3.1505 \(\int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=240 \[ \frac{4 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac{(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^6}+\frac{2 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac{5 c^2 (d+e x)^8 (2 c d-b e)}{8 e^6}+\frac{2 c^3 (d+e x)^9}{9 e^6} \]

[Out]

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

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Rubi [A]  time = 0.795203, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{4 c (d+e x)^7 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac{(d+e x)^6 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{6 e^6}+\frac{2 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac{(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6}-\frac{5 c^2 (d+e x)^8 (2 c d-b e)}{8 e^6}+\frac{2 c^3 (d+e x)^9}{9 e^6} \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 92.4961, size = 236, normalized size = 0.98 \[ \frac{2 c^{3} \left (d + e x\right )^{9}}{9 e^{6}} + \frac{5 c^{2} \left (d + e x\right )^{8} \left (b e - 2 c d\right )}{8 e^{6}} + \frac{4 c \left (d + e x\right )^{7} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{6} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{6 e^{6}} + \frac{2 \left (d + e x\right )^{5} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{4} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{4 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.249231, size = 351, normalized size = 1.46 \[ \frac{1}{3} d x^3 \left (6 a^2 c d e+6 a b^2 d e+3 a b \left (a e^2+2 c d^2\right )+b^3 d^2\right )+a^2 b d^3 x+\frac{1}{6} x^6 \left (3 c^2 d e (4 a e+5 b d)+6 b c e^2 (a e+2 b d)+b^3 e^3+2 c^3 d^3\right )+\frac{1}{7} c e x^7 \left (c e (4 a e+15 b d)+4 b^2 e^2+6 c^2 d^2\right )+\frac{1}{2} a d^2 x^2 \left (3 a b e+2 a c d+2 b^2 d\right )+\frac{1}{5} x^5 \left (2 b^2 \left (a e^3+6 c d^2 e\right )+b c d \left (18 a e^2+5 c d^2\right )+2 a c e \left (a e^2+6 c d^2\right )+3 b^3 d e^2\right )+\frac{1}{4} x^4 \left (b^2 \left (6 a d e^2+4 c d^3\right )+a b e \left (a e^2+18 c d^2\right )+2 a c d \left (3 a e^2+2 c d^2\right )+3 b^3 d^2 e\right )+\frac{1}{8} c^2 e^2 x^8 (5 b e+6 c d)+\frac{2}{9} c^3 e^3 x^9 \]

Antiderivative was successfully verified.

[In]  Integrate[(b + 2*c*x)*(d + e*x)^3*(a + b*x + c*x^2)^2,x]

[Out]

a^2*b*d^3*x + (a*d^2*(2*b^2*d + 2*a*c*d + 3*a*b*e)*x^2)/2 + (d*(b^3*d^2 + 6*a*b^
2*d*e + 6*a^2*c*d*e + 3*a*b*(2*c*d^2 + a*e^2))*x^3)/3 + ((3*b^3*d^2*e + a*b*e*(1
8*c*d^2 + a*e^2) + 2*a*c*d*(2*c*d^2 + 3*a*e^2) + b^2*(4*c*d^3 + 6*a*d*e^2))*x^4)
/4 + ((3*b^3*d*e^2 + 2*a*c*e*(6*c*d^2 + a*e^2) + b*c*d*(5*c*d^2 + 18*a*e^2) + 2*
b^2*(6*c*d^2*e + a*e^3))*x^5)/5 + ((2*c^3*d^3 + b^3*e^3 + 6*b*c*e^2*(2*b*d + a*e
) + 3*c^2*d*e*(5*b*d + 4*a*e))*x^6)/6 + (c*e*(6*c^2*d^2 + 4*b^2*e^2 + c*e*(15*b*
d + 4*a*e))*x^7)/7 + (c^2*e^2*(6*c*d + 5*b*e)*x^8)/8 + (2*c^3*e^3*x^9)/9

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Maple [A]  time = 0.001, size = 428, normalized size = 1.8 \[{\frac{2\,{e}^{3}{c}^{3}{x}^{9}}{9}}+{\frac{ \left ( \left ( b{e}^{3}+6\,d{e}^{2}c \right ){c}^{2}+4\,b{e}^{3}{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,bd{e}^{2}+6\,{d}^{2}ec \right ){c}^{2}+2\, \left ( b{e}^{3}+6\,d{e}^{2}c \right ) bc+2\,{e}^{3}c \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ){c}^{2}+2\, \left ( 3\,bd{e}^{2}+6\,{d}^{2}ec \right ) bc+ \left ( b{e}^{3}+6\,d{e}^{2}c \right ) \left ( 2\,ac+{b}^{2} \right ) +4\,abc{e}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( b{d}^{3}{c}^{2}+2\, \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ) bc+ \left ( 3\,bd{e}^{2}+6\,{d}^{2}ec \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( b{e}^{3}+6\,d{e}^{2}c \right ) ab+2\,{e}^{3}c{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{2}{d}^{3}c+ \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 3\,bd{e}^{2}+6\,{d}^{2}ec \right ) ab+ \left ( b{e}^{3}+6\,d{e}^{2}c \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( b{d}^{3} \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ) ab+ \left ( 3\,bd{e}^{2}+6\,{d}^{2}ec \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{b}^{2}{d}^{3}a+ \left ( 3\,b{d}^{2}e+2\,c{d}^{3} \right ){a}^{2} \right ){x}^{2}}{2}}+b{d}^{3}{a}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.709257, size = 463, normalized size = 1.93 \[ \frac{2}{9} \, c^{3} e^{3} x^{9} + \frac{1}{8} \,{\left (6 \, c^{3} d e^{2} + 5 \, b c^{2} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (6 \, c^{3} d^{2} e + 15 \, b c^{2} d e^{2} + 4 \,{\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{7} + a^{2} b d^{3} x + \frac{1}{6} \,{\left (2 \, c^{3} d^{3} + 15 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} +{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (5 \, b c^{2} d^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e + 3 \,{\left (b^{3} + 6 \, a b c\right )} d e^{2} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (a^{2} b e^{3} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} + 3 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e + 6 \,{\left (a b^{2} + a^{2} c\right )} d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a^{2} b d e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{3} + 6 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{2} e + 2 \,{\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)*(e*x + d)^3,x, algorithm="maxima")

[Out]

2/9*c^3*e^3*x^9 + 1/8*(6*c^3*d*e^2 + 5*b*c^2*e^3)*x^8 + 1/7*(6*c^3*d^2*e + 15*b*
c^2*d*e^2 + 4*(b^2*c + a*c^2)*e^3)*x^7 + a^2*b*d^3*x + 1/6*(2*c^3*d^3 + 15*b*c^2
*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 + (b^3 + 6*a*b*c)*e^3)*x^6 + 1/5*(5*b*c^2*d^3
+ 12*(b^2*c + a*c^2)*d^2*e + 3*(b^3 + 6*a*b*c)*d*e^2 + 2*(a*b^2 + a^2*c)*e^3)*x^
5 + 1/4*(a^2*b*e^3 + 4*(b^2*c + a*c^2)*d^3 + 3*(b^3 + 6*a*b*c)*d^2*e + 6*(a*b^2
+ a^2*c)*d*e^2)*x^4 + 1/3*(3*a^2*b*d*e^2 + (b^3 + 6*a*b*c)*d^3 + 6*(a*b^2 + a^2*
c)*d^2*e)*x^3 + 1/2*(3*a^2*b*d^2*e + 2*(a*b^2 + a^2*c)*d^3)*x^2

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Fricas [A]  time = 0.240623, size = 1, normalized size = 0. \[ \frac{2}{9} x^{9} e^{3} c^{3} + \frac{3}{4} x^{8} e^{2} d c^{3} + \frac{5}{8} x^{8} e^{3} c^{2} b + \frac{6}{7} x^{7} e d^{2} c^{3} + \frac{15}{7} x^{7} e^{2} d c^{2} b + \frac{4}{7} x^{7} e^{3} c b^{2} + \frac{4}{7} x^{7} e^{3} c^{2} a + \frac{1}{3} x^{6} d^{3} c^{3} + \frac{5}{2} x^{6} e d^{2} c^{2} b + 2 x^{6} e^{2} d c b^{2} + \frac{1}{6} x^{6} e^{3} b^{3} + 2 x^{6} e^{2} d c^{2} a + x^{6} e^{3} c b a + x^{5} d^{3} c^{2} b + \frac{12}{5} x^{5} e d^{2} c b^{2} + \frac{3}{5} x^{5} e^{2} d b^{3} + \frac{12}{5} x^{5} e d^{2} c^{2} a + \frac{18}{5} x^{5} e^{2} d c b a + \frac{2}{5} x^{5} e^{3} b^{2} a + \frac{2}{5} x^{5} e^{3} c a^{2} + x^{4} d^{3} c b^{2} + \frac{3}{4} x^{4} e d^{2} b^{3} + x^{4} d^{3} c^{2} a + \frac{9}{2} x^{4} e d^{2} c b a + \frac{3}{2} x^{4} e^{2} d b^{2} a + \frac{3}{2} x^{4} e^{2} d c a^{2} + \frac{1}{4} x^{4} e^{3} b a^{2} + \frac{1}{3} x^{3} d^{3} b^{3} + 2 x^{3} d^{3} c b a + 2 x^{3} e d^{2} b^{2} a + 2 x^{3} e d^{2} c a^{2} + x^{3} e^{2} d b a^{2} + x^{2} d^{3} b^{2} a + x^{2} d^{3} c a^{2} + \frac{3}{2} x^{2} e d^{2} b a^{2} + x d^{3} b a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)*(e*x + d)^3,x, algorithm="fricas")

[Out]

2/9*x^9*e^3*c^3 + 3/4*x^8*e^2*d*c^3 + 5/8*x^8*e^3*c^2*b + 6/7*x^7*e*d^2*c^3 + 15
/7*x^7*e^2*d*c^2*b + 4/7*x^7*e^3*c*b^2 + 4/7*x^7*e^3*c^2*a + 1/3*x^6*d^3*c^3 + 5
/2*x^6*e*d^2*c^2*b + 2*x^6*e^2*d*c*b^2 + 1/6*x^6*e^3*b^3 + 2*x^6*e^2*d*c^2*a + x
^6*e^3*c*b*a + x^5*d^3*c^2*b + 12/5*x^5*e*d^2*c*b^2 + 3/5*x^5*e^2*d*b^3 + 12/5*x
^5*e*d^2*c^2*a + 18/5*x^5*e^2*d*c*b*a + 2/5*x^5*e^3*b^2*a + 2/5*x^5*e^3*c*a^2 +
x^4*d^3*c*b^2 + 3/4*x^4*e*d^2*b^3 + x^4*d^3*c^2*a + 9/2*x^4*e*d^2*c*b*a + 3/2*x^
4*e^2*d*b^2*a + 3/2*x^4*e^2*d*c*a^2 + 1/4*x^4*e^3*b*a^2 + 1/3*x^3*d^3*b^3 + 2*x^
3*d^3*c*b*a + 2*x^3*e*d^2*b^2*a + 2*x^3*e*d^2*c*a^2 + x^3*e^2*d*b*a^2 + x^2*d^3*
b^2*a + x^2*d^3*c*a^2 + 3/2*x^2*e*d^2*b*a^2 + x*d^3*b*a^2

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Sympy [A]  time = 0.302114, size = 430, normalized size = 1.79 \[ a^{2} b d^{3} x + \frac{2 c^{3} e^{3} x^{9}}{9} + x^{8} \left (\frac{5 b c^{2} e^{3}}{8} + \frac{3 c^{3} d e^{2}}{4}\right ) + x^{7} \left (\frac{4 a c^{2} e^{3}}{7} + \frac{4 b^{2} c e^{3}}{7} + \frac{15 b c^{2} d e^{2}}{7} + \frac{6 c^{3} d^{2} e}{7}\right ) + x^{6} \left (a b c e^{3} + 2 a c^{2} d e^{2} + \frac{b^{3} e^{3}}{6} + 2 b^{2} c d e^{2} + \frac{5 b c^{2} d^{2} e}{2} + \frac{c^{3} d^{3}}{3}\right ) + x^{5} \left (\frac{2 a^{2} c e^{3}}{5} + \frac{2 a b^{2} e^{3}}{5} + \frac{18 a b c d e^{2}}{5} + \frac{12 a c^{2} d^{2} e}{5} + \frac{3 b^{3} d e^{2}}{5} + \frac{12 b^{2} c d^{2} e}{5} + b c^{2} d^{3}\right ) + x^{4} \left (\frac{a^{2} b e^{3}}{4} + \frac{3 a^{2} c d e^{2}}{2} + \frac{3 a b^{2} d e^{2}}{2} + \frac{9 a b c d^{2} e}{2} + a c^{2} d^{3} + \frac{3 b^{3} d^{2} e}{4} + b^{2} c d^{3}\right ) + x^{3} \left (a^{2} b d e^{2} + 2 a^{2} c d^{2} e + 2 a b^{2} d^{2} e + 2 a b c d^{3} + \frac{b^{3} d^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{2} b d^{2} e}{2} + a^{2} c d^{3} + a b^{2} d^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a)**2,x)

[Out]

a**2*b*d**3*x + 2*c**3*e**3*x**9/9 + x**8*(5*b*c**2*e**3/8 + 3*c**3*d*e**2/4) +
x**7*(4*a*c**2*e**3/7 + 4*b**2*c*e**3/7 + 15*b*c**2*d*e**2/7 + 6*c**3*d**2*e/7)
+ x**6*(a*b*c*e**3 + 2*a*c**2*d*e**2 + b**3*e**3/6 + 2*b**2*c*d*e**2 + 5*b*c**2*
d**2*e/2 + c**3*d**3/3) + x**5*(2*a**2*c*e**3/5 + 2*a*b**2*e**3/5 + 18*a*b*c*d*e
**2/5 + 12*a*c**2*d**2*e/5 + 3*b**3*d*e**2/5 + 12*b**2*c*d**2*e/5 + b*c**2*d**3)
+ x**4*(a**2*b*e**3/4 + 3*a**2*c*d*e**2/2 + 3*a*b**2*d*e**2/2 + 9*a*b*c*d**2*e/
2 + a*c**2*d**3 + 3*b**3*d**2*e/4 + b**2*c*d**3) + x**3*(a**2*b*d*e**2 + 2*a**2*
c*d**2*e + 2*a*b**2*d**2*e + 2*a*b*c*d**3 + b**3*d**3/3) + x**2*(3*a**2*b*d**2*e
/2 + a**2*c*d**3 + a*b**2*d**3)

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GIAC/XCAS [A]  time = 0.269771, size = 567, normalized size = 2.36 \[ \frac{2}{9} \, c^{3} x^{9} e^{3} + \frac{3}{4} \, c^{3} d x^{8} e^{2} + \frac{6}{7} \, c^{3} d^{2} x^{7} e + \frac{1}{3} \, c^{3} d^{3} x^{6} + \frac{5}{8} \, b c^{2} x^{8} e^{3} + \frac{15}{7} \, b c^{2} d x^{7} e^{2} + \frac{5}{2} \, b c^{2} d^{2} x^{6} e + b c^{2} d^{3} x^{5} + \frac{4}{7} \, b^{2} c x^{7} e^{3} + \frac{4}{7} \, a c^{2} x^{7} e^{3} + 2 \, b^{2} c d x^{6} e^{2} + 2 \, a c^{2} d x^{6} e^{2} + \frac{12}{5} \, b^{2} c d^{2} x^{5} e + \frac{12}{5} \, a c^{2} d^{2} x^{5} e + b^{2} c d^{3} x^{4} + a c^{2} d^{3} x^{4} + \frac{1}{6} \, b^{3} x^{6} e^{3} + a b c x^{6} e^{3} + \frac{3}{5} \, b^{3} d x^{5} e^{2} + \frac{18}{5} \, a b c d x^{5} e^{2} + \frac{3}{4} \, b^{3} d^{2} x^{4} e + \frac{9}{2} \, a b c d^{2} x^{4} e + \frac{1}{3} \, b^{3} d^{3} x^{3} + 2 \, a b c d^{3} x^{3} + \frac{2}{5} \, a b^{2} x^{5} e^{3} + \frac{2}{5} \, a^{2} c x^{5} e^{3} + \frac{3}{2} \, a b^{2} d x^{4} e^{2} + \frac{3}{2} \, a^{2} c d x^{4} e^{2} + 2 \, a b^{2} d^{2} x^{3} e + 2 \, a^{2} c d^{2} x^{3} e + a b^{2} d^{3} x^{2} + a^{2} c d^{3} x^{2} + \frac{1}{4} \, a^{2} b x^{4} e^{3} + a^{2} b d x^{3} e^{2} + \frac{3}{2} \, a^{2} b d^{2} x^{2} e + a^{2} b d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)*(e*x + d)^3,x, algorithm="giac")

[Out]

2/9*c^3*x^9*e^3 + 3/4*c^3*d*x^8*e^2 + 6/7*c^3*d^2*x^7*e + 1/3*c^3*d^3*x^6 + 5/8*
b*c^2*x^8*e^3 + 15/7*b*c^2*d*x^7*e^2 + 5/2*b*c^2*d^2*x^6*e + b*c^2*d^3*x^5 + 4/7
*b^2*c*x^7*e^3 + 4/7*a*c^2*x^7*e^3 + 2*b^2*c*d*x^6*e^2 + 2*a*c^2*d*x^6*e^2 + 12/
5*b^2*c*d^2*x^5*e + 12/5*a*c^2*d^2*x^5*e + b^2*c*d^3*x^4 + a*c^2*d^3*x^4 + 1/6*b
^3*x^6*e^3 + a*b*c*x^6*e^3 + 3/5*b^3*d*x^5*e^2 + 18/5*a*b*c*d*x^5*e^2 + 3/4*b^3*
d^2*x^4*e + 9/2*a*b*c*d^2*x^4*e + 1/3*b^3*d^3*x^3 + 2*a*b*c*d^3*x^3 + 2/5*a*b^2*
x^5*e^3 + 2/5*a^2*c*x^5*e^3 + 3/2*a*b^2*d*x^4*e^2 + 3/2*a^2*c*d*x^4*e^2 + 2*a*b^
2*d^2*x^3*e + 2*a^2*c*d^2*x^3*e + a*b^2*d^3*x^2 + a^2*c*d^3*x^2 + 1/4*a^2*b*x^4*
e^3 + a^2*b*d*x^3*e^2 + 3/2*a^2*b*d^2*x^2*e + a^2*b*d^3*x