3.9.97 \(\int (a+b x)^3 (A+B x) (d+e x)^4 \, dx\)

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

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Rubi [A]  time = 0.29, 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)^8 (-3 a B e-A b e+4 b B d)}{8 e^5}+\frac {3 b (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac {(d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5}+\frac {(d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5}+\frac {b^3 B (d+e x)^9}{9 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(A + B*x)*(d + e*x)^4,x]

[Out]

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

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Mathematica [B]  time = 0.14, size = 397, normalized size = 2.44 \begin {gather*} a^3 A d^4 x+\frac {1}{3} a d^2 x^3 \left (3 A \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+a B d (4 a e+3 b d)\right )+\frac {1}{7} b e^2 x^7 \left (3 a^2 B e^2+3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+\frac {1}{2} a^2 d^3 x^2 (4 a A e+a B d+3 A b d)+\frac {1}{6} e x^6 \left (a^3 B e^3+3 a^2 b e^2 (A e+4 B d)+6 a b^2 d e (2 A e+3 B d)+2 b^3 d^2 (3 A e+2 B d)\right )+\frac {1}{5} x^5 \left (a^3 e^3 (A e+4 B d)+6 a^2 b d e^2 (2 A e+3 B d)+6 a b^2 d^2 e (3 A e+2 B d)+b^3 d^3 (4 A e+B d)\right )+\frac {1}{4} d x^4 \left (3 a B d \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+A \left (4 a^3 e^3+18 a^2 b d e^2+12 a b^2 d^2 e+b^3 d^3\right )\right )+\frac {1}{8} b^2 e^3 x^8 (3 a B e+A b e+4 b B d)+\frac {1}{9} b^3 B e^4 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^4,x]

[Out]

a^3*A*d^4*x + (a^2*d^3*(3*A*b*d + a*B*d + 4*a*A*e)*x^2)/2 + (a*d^2*(a*B*d*(3*b*d + 4*a*e) + 3*A*(b^2*d^2 + 4*a
*b*d*e + 2*a^2*e^2))*x^3)/3 + (d*(3*a*B*d*(b^2*d^2 + 4*a*b*d*e + 2*a^2*e^2) + A*(b^3*d^3 + 12*a*b^2*d^2*e + 18
*a^2*b*d*e^2 + 4*a^3*e^3))*x^4)/4 + ((a^3*e^3*(4*B*d + A*e) + 6*a^2*b*d*e^2*(3*B*d + 2*A*e) + 6*a*b^2*d^2*e*(2
*B*d + 3*A*e) + b^3*d^3*(B*d + 4*A*e))*x^5)/5 + (e*(a^3*B*e^3 + 3*a^2*b*e^2*(4*B*d + A*e) + 6*a*b^2*d*e*(3*B*d
+ 2*A*e) + 2*b^3*d^2*(2*B*d + 3*A*e))*x^6)/6 + (b*e^2*(3*a^2*B*e^2 + 3*a*b*e*(4*B*d + A*e) + 2*b^2*d*(3*B*d +
2*A*e))*x^7)/7 + (b^2*e^3*(4*b*B*d + A*b*e + 3*a*B*e)*x^8)/8 + (b^3*B*e^4*x^9)/9

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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)^4 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^3*(A + B*x)*(d + e*x)^4,x]

[Out]

IntegrateAlgebraic[(a + b*x)^3*(A + B*x)*(d + e*x)^4, x]

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fricas [B]  time = 1.00, size = 534, normalized size = 3.28 \begin {gather*} \frac {1}{9} x^{9} e^{4} b^{3} B + \frac {1}{2} x^{8} e^{3} d b^{3} B + \frac {3}{8} x^{8} e^{4} b^{2} a B + \frac {1}{8} x^{8} e^{4} b^{3} A + \frac {6}{7} x^{7} e^{2} d^{2} b^{3} B + \frac {12}{7} x^{7} e^{3} d b^{2} a B + \frac {3}{7} x^{7} e^{4} b a^{2} B + \frac {4}{7} x^{7} e^{3} d b^{3} A + \frac {3}{7} x^{7} e^{4} b^{2} a A + \frac {2}{3} x^{6} e d^{3} b^{3} B + 3 x^{6} e^{2} d^{2} b^{2} a B + 2 x^{6} e^{3} d b a^{2} B + \frac {1}{6} x^{6} e^{4} a^{3} B + x^{6} e^{2} d^{2} b^{3} A + 2 x^{6} e^{3} d b^{2} a A + \frac {1}{2} x^{6} e^{4} b a^{2} A + \frac {1}{5} x^{5} d^{4} b^{3} B + \frac {12}{5} x^{5} e d^{3} b^{2} a B + \frac {18}{5} x^{5} e^{2} d^{2} b a^{2} B + \frac {4}{5} x^{5} e^{3} d a^{3} B + \frac {4}{5} x^{5} e d^{3} b^{3} A + \frac {18}{5} x^{5} e^{2} d^{2} b^{2} a A + \frac {12}{5} x^{5} e^{3} d b a^{2} A + \frac {1}{5} x^{5} e^{4} a^{3} A + \frac {3}{4} x^{4} d^{4} b^{2} a B + 3 x^{4} e d^{3} b a^{2} B + \frac {3}{2} x^{4} e^{2} d^{2} a^{3} B + \frac {1}{4} x^{4} d^{4} b^{3} A + 3 x^{4} e d^{3} b^{2} a A + \frac {9}{2} x^{4} e^{2} d^{2} b a^{2} A + x^{4} e^{3} d a^{3} A + x^{3} d^{4} b a^{2} B + \frac {4}{3} x^{3} e d^{3} a^{3} B + x^{3} d^{4} b^{2} a A + 4 x^{3} e d^{3} b a^{2} A + 2 x^{3} e^{2} d^{2} a^{3} A + \frac {1}{2} x^{2} d^{4} a^{3} B + \frac {3}{2} x^{2} d^{4} b a^{2} A + 2 x^{2} e d^{3} a^{3} A + x d^{4} 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)^4,x, algorithm="fricas")

[Out]

1/9*x^9*e^4*b^3*B + 1/2*x^8*e^3*d*b^3*B + 3/8*x^8*e^4*b^2*a*B + 1/8*x^8*e^4*b^3*A + 6/7*x^7*e^2*d^2*b^3*B + 12
/7*x^7*e^3*d*b^2*a*B + 3/7*x^7*e^4*b*a^2*B + 4/7*x^7*e^3*d*b^3*A + 3/7*x^7*e^4*b^2*a*A + 2/3*x^6*e*d^3*b^3*B +
3*x^6*e^2*d^2*b^2*a*B + 2*x^6*e^3*d*b*a^2*B + 1/6*x^6*e^4*a^3*B + x^6*e^2*d^2*b^3*A + 2*x^6*e^3*d*b^2*a*A + 1
/2*x^6*e^4*b*a^2*A + 1/5*x^5*d^4*b^3*B + 12/5*x^5*e*d^3*b^2*a*B + 18/5*x^5*e^2*d^2*b*a^2*B + 4/5*x^5*e^3*d*a^3
*B + 4/5*x^5*e*d^3*b^3*A + 18/5*x^5*e^2*d^2*b^2*a*A + 12/5*x^5*e^3*d*b*a^2*A + 1/5*x^5*e^4*a^3*A + 3/4*x^4*d^4
*b^2*a*B + 3*x^4*e*d^3*b*a^2*B + 3/2*x^4*e^2*d^2*a^3*B + 1/4*x^4*d^4*b^3*A + 3*x^4*e*d^3*b^2*a*A + 9/2*x^4*e^2
*d^2*b*a^2*A + x^4*e^3*d*a^3*A + x^3*d^4*b*a^2*B + 4/3*x^3*e*d^3*a^3*B + x^3*d^4*b^2*a*A + 4*x^3*e*d^3*b*a^2*A
+ 2*x^3*e^2*d^2*a^3*A + 1/2*x^2*d^4*a^3*B + 3/2*x^2*d^4*b*a^2*A + 2*x^2*e*d^3*a^3*A + x*d^4*a^3*A

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giac [B]  time = 1.20, size = 518, normalized size = 3.18 \begin {gather*} \frac {1}{9} \, B b^{3} x^{9} e^{4} + \frac {1}{2} \, B b^{3} d x^{8} e^{3} + \frac {6}{7} \, B b^{3} d^{2} x^{7} e^{2} + \frac {2}{3} \, B b^{3} d^{3} x^{6} e + \frac {1}{5} \, B b^{3} d^{4} x^{5} + \frac {3}{8} \, B a b^{2} x^{8} e^{4} + \frac {1}{8} \, A b^{3} x^{8} e^{4} + \frac {12}{7} \, B a b^{2} d x^{7} e^{3} + \frac {4}{7} \, A b^{3} d x^{7} e^{3} + 3 \, B a b^{2} d^{2} x^{6} e^{2} + A b^{3} d^{2} x^{6} e^{2} + \frac {12}{5} \, B a b^{2} d^{3} x^{5} e + \frac {4}{5} \, A b^{3} d^{3} x^{5} e + \frac {3}{4} \, B a b^{2} d^{4} x^{4} + \frac {1}{4} \, A b^{3} d^{4} x^{4} + \frac {3}{7} \, B a^{2} b x^{7} e^{4} + \frac {3}{7} \, A a b^{2} x^{7} e^{4} + 2 \, B a^{2} b d x^{6} e^{3} + 2 \, A a b^{2} d x^{6} e^{3} + \frac {18}{5} \, B a^{2} b d^{2} x^{5} e^{2} + \frac {18}{5} \, A a b^{2} d^{2} x^{5} e^{2} + 3 \, B a^{2} b d^{3} x^{4} e + 3 \, A a b^{2} d^{3} x^{4} e + B a^{2} b d^{4} x^{3} + A a b^{2} d^{4} x^{3} + \frac {1}{6} \, B a^{3} x^{6} e^{4} + \frac {1}{2} \, A a^{2} b x^{6} e^{4} + \frac {4}{5} \, B a^{3} d x^{5} e^{3} + \frac {12}{5} \, A a^{2} b d x^{5} e^{3} + \frac {3}{2} \, B a^{3} d^{2} x^{4} e^{2} + \frac {9}{2} \, A a^{2} b d^{2} x^{4} e^{2} + \frac {4}{3} \, B a^{3} d^{3} x^{3} e + 4 \, A a^{2} b d^{3} x^{3} e + \frac {1}{2} \, B a^{3} d^{4} x^{2} + \frac {3}{2} \, A a^{2} b d^{4} x^{2} + \frac {1}{5} \, A a^{3} x^{5} e^{4} + A a^{3} d x^{4} e^{3} + 2 \, A a^{3} d^{2} x^{3} e^{2} + 2 \, A a^{3} d^{3} x^{2} e + A a^{3} d^{4} 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)^4,x, algorithm="giac")

[Out]

1/9*B*b^3*x^9*e^4 + 1/2*B*b^3*d*x^8*e^3 + 6/7*B*b^3*d^2*x^7*e^2 + 2/3*B*b^3*d^3*x^6*e + 1/5*B*b^3*d^4*x^5 + 3/
8*B*a*b^2*x^8*e^4 + 1/8*A*b^3*x^8*e^4 + 12/7*B*a*b^2*d*x^7*e^3 + 4/7*A*b^3*d*x^7*e^3 + 3*B*a*b^2*d^2*x^6*e^2 +
A*b^3*d^2*x^6*e^2 + 12/5*B*a*b^2*d^3*x^5*e + 4/5*A*b^3*d^3*x^5*e + 3/4*B*a*b^2*d^4*x^4 + 1/4*A*b^3*d^4*x^4 +
3/7*B*a^2*b*x^7*e^4 + 3/7*A*a*b^2*x^7*e^4 + 2*B*a^2*b*d*x^6*e^3 + 2*A*a*b^2*d*x^6*e^3 + 18/5*B*a^2*b*d^2*x^5*e
^2 + 18/5*A*a*b^2*d^2*x^5*e^2 + 3*B*a^2*b*d^3*x^4*e + 3*A*a*b^2*d^3*x^4*e + B*a^2*b*d^4*x^3 + A*a*b^2*d^4*x^3
+ 1/6*B*a^3*x^6*e^4 + 1/2*A*a^2*b*x^6*e^4 + 4/5*B*a^3*d*x^5*e^3 + 12/5*A*a^2*b*d*x^5*e^3 + 3/2*B*a^3*d^2*x^4*e
^2 + 9/2*A*a^2*b*d^2*x^4*e^2 + 4/3*B*a^3*d^3*x^3*e + 4*A*a^2*b*d^3*x^3*e + 1/2*B*a^3*d^4*x^2 + 3/2*A*a^2*b*d^4
*x^2 + 1/5*A*a^3*x^5*e^4 + A*a^3*d*x^4*e^3 + 2*A*a^3*d^2*x^3*e^2 + 2*A*a^3*d^3*x^2*e + A*a^3*d^4*x

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maple [B]  time = 0.00, size = 434, normalized size = 2.66 \begin {gather*} \frac {B \,b^{3} e^{4} x^{9}}{9}+A \,a^{3} d^{4} x +\frac {\left (4 B \,b^{3} d \,e^{3}+\left (b^{3} A +3 a \,b^{2} B \right ) e^{4}\right ) x^{8}}{8}+\frac {\left (6 B \,b^{3} d^{2} e^{2}+4 \left (b^{3} A +3 a \,b^{2} B \right ) d \,e^{3}+\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{4}\right ) x^{7}}{7}+\frac {\left (4 B \,b^{3} d^{3} e +6 \left (b^{3} A +3 a \,b^{2} B \right ) d^{2} e^{2}+4 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d \,e^{3}+\left (3 A \,a^{2} b +B \,a^{3}\right ) e^{4}\right ) x^{6}}{6}+\frac {\left (A \,a^{3} e^{4}+B \,b^{3} d^{4}+4 \left (b^{3} A +3 a \,b^{2} B \right ) d^{3} e +6 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2} e^{2}+4 \left (3 A \,a^{2} b +B \,a^{3}\right ) d \,e^{3}\right ) x^{5}}{5}+\frac {\left (4 A \,a^{3} d \,e^{3}+\left (b^{3} A +3 a \,b^{2} B \right ) d^{4}+4 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{3} e +6 \left (3 A \,a^{2} b +B \,a^{3}\right ) d^{2} e^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{3} d^{2} e^{2}+\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{4}+4 \left (3 A \,a^{2} b +B \,a^{3}\right ) d^{3} e \right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} d^{3} e +\left (3 A \,a^{2} b +B \,a^{3}\right ) d^{4}\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)^4,x)

[Out]

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

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maxima [B]  time = 0.65, size = 425, normalized size = 2.61 \begin {gather*} \frac {1}{9} \, B b^{3} e^{4} x^{9} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (4 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, B b^{3} d^{2} e^{2} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (4 \, B b^{3} d^{3} e + 6 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{4} + A a^{3} e^{4} + 4 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 18 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e + 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{3} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} + 4 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{3} d^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\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)^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.17, size = 439, normalized size = 2.69 \begin {gather*} x^3\,\left (\frac {4\,B\,a^3\,d^3\,e}{3}+2\,A\,a^3\,d^2\,e^2+B\,a^2\,b\,d^4+4\,A\,a^2\,b\,d^3\,e+A\,a\,b^2\,d^4\right )+x^7\,\left (\frac {3\,B\,a^2\,b\,e^4}{7}+\frac {12\,B\,a\,b^2\,d\,e^3}{7}+\frac {3\,A\,a\,b^2\,e^4}{7}+\frac {6\,B\,b^3\,d^2\,e^2}{7}+\frac {4\,A\,b^3\,d\,e^3}{7}\right )+x^5\,\left (\frac {4\,B\,a^3\,d\,e^3}{5}+\frac {A\,a^3\,e^4}{5}+\frac {18\,B\,a^2\,b\,d^2\,e^2}{5}+\frac {12\,A\,a^2\,b\,d\,e^3}{5}+\frac {12\,B\,a\,b^2\,d^3\,e}{5}+\frac {18\,A\,a\,b^2\,d^2\,e^2}{5}+\frac {B\,b^3\,d^4}{5}+\frac {4\,A\,b^3\,d^3\,e}{5}\right )+x^4\,\left (\frac {3\,B\,a^3\,d^2\,e^2}{2}+A\,a^3\,d\,e^3+3\,B\,a^2\,b\,d^3\,e+\frac {9\,A\,a^2\,b\,d^2\,e^2}{2}+\frac {3\,B\,a\,b^2\,d^4}{4}+3\,A\,a\,b^2\,d^3\,e+\frac {A\,b^3\,d^4}{4}\right )+x^6\,\left (\frac {B\,a^3\,e^4}{6}+2\,B\,a^2\,b\,d\,e^3+\frac {A\,a^2\,b\,e^4}{2}+3\,B\,a\,b^2\,d^2\,e^2+2\,A\,a\,b^2\,d\,e^3+\frac {2\,B\,b^3\,d^3\,e}{3}+A\,b^3\,d^2\,e^2\right )+\frac {a^2\,d^3\,x^2\,\left (4\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e^3\,x^8\,\left (A\,b\,e+3\,B\,a\,e+4\,B\,b\,d\right )}{8}+A\,a^3\,d^4\,x+\frac {B\,b^3\,e^4\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^3*(d + e*x)^4,x)

[Out]

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

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sympy [B]  time = 0.14, size = 546, normalized size = 3.35 \begin {gather*} A a^{3} d^{4} x + \frac {B b^{3} e^{4} x^{9}}{9} + x^{8} \left (\frac {A b^{3} e^{4}}{8} + \frac {3 B a b^{2} e^{4}}{8} + \frac {B b^{3} d e^{3}}{2}\right ) + x^{7} \left (\frac {3 A a b^{2} e^{4}}{7} + \frac {4 A b^{3} d e^{3}}{7} + \frac {3 B a^{2} b e^{4}}{7} + \frac {12 B a b^{2} d e^{3}}{7} + \frac {6 B b^{3} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac {A a^{2} b e^{4}}{2} + 2 A a b^{2} d e^{3} + A b^{3} d^{2} e^{2} + \frac {B a^{3} e^{4}}{6} + 2 B a^{2} b d e^{3} + 3 B a b^{2} d^{2} e^{2} + \frac {2 B b^{3} d^{3} e}{3}\right ) + x^{5} \left (\frac {A a^{3} e^{4}}{5} + \frac {12 A a^{2} b d e^{3}}{5} + \frac {18 A a b^{2} d^{2} e^{2}}{5} + \frac {4 A b^{3} d^{3} e}{5} + \frac {4 B a^{3} d e^{3}}{5} + \frac {18 B a^{2} b d^{2} e^{2}}{5} + \frac {12 B a b^{2} d^{3} e}{5} + \frac {B b^{3} d^{4}}{5}\right ) + x^{4} \left (A a^{3} d e^{3} + \frac {9 A a^{2} b d^{2} e^{2}}{2} + 3 A a b^{2} d^{3} e + \frac {A b^{3} d^{4}}{4} + \frac {3 B a^{3} d^{2} e^{2}}{2} + 3 B a^{2} b d^{3} e + \frac {3 B a b^{2} d^{4}}{4}\right ) + x^{3} \left (2 A a^{3} d^{2} e^{2} + 4 A a^{2} b d^{3} e + A a b^{2} d^{4} + \frac {4 B a^{3} d^{3} e}{3} + B a^{2} b d^{4}\right ) + x^{2} \left (2 A a^{3} d^{3} e + \frac {3 A a^{2} b d^{4}}{2} + \frac {B a^{3} d^{4}}{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)**4,x)

[Out]

A*a**3*d**4*x + B*b**3*e**4*x**9/9 + x**8*(A*b**3*e**4/8 + 3*B*a*b**2*e**4/8 + B*b**3*d*e**3/2) + x**7*(3*A*a*
b**2*e**4/7 + 4*A*b**3*d*e**3/7 + 3*B*a**2*b*e**4/7 + 12*B*a*b**2*d*e**3/7 + 6*B*b**3*d**2*e**2/7) + x**6*(A*a
**2*b*e**4/2 + 2*A*a*b**2*d*e**3 + A*b**3*d**2*e**2 + B*a**3*e**4/6 + 2*B*a**2*b*d*e**3 + 3*B*a*b**2*d**2*e**2
+ 2*B*b**3*d**3*e/3) + x**5*(A*a**3*e**4/5 + 12*A*a**2*b*d*e**3/5 + 18*A*a*b**2*d**2*e**2/5 + 4*A*b**3*d**3*e
/5 + 4*B*a**3*d*e**3/5 + 18*B*a**2*b*d**2*e**2/5 + 12*B*a*b**2*d**3*e/5 + B*b**3*d**4/5) + x**4*(A*a**3*d*e**3
+ 9*A*a**2*b*d**2*e**2/2 + 3*A*a*b**2*d**3*e + A*b**3*d**4/4 + 3*B*a**3*d**2*e**2/2 + 3*B*a**2*b*d**3*e + 3*B
*a*b**2*d**4/4) + x**3*(2*A*a**3*d**2*e**2 + 4*A*a**2*b*d**3*e + A*a*b**2*d**4 + 4*B*a**3*d**3*e/3 + B*a**2*b*
d**4) + x**2*(2*A*a**3*d**3*e + 3*A*a**2*b*d**4/2 + B*a**3*d**4/2)

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