3.1019 \(\int (a+b x)^3 (A+B x) (d+e x)^4 \, dx\)

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

[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)

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Rubi [A]  time = 0.844321, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{b^2 (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5}+\frac{3 b (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac{(d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5}+\frac{(d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5}+\frac{b^3 B (d+e x)^9}{9 e^5} \]

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)

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Rubi in Sympy [A]  time = 80.5366, size = 155, normalized size = 0.95 \[ \frac{B b^{3} \left (d + e x\right )^{9}}{9 e^{5}} + \frac{b^{2} \left (d + e x\right )^{8} \left (A b e + 3 B a e - 4 B b d\right )}{8 e^{5}} + \frac{3 b \left (d + e x\right )^{7} \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{7 e^{5}} + \frac{\left (d + e x\right )^{6} \left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{6 e^{5}} + \frac{\left (d + e x\right )^{5} \left (A e - B d\right ) \left (a e - b d\right )^{3}}{5 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.280052, size = 397, normalized size = 2.44 \[ a^3 A d^4 x+\frac{1}{3} a d^2 x^3 \left (3 A \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+a B d (4 a e+3 b d)\right )+\frac{1}{7} b e^2 x^7 \left (3 a^2 B e^2+3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+\frac{1}{2} a^2 d^3 x^2 (4 a A e+a B d+3 A b d)+\frac{1}{6} e x^6 \left (a^3 B e^3+3 a^2 b e^2 (A e+4 B d)+6 a b^2 d e (2 A e+3 B d)+2 b^3 d^2 (3 A e+2 B d)\right )+\frac{1}{5} x^5 \left (a^3 e^3 (A e+4 B d)+6 a^2 b d e^2 (2 A e+3 B d)+6 a b^2 d^2 e (3 A e+2 B d)+b^3 d^3 (4 A e+B d)\right )+\frac{1}{4} d x^4 \left (3 a B d \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+A \left (4 a^3 e^3+18 a^2 b d e^2+12 a b^2 d^2 e+b^3 d^3\right )\right )+\frac{1}{8} b^2 e^3 x^8 (3 a B e+A b e+4 b B d)+\frac{1}{9} b^3 B e^4 x^9 \]

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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Maple [B]  time = 0.001, size = 434, normalized size = 2.7 \[{\frac{{b}^{3}B{e}^{4}{x}^{9}}{9}}+{\frac{ \left ( \left ({b}^{3}A+3\,a{b}^{2}B \right ){e}^{4}+4\,{b}^{3}Bd{e}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){e}^{4}+4\, \left ({b}^{3}A+3\,a{b}^{2}B \right ) d{e}^{3}+6\,{b}^{3}B{d}^{2}{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){e}^{4}+4\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ) d{e}^{3}+6\, \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{2}{e}^{2}+4\,{b}^{3}B{d}^{3}e \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{3}A{e}^{4}+4\, \left ( 3\,A{a}^{2}b+B{a}^{3} \right ) d{e}^{3}+6\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{2}{e}^{2}+4\, \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{3}e+{b}^{3}B{d}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{3}Ad{e}^{3}+6\, \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){d}^{2}{e}^{2}+4\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{3}e+ \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{a}^{3}A{d}^{2}{e}^{2}+4\, \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){d}^{3}e+ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{3}A{d}^{3}e+ \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){d}^{4} \right ){x}^{2}}{2}}+{a}^{3}A{d}^{4}x \]

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 [A]  time = 1.3632, size = 574, normalized size = 3.52 \[ \frac{1}{9} \, B b^{3} e^{4} x^{9} + A a^{3} d^{4} x + \frac{1}{8} \,{\left (4 \, B b^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (6 \, B b^{3} d^{2} e^{2} + 4 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (4 \, B b^{3} d^{3} e + 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{4} + A a^{3} e^{4} + 4 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 18 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 4 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, A a^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e + 6 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, A a^{3} d^{2} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} + 4 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, A a^{3} d^{3} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(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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Fricas [A]  time = 0.194733, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e^{4} b^{3} B + \frac{1}{2} x^{8} e^{3} d b^{3} B + \frac{3}{8} x^{8} e^{4} b^{2} a B + \frac{1}{8} x^{8} e^{4} b^{3} A + \frac{6}{7} x^{7} e^{2} d^{2} b^{3} B + \frac{12}{7} x^{7} e^{3} d b^{2} a B + \frac{3}{7} x^{7} e^{4} b a^{2} B + \frac{4}{7} x^{7} e^{3} d b^{3} A + \frac{3}{7} x^{7} e^{4} b^{2} a A + \frac{2}{3} x^{6} e d^{3} b^{3} B + 3 x^{6} e^{2} d^{2} b^{2} a B + 2 x^{6} e^{3} d b a^{2} B + \frac{1}{6} x^{6} e^{4} a^{3} B + x^{6} e^{2} d^{2} b^{3} A + 2 x^{6} e^{3} d b^{2} a A + \frac{1}{2} x^{6} e^{4} b a^{2} A + \frac{1}{5} x^{5} d^{4} b^{3} B + \frac{12}{5} x^{5} e d^{3} b^{2} a B + \frac{18}{5} x^{5} e^{2} d^{2} b a^{2} B + \frac{4}{5} x^{5} e^{3} d a^{3} B + \frac{4}{5} x^{5} e d^{3} b^{3} A + \frac{18}{5} x^{5} e^{2} d^{2} b^{2} a A + \frac{12}{5} x^{5} e^{3} d b a^{2} A + \frac{1}{5} x^{5} e^{4} a^{3} A + \frac{3}{4} x^{4} d^{4} b^{2} a B + 3 x^{4} e d^{3} b a^{2} B + \frac{3}{2} x^{4} e^{2} d^{2} a^{3} B + \frac{1}{4} x^{4} d^{4} b^{3} A + 3 x^{4} e d^{3} b^{2} a A + \frac{9}{2} x^{4} e^{2} d^{2} b a^{2} A + x^{4} e^{3} d a^{3} A + x^{3} d^{4} b a^{2} B + \frac{4}{3} x^{3} e d^{3} a^{3} B + x^{3} d^{4} b^{2} a A + 4 x^{3} e d^{3} b a^{2} A + 2 x^{3} e^{2} d^{2} a^{3} A + \frac{1}{2} x^{2} d^{4} a^{3} B + \frac{3}{2} x^{2} d^{4} b a^{2} A + 2 x^{2} e d^{3} a^{3} A + x d^{4} a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(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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Sympy [A]  time = 0.322296, size = 546, normalized size = 3.35 \[ A a^{3} d^{4} x + \frac{B b^{3} e^{4} x^{9}}{9} + x^{8} \left (\frac{A b^{3} e^{4}}{8} + \frac{3 B a b^{2} e^{4}}{8} + \frac{B b^{3} d e^{3}}{2}\right ) + x^{7} \left (\frac{3 A a b^{2} e^{4}}{7} + \frac{4 A b^{3} d e^{3}}{7} + \frac{3 B a^{2} b e^{4}}{7} + \frac{12 B a b^{2} d e^{3}}{7} + \frac{6 B b^{3} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac{A a^{2} b e^{4}}{2} + 2 A a b^{2} d e^{3} + A b^{3} d^{2} e^{2} + \frac{B a^{3} e^{4}}{6} + 2 B a^{2} b d e^{3} + 3 B a b^{2} d^{2} e^{2} + \frac{2 B b^{3} d^{3} e}{3}\right ) + x^{5} \left (\frac{A a^{3} e^{4}}{5} + \frac{12 A a^{2} b d e^{3}}{5} + \frac{18 A a b^{2} d^{2} e^{2}}{5} + \frac{4 A b^{3} d^{3} e}{5} + \frac{4 B a^{3} d e^{3}}{5} + \frac{18 B a^{2} b d^{2} e^{2}}{5} + \frac{12 B a b^{2} d^{3} e}{5} + \frac{B b^{3} d^{4}}{5}\right ) + x^{4} \left (A a^{3} d e^{3} + \frac{9 A a^{2} b d^{2} e^{2}}{2} + 3 A a b^{2} d^{3} e + \frac{A b^{3} d^{4}}{4} + \frac{3 B a^{3} d^{2} e^{2}}{2} + 3 B a^{2} b d^{3} e + \frac{3 B a b^{2} d^{4}}{4}\right ) + x^{3} \left (2 A a^{3} d^{2} e^{2} + 4 A a^{2} b d^{3} e + A a b^{2} d^{4} + \frac{4 B a^{3} d^{3} e}{3} + B a^{2} b d^{4}\right ) + x^{2} \left (2 A a^{3} d^{3} e + \frac{3 A a^{2} b d^{4}}{2} + \frac{B a^{3} d^{4}}{2}\right ) \]

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 + 1
8*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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GIAC/XCAS [A]  time = 0.221113, size = 699, normalized size = 4.29 \[ \frac{1}{9} \, B b^{3} x^{9} e^{4} + \frac{1}{2} \, B b^{3} d x^{8} e^{3} + \frac{6}{7} \, B b^{3} d^{2} x^{7} e^{2} + \frac{2}{3} \, B b^{3} d^{3} x^{6} e + \frac{1}{5} \, B b^{3} d^{4} x^{5} + \frac{3}{8} \, B a b^{2} x^{8} e^{4} + \frac{1}{8} \, A b^{3} x^{8} e^{4} + \frac{12}{7} \, B a b^{2} d x^{7} e^{3} + \frac{4}{7} \, A b^{3} d x^{7} e^{3} + 3 \, B a b^{2} d^{2} x^{6} e^{2} + A b^{3} d^{2} x^{6} e^{2} + \frac{12}{5} \, B a b^{2} d^{3} x^{5} e + \frac{4}{5} \, A b^{3} d^{3} x^{5} e + \frac{3}{4} \, B a b^{2} d^{4} x^{4} + \frac{1}{4} \, A b^{3} d^{4} x^{4} + \frac{3}{7} \, B a^{2} b x^{7} e^{4} + \frac{3}{7} \, A a b^{2} x^{7} e^{4} + 2 \, B a^{2} b d x^{6} e^{3} + 2 \, A a b^{2} d x^{6} e^{3} + \frac{18}{5} \, B a^{2} b d^{2} x^{5} e^{2} + \frac{18}{5} \, A a b^{2} d^{2} x^{5} e^{2} + 3 \, B a^{2} b d^{3} x^{4} e + 3 \, A a b^{2} d^{3} x^{4} e + B a^{2} b d^{4} x^{3} + A a b^{2} d^{4} x^{3} + \frac{1}{6} \, B a^{3} x^{6} e^{4} + \frac{1}{2} \, A a^{2} b x^{6} e^{4} + \frac{4}{5} \, B a^{3} d x^{5} e^{3} + \frac{12}{5} \, A a^{2} b d x^{5} e^{3} + \frac{3}{2} \, B a^{3} d^{2} x^{4} e^{2} + \frac{9}{2} \, A a^{2} b d^{2} x^{4} e^{2} + \frac{4}{3} \, B a^{3} d^{3} x^{3} e + 4 \, A a^{2} b d^{3} x^{3} e + \frac{1}{2} \, B a^{3} d^{4} x^{2} + \frac{3}{2} \, A a^{2} b d^{4} x^{2} + \frac{1}{5} \, A a^{3} x^{5} e^{4} + A a^{3} d x^{4} e^{3} + 2 \, A a^{3} d^{2} x^{3} e^{2} + 2 \, A a^{3} d^{3} x^{2} e + A a^{3} d^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(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