3.11.36 \(\int (a+b x)^3 (A+B x) (d+e x)^4 \, dx\) [1036]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.20, 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 = {78} \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 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)^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*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(397\) vs. \(2(163)=326\).
time = 0.09, size = 397, normalized size = 2.44 \begin {gather*} a^3 A d^4 x+\frac {1}{2} a^2 d^3 (3 A b d+a B d+4 a A e) x^2+\frac {1}{3} a d^2 \left (a B d (3 b d+4 a e)+3 A \left (b^2 d^2+4 a b d e+2 a^2 e^2\right )\right ) x^3+\frac {1}{4} d \left (3 a B d \left (b^2 d^2+4 a b d e+2 a^2 e^2\right )+A \left (b^3 d^3+12 a b^2 d^2 e+18 a^2 b d e^2+4 a^3 e^3\right )\right ) x^4+\frac {1}{5} \left (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)\right ) x^5+\frac {1}{6} e \left (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)\right ) x^6+\frac {1}{7} b e^2 \left (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)\right ) x^7+\frac {1}{8} b^2 e^3 (4 b B d+A b e+3 a B e) x^8+\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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(433\) vs. \(2(153)=306\).
time = 0.07, size = 434, normalized size = 2.66 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*e*d^3)*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)*e*d^3+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)*e*d^3+(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)*e*d^3+(3*A*a*b^2+3*B*a^2*b)*d^4)*x^3+1/2*(4*a
^3*A*e*d^3+(3*A*a^2*b+B*a^3)*d^4)*x^2+a^3*A*d^4*x

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (163) = 326\).
time = 0.28, size = 434, normalized size = 2.66 \begin {gather*} \frac {1}{9} \, B b^{3} x^{9} e^{4} + A a^{3} d^{4} x + \frac {1}{8} \, {\left (4 \, B b^{3} d e^{3} + 3 \, B a b^{2} e^{4} + A b^{3} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, B b^{3} d^{2} e^{2} + 3 \, B a^{2} b e^{4} + 3 \, A a b^{2} e^{4} + 4 \, {\left (3 \, B a b^{2} e^{3} + A b^{3} e^{3}\right )} d\right )} x^{7} + \frac {1}{6} \, {\left (4 \, B b^{3} d^{3} e + B a^{3} e^{4} + 3 \, A a^{2} b e^{4} + 6 \, {\left (3 \, B a b^{2} e^{2} + A b^{3} e^{2}\right )} d^{2} + 12 \, {\left (B a^{2} b e^{3} + A a b^{2} e^{3}\right )} d\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{4} + A a^{3} e^{4} + 4 \, {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{3} + 18 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d^{2} + 4 \, {\left (B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} d\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 e + A a b^{2} e\right )} d^{3} + 6 \, {\left (B a^{3} e^{2} + 3 \, A a^{2} b e^{2}\right )} d^{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} e + 3 \, A a^{2} b e\right )} d^{3}\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*x^9*e^4 + A*a^3*d^4*x + 1/8*(4*B*b^3*d*e^3 + 3*B*a*b^2*e^4 + A*b^3*e^4)*x^8 + 1/7*(6*B*b^3*d^2*e^2 +
3*B*a^2*b*e^4 + 3*A*a*b^2*e^4 + 4*(3*B*a*b^2*e^3 + A*b^3*e^3)*d)*x^7 + 1/6*(4*B*b^3*d^3*e + B*a^3*e^4 + 3*A*a
^2*b*e^4 + 6*(3*B*a*b^2*e^2 + A*b^3*e^2)*d^2 + 12*(B*a^2*b*e^3 + A*a*b^2*e^3)*d)*x^6 + 1/5*(B*b^3*d^4 + A*a^3*
e^4 + 4*(3*B*a*b^2*e + A*b^3*e)*d^3 + 18*(B*a^2*b*e^2 + A*a*b^2*e^2)*d^2 + 4*(B*a^3*e^3 + 3*A*a^2*b*e^3)*d)*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*e + A*a*b^2*e)*d^3 + 6*(B*a^3*e^2 + 3*A*a^2*b*e
^2)*d^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*e + 3*A*a^2*b*e)*d^3)*x^3 + 1/2*(4*
A*a^3*d^3*e + (B*a^3 + 3*A*a^2*b)*d^4)*x^2

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (163) = 326\).
time = 0.75, size = 427, normalized size = 2.62 \begin {gather*} \frac {1}{5} \, B b^{3} d^{4} x^{5} + A a^{3} d^{4} x + \frac {1}{4} \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} x^{4} + {\left (B a^{2} b + A a b^{2}\right )} d^{4} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} x^{2} + \frac {1}{2520} \, {\left (280 \, B b^{3} x^{9} + 504 \, A a^{3} x^{5} + 315 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{8} + 1080 \, {\left (B a^{2} b + A a b^{2}\right )} x^{7} + 420 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6}\right )} e^{4} + \frac {1}{70} \, {\left (35 \, B b^{3} d x^{8} + 70 \, A a^{3} d x^{4} + 40 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{7} + 140 \, {\left (B a^{2} b + A a b^{2}\right )} d x^{6} + 56 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d x^{5}\right )} e^{3} + \frac {1}{70} \, {\left (60 \, B b^{3} d^{2} x^{7} + 140 \, A a^{3} d^{2} x^{3} + 70 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x^{6} + 252 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} x^{5} + 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} x^{4}\right )} e^{2} + \frac {1}{15} \, {\left (10 \, B b^{3} d^{3} x^{6} + 30 \, A a^{3} d^{3} x^{2} + 12 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} x^{5} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} x^{4} + 20 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} x^{3}\right )} e \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]

1/5*B*b^3*d^4*x^5 + A*a^3*d^4*x + 1/4*(3*B*a*b^2 + A*b^3)*d^4*x^4 + (B*a^2*b + A*a*b^2)*d^4*x^3 + 1/2*(B*a^3 +
3*A*a^2*b)*d^4*x^2 + 1/2520*(280*B*b^3*x^9 + 504*A*a^3*x^5 + 315*(3*B*a*b^2 + A*b^3)*x^8 + 1080*(B*a^2*b + A*
a*b^2)*x^7 + 420*(B*a^3 + 3*A*a^2*b)*x^6)*e^4 + 1/70*(35*B*b^3*d*x^8 + 70*A*a^3*d*x^4 + 40*(3*B*a*b^2 + A*b^3)
*d*x^7 + 140*(B*a^2*b + A*a*b^2)*d*x^6 + 56*(B*a^3 + 3*A*a^2*b)*d*x^5)*e^3 + 1/70*(60*B*b^3*d^2*x^7 + 140*A*a^
3*d^2*x^3 + 70*(3*B*a*b^2 + A*b^3)*d^2*x^6 + 252*(B*a^2*b + A*a*b^2)*d^2*x^5 + 105*(B*a^3 + 3*A*a^2*b)*d^2*x^4
)*e^2 + 1/15*(10*B*b^3*d^3*x^6 + 30*A*a^3*d^3*x^2 + 12*(3*B*a*b^2 + A*b^3)*d^3*x^5 + 45*(B*a^2*b + A*a*b^2)*d^
3*x^4 + 20*(B*a^3 + 3*A*a^2*b)*d^3*x^3)*e

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (155) = 310\).
time = 0.04, 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} \cdot \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} \cdot \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} \cdot \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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (163) = 326\).
time = 4.18, 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

________________________________________________________________________________________

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

________________________________________________________________________________________