 5*x^3*e*d^4*b*a^2*A + 10/3*x^3*e^2*d^3*
a^3*A + 1/2*x^2*d^5*a^3*B + 3/2*x^2*d^5*b*a^2*A + 5/2*x^2*e*d^4*a^3*A + x*d^5*a^3*A

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giac [B]  time = 1.33, size = 634, normalized size = 3.89 \begin {gather*} \frac {1}{10} \, B b^{3} x^{10} e^{5} + \frac {5}{9} \, B b^{3} d x^{9} e^{4} + \frac {5}{4} \, B b^{3} d^{2} x^{8} e^{3} + \frac {10}{7} \, B b^{3} d^{3} x^{7} e^{2} + \frac {5}{6} \, B b^{3} d^{4} x^{6} e + \frac {1}{5} \, B b^{3} d^{5} x^{5} + \frac {1}{3} \, B a b^{2} x^{9} e^{5} + \frac {1}{9} \, A b^{3} x^{9} e^{5} + \frac {15}{8} \, B a b^{2} d x^{8} e^{4} + \frac {5}{8} \, A b^{3} d x^{8} e^{4} + \frac {30}{7} \, B a b^{2} d^{2} x^{7} e^{3} + \frac {10}{7} \, A b^{3} d^{2} x^{7} e^{3} + 5 \, B a b^{2} d^{3} x^{6} e^{2} + \frac {5}{3} \, A b^{3} d^{3} x^{6} e^{2} + 3 \, B a b^{2} d^{4} x^{5} e + A b^{3} d^{4} x^{5} e + \frac {3}{4} \, B a b^{2} d^{5} x^{4} + \frac {1}{4} \, A b^{3} d^{5} x^{4} + \frac {3}{8} \, B a^{2} b x^{8} e^{5} + \frac {3}{8} \, A a b^{2} x^{8} e^{5} + \frac {15}{7} \, B a^{2} b d x^{7} e^{4} + \frac {15}{7} \, A a b^{2} d x^{7} e^{4} + 5 \, B a^{2} b d^{2} x^{6} e^{3} + 5 \, A a b^{2} d^{2} x^{6} e^{3} + 6 \, B a^{2} b d^{3} x^{5} e^{2} + 6 \, A a b^{2} d^{3} x^{5} e^{2} + \frac {15}{4} \, B a^{2} b d^{4} x^{4} e + \frac {15}{4} \, A a b^{2} d^{4} x^{4} e + B a^{2} b d^{5} x^{3} + A a b^{2} d^{5} x^{3} + \frac {1}{7} \, B a^{3} x^{7} e^{5} + \frac {3}{7} \, A a^{2} b x^{7} e^{5} + \frac {5}{6} \, B a^{3} d x^{6} e^{4} + \frac {5}{2} \, A a^{2} b d x^{6} e^{4} + 2 \, B a^{3} d^{2} x^{5} e^{3} + 6 \, A a^{2} b d^{2} x^{5} e^{3} + \frac {5}{2} \, B a^{3} d^{3} x^{4} e^{2} + \frac {15}{2} \, A a^{2} b d^{3} x^{4} e^{2} + \frac {5}{3} \, B a^{3} d^{4} x^{3} e + 5 \, A a^{2} b d^{4} x^{3} e + \frac {1}{2} \, B a^{3} d^{5} x^{2} + \frac {3}{2} \, A a^{2} b d^{5} x^{2} + \frac {1}{6} \, A a^{3} x^{6} e^{5} + A a^{3} d x^{5} e^{4} + \frac {5}{2} \, A a^{3} d^{2} x^{4} e^{3} + \frac {10}{3} \, A a^{3} d^{3} x^{3} e^{2} + \frac {5}{2} \, A a^{3} d^{4} x^{2} e + A a^{3} d^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^5,x, algorithm="giac")

[Out]

1/10*B*b^3*x^10*e^5 + 5/9*B*b^3*d*x^9*e^4 + 5/4*B*b^3*d^2*x^8*e^3 + 10/7*B*b^3*d^3*x^7*e^2 + 5/6*B*b^3*d^4*x^6
*e + 1/5*B*b^3*d^5*x^5 + 1/3*B*a*b^2*x^9*e^5 + 1/9*A*b^3*x^9*e^5 + 15/8*B*a*b^2*d*x^8*e^4 + 5/8*A*b^3*d*x^8*e^
4 + 30/7*B*a*b^2*d^2*x^7*e^3 + 10/7*A*b^3*d^2*x^7*e^3 + 5*B*a*b^2*d^3*x^6*e^2 + 5/3*A*b^3*d^3*x^6*e^2 + 3*B*a*
b^2*d^4*x^5*e + A*b^3*d^4*x^5*e + 3/4*B*a*b^2*d^5*x^4 + 1/4*A*b^3*d^5*x^4 + 3/8*B*a^2*b*x^8*e^5 + 3/8*A*a*b^2*
x^8*e^5 + 15/7*B*a^2*b*d*x^7*e^4 + 15/7*A*a*b^2*d*x^7*e^4 + 5*B*a^2*b*d^2*x^6*e^3 + 5*A*a*b^2*d^2*x^6*e^3 + 6*
B*a^2*b*d^3*x^5*e^2 + 6*A*a*b^2*d^3*x^5*e^2 + 15/4*B*a^2*b*d^4*x^4*e + 15/4*A*a*b^2*d^4*x^4*e + B*a^2*b*d^5*x^
3 + A*a*b^2*d^5*x^3 + 1/7*B*a^3*x^7*e^5 + 3/7*A*a^2*b*x^7*e^5 + 5/6*B*a^3*d*x^6*e^4 + 5/2*A*a^2*b*d*x^6*e^4 +
2*B*a^3*d^2*x^5*e^3 + 6*A*a^2*b*d^2*x^5*e^3 + 5/2*B*a^3*d^3*x^4*e^2 + 15/2*A*a^2*b*d^3*x^4*e^2 + 5/3*B*a^3*d^4
*x^3*e + 5*A*a^2*b*d^4*x^3*e + 1/2*B*a^3*d^5*x^2 + 3/2*A*a^2*b*d^5*x^2 + 1/6*A*a^3*x^6*e^5 + A*a^3*d*x^5*e^4 +
5/2*A*a^3*d^2*x^4*e^3 + 10/3*A*a^3*d^3*x^3*e^2 + 5/2*A*a^3*d^4*x^2*e + A*a^3*d^5*x

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maple [B]  time = 0.00, size = 529, normalized size = 3.25 \begin {gather*} \frac {B \,b^{3} e^{5} x^{10}}{10}+A \,a^{3} d^{5} x +\frac {\left (5 B \,b^{3} d \,e^{4}+\left (b^{3} A +3 a \,b^{2} B \right ) e^{5}\right ) x^{9}}{9}+\frac {\left (10 B \,b^{3} d^{2} e^{3}+5 \left (b^{3} A +3 a \,b^{2} B \right ) d \,e^{4}+\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{5}\right ) x^{8}}{8}+\frac {\left (10 B \,b^{3} d^{3} e^{2}+10 \left (b^{3} A +3 a \,b^{2} B \right ) d^{2} e^{3}+5 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d \,e^{4}+\left (3 A \,a^{2} b +B \,a^{3}\right ) e^{5}\right ) x^{7}}{7}+\frac {\left (A \,a^{3} e^{5}+5 B \,b^{3} d^{4} e +10 \left (b^{3} A +3 a \,b^{2} B \right ) d^{3} e^{2}+10 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2} e^{3}+5 \left (3 A \,a^{2} b +B \,a^{3}\right ) d \,e^{4}\right ) x^{6}}{6}+\frac {\left (5 A \,a^{3} d \,e^{4}+B \,b^{3} d^{5}+5 \left (b^{3} A +3 a \,b^{2} B \right ) d^{4} e +10 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{3} e^{2}+10 \left (3 A \,a^{2} b +B \,a^{3}\right ) d^{2} e^{3}\right ) x^{5}}{5}+\frac {\left (10 A \,a^{3} d^{2} e^{3}+\left (b^{3} A +3 a \,b^{2} B \right ) d^{5}+5 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{4} e +10 \left (3 A \,a^{2} b +B \,a^{3}\right ) d^{3} e^{2}\right ) x^{4}}{4}+\frac {\left (10 A \,a^{3} d^{3} e^{2}+\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{5}+5 \left (3 A \,a^{2} b +B \,a^{3}\right ) d^{4} e \right ) x^{3}}{3}+\frac {\left (5 A \,a^{3} d^{4} e +\left (3 A \,a^{2} b +B \,a^{3}\right ) d^{5}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)*(e*x+d)^5,x)

[Out]

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

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maxima [B]  time = 0.70, size = 518, normalized size = 3.18 \begin {gather*} \frac {1}{10} \, B b^{3} e^{5} x^{10} + A a^{3} d^{5} x + \frac {1}{9} \, {\left (5 \, B b^{3} d e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (10 \, B b^{3} d^{2} e^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, B b^{3} d^{3} e^{2} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 15 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{4} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (5 \, B b^{3} d^{4} e + A a^{3} e^{5} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{5} + 5 \, A a^{3} d e^{4} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 30 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, A a^{3} d^{2} e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} + 15 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e + 10 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, A a^{3} d^{3} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{3} d^{4} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.20, size = 544, normalized size = 3.34 \begin {gather*} x^3\,\left (\frac {5\,B\,a^3\,d^4\,e}{3}+\frac {10\,A\,a^3\,d^3\,e^2}{3}+B\,a^2\,b\,d^5+5\,A\,a^2\,b\,d^4\,e+A\,a\,b^2\,d^5\right )+x^8\,\left (\frac {3\,B\,a^2\,b\,e^5}{8}+\frac {15\,B\,a\,b^2\,d\,e^4}{8}+\frac {3\,A\,a\,b^2\,e^5}{8}+\frac {5\,B\,b^3\,d^2\,e^3}{4}+\frac {5\,A\,b^3\,d\,e^4}{8}\right )+x^4\,\left (\frac {5\,B\,a^3\,d^3\,e^2}{2}+\frac {5\,A\,a^3\,d^2\,e^3}{2}+\frac {15\,B\,a^2\,b\,d^4\,e}{4}+\frac {15\,A\,a^2\,b\,d^3\,e^2}{2}+\frac {3\,B\,a\,b^2\,d^5}{4}+\frac {15\,A\,a\,b^2\,d^4\,e}{4}+\frac {A\,b^3\,d^5}{4}\right )+x^7\,\left (\frac {B\,a^3\,e^5}{7}+\frac {15\,B\,a^2\,b\,d\,e^4}{7}+\frac {3\,A\,a^2\,b\,e^5}{7}+\frac {30\,B\,a\,b^2\,d^2\,e^3}{7}+\frac {15\,A\,a\,b^2\,d\,e^4}{7}+\frac {10\,B\,b^3\,d^3\,e^2}{7}+\frac {10\,A\,b^3\,d^2\,e^3}{7}\right )+x^5\,\left (2\,B\,a^3\,d^2\,e^3+A\,a^3\,d\,e^4+6\,B\,a^2\,b\,d^3\,e^2+6\,A\,a^2\,b\,d^2\,e^3+3\,B\,a\,b^2\,d^4\,e+6\,A\,a\,b^2\,d^3\,e^2+\frac {B\,b^3\,d^5}{5}+A\,b^3\,d^4\,e\right )+x^6\,\left (\frac {5\,B\,a^3\,d\,e^4}{6}+\frac {A\,a^3\,e^5}{6}+5\,B\,a^2\,b\,d^2\,e^3+\frac {5\,A\,a^2\,b\,d\,e^4}{2}+5\,B\,a\,b^2\,d^3\,e^2+5\,A\,a\,b^2\,d^2\,e^3+\frac {5\,B\,b^3\,d^4\,e}{6}+\frac {5\,A\,b^3\,d^3\,e^2}{3}\right )+\frac {a^2\,d^4\,x^2\,\left (5\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e^4\,x^9\,\left (A\,b\,e+3\,B\,a\,e+5\,B\,b\,d\right )}{9}+A\,a^3\,d^5\,x+\frac {B\,b^3\,e^5\,x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^3*(d + e*x)^5,x)

[Out]

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

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sympy [B]  time = 0.16, size = 678, normalized size = 4.16 \begin {gather*} A a^{3} d^{5} x + \frac {B b^{3} e^{5} x^{10}}{10} + x^{9} \left (\frac {A b^{3} e^{5}}{9} + \frac {B a b^{2} e^{5}}{3} + \frac {5 B b^{3} d e^{4}}{9}\right ) + x^{8} \left (\frac {3 A a b^{2} e^{5}}{8} + \frac {5 A b^{3} d e^{4}}{8} + \frac {3 B a^{2} b e^{5}}{8} + \frac {15 B a b^{2} d e^{4}}{8} + \frac {5 B b^{3} d^{2} e^{3}}{4}\right ) + x^{7} \left (\frac {3 A a^{2} b e^{5}}{7} + \frac {15 A a b^{2} d e^{4}}{7} + \frac {10 A b^{3} d^{2} e^{3}}{7} + \frac {B a^{3} e^{5}}{7} + \frac {15 B a^{2} b d e^{4}}{7} + \frac {30 B a b^{2} d^{2} e^{3}}{7} + \frac {10 B b^{3} d^{3} e^{2}}{7}\right ) + x^{6} \left (\frac {A a^{3} e^{5}}{6} + \frac {5 A a^{2} b d e^{4}}{2} + 5 A a b^{2} d^{2} e^{3} + \frac {5 A b^{3} d^{3} e^{2}}{3} + \frac {5 B a^{3} d e^{4}}{6} + 5 B a^{2} b d^{2} e^{3} + 5 B a b^{2} d^{3} e^{2} + \frac {5 B b^{3} d^{4} e}{6}\right ) + x^{5} \left (A a^{3} d e^{4} + 6 A a^{2} b d^{2} e^{3} + 6 A a b^{2} d^{3} e^{2} + A b^{3} d^{4} e + 2 B a^{3} d^{2} e^{3} + 6 B a^{2} b d^{3} e^{2} + 3 B a b^{2} d^{4} e + \frac {B b^{3} d^{5}}{5}\right ) + x^{4} \left (\frac {5 A a^{3} d^{2} e^{3}}{2} + \frac {15 A a^{2} b d^{3} e^{2}}{2} + \frac {15 A a b^{2} d^{4} e}{4} + \frac {A b^{3} d^{5}}{4} + \frac {5 B a^{3} d^{3} e^{2}}{2} + \frac {15 B a^{2} b d^{4} e}{4} + \frac {3 B a b^{2} d^{5}}{4}\right ) + x^{3} \left (\frac {10 A a^{3} d^{3} e^{2}}{3} + 5 A a^{2} b d^{4} e + A a b^{2} d^{5} + \frac {5 B a^{3} d^{4} e}{3} + B a^{2} b d^{5}\right ) + x^{2} \left (\frac {5 A a^{3} d^{4} e}{2} + \frac {3 A a^{2} b d^{5}}{2} + \frac {B a^{3} d^{5}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)*(e*x+d)**5,x)

[Out]

A*a**3*d**5*x + B*b**3*e**5*x**10/10 + x**9*(A*b**3*e**5/9 + B*a*b**2*e**5/3 + 5*B*b**3*d*e**4/9) + x**8*(3*A*
a*b**2*e**5/8 + 5*A*b**3*d*e**4/8 + 3*B*a**2*b*e**5/8 + 15*B*a*b**2*d*e**4/8 + 5*B*b**3*d**2*e**3/4) + x**7*(3
*A*a**2*b*e**5/7 + 15*A*a*b**2*d*e**4/7 + 10*A*b**3*d**2*e**3/7 + B*a**3*e**5/7 + 15*B*a**2*b*d*e**4/7 + 30*B*
a*b**2*d**2*e**3/7 + 10*B*b**3*d**3*e**2/7) + x**6*(A*a**3*e**5/6 + 5*A*a**2*b*d*e**4/2 + 5*A*a*b**2*d**2*e**3
+ 5*A*b**3*d**3*e**2/3 + 5*B*a**3*d*e**4/6 + 5*B*a**2*b*d**2*e**3 + 5*B*a*b**2*d**3*e**2 + 5*B*b**3*d**4*e/6)
+ x**5*(A*a**3*d*e**4 + 6*A*a**2*b*d**2*e**3 + 6*A*a*b**2*d**3*e**2 + A*b**3*d**4*e + 2*B*a**3*d**2*e**3 + 6*
B*a**2*b*d**3*e**2 + 3*B*a*b**2*d**4*e + B*b**3*d**5/5) + x**4*(5*A*a**3*d**2*e**3/2 + 15*A*a**2*b*d**3*e**2/2
+ 15*A*a*b**2*d**4*e/4 + A*b**3*d**5/4 + 5*B*a**3*d**3*e**2/2 + 15*B*a**2*b*d**4*e/4 + 3*B*a*b**2*d**5/4) + x
**3*(10*A*a**3*d**3*e**2/3 + 5*A*a**2*b*d**4*e + A*a*b**2*d**5 + 5*B*a**3*d**4*e/3 + B*a**2*b*d**5) + x**2*(5*
A*a**3*d**4*e/2 + 3*A*a**2*b*d**5/2 + B*a**3*d**5/2)

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