3.1506 \(\int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.637643, 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 \[ \frac{2 c (d+e x)^6 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac{(d+e x)^5 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^6}-\frac{(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac{5 c^2 (d+e x)^7 (2 c d-b e)}{7 e^6}+\frac{c^3 (d+e x)^8}{4 e^6} \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x)*(d + e*x)^2*(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)^3)/(3*e^6) + ((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)^4)/(2*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)^5)/(5*e^6) +
(2*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^6)/(3*e^6) - (5*c^2*(2*
c*d - b*e)*(d + e*x)^7)/(7*e^6) + (c^3*(d + e*x)^8)/(4*e^6)

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Rubi in Sympy [A]  time = 88.3889, size = 233, normalized size = 0.97 \[ \frac{c^{3} \left (d + e x\right )^{8}}{4 e^{6}} + \frac{5 c^{2} \left (d + e x\right )^{7} \left (b e - 2 c d\right )}{7 e^{6}} + \frac{2 c \left (d + e x\right )^{6} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{3 e^{6}} + \frac{\left (d + e x\right )^{5} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{5 e^{6}} + \frac{\left (d + e x\right )^{4} \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{2 e^{6}} + \frac{\left (d + e x\right )^{3} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{3 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.148205, size = 244, normalized size = 1.02 \[ \frac{1}{3} x^3 \left (4 a^2 c d e+4 a b^2 d e+a b \left (a e^2+6 c d^2\right )+b^3 d^2\right )+a^2 b d^2 x+\frac{1}{3} c x^6 \left (c e (2 a e+5 b d)+2 b^2 e^2+c^2 d^2\right )+a d x^2 \left (a b e+a c d+b^2 d\right )+\frac{1}{5} x^5 \left (b c \left (6 a e^2+5 c d^2\right )+8 a c^2 d e+b^3 e^2+8 b^2 c d e\right )+\frac{1}{2} x^4 \left (b^2 \left (a e^2+2 c d^2\right )+6 a b c d e+a c \left (a e^2+2 c d^2\right )+b^3 d e\right )+\frac{1}{7} c^2 e x^7 (5 b e+4 c d)+\frac{1}{4} c^3 e^2 x^8 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 302, normalized size = 1.3 \[{\frac{{c}^{3}{e}^{2}{x}^{8}}{4}}+{\frac{ \left ( \left ( b{e}^{2}+4\,dec \right ){c}^{2}+4\,{c}^{2}{e}^{2}b \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,bde+2\,c{d}^{2} \right ){c}^{2}+2\, \left ( b{e}^{2}+4\,dec \right ) bc+2\,c{e}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( b{c}^{2}{d}^{2}+2\, \left ( 2\,bde+2\,c{d}^{2} \right ) bc+ \left ( b{e}^{2}+4\,dec \right ) \left ( 2\,ac+{b}^{2} \right ) +4\,c{e}^{2}ab \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,c{b}^{2}{d}^{2}+ \left ( 2\,bde+2\,c{d}^{2} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( b{e}^{2}+4\,dec \right ) ab+2\,c{e}^{2}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( b{d}^{2} \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 2\,bde+2\,c{d}^{2} \right ) ab+ \left ( b{e}^{2}+4\,dec \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{b}^{2}{d}^{2}a+ \left ( 2\,bde+2\,c{d}^{2} \right ){a}^{2} \right ){x}^{2}}{2}}+b{d}^{2}{a}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.706907, size = 325, normalized size = 1.35 \[ \frac{1}{4} \, c^{3} e^{2} x^{8} + \frac{1}{7} \,{\left (4 \, c^{3} d e + 5 \, b c^{2} e^{2}\right )} x^{7} + \frac{1}{3} \,{\left (c^{3} d^{2} + 5 \, b c^{2} d e + 2 \,{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x^{6} + a^{2} b d^{2} x + \frac{1}{5} \,{\left (5 \, b c^{2} d^{2} + 8 \,{\left (b^{2} c + a c^{2}\right )} d e +{\left (b^{3} + 6 \, a b c\right )} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (2 \,{\left (b^{2} c + a c^{2}\right )} d^{2} +{\left (b^{3} + 6 \, a b c\right )} d e +{\left (a b^{2} + a^{2} c\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (a^{2} b e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{2} + 4 \,{\left (a b^{2} + a^{2} c\right )} d e\right )} x^{3} +{\left (a^{2} b d e +{\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{2} \]

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.244205, size = 1, normalized size = 0. \[ \frac{1}{4} x^{8} e^{2} c^{3} + \frac{4}{7} x^{7} e d c^{3} + \frac{5}{7} x^{7} e^{2} c^{2} b + \frac{1}{3} x^{6} d^{2} c^{3} + \frac{5}{3} x^{6} e d c^{2} b + \frac{2}{3} x^{6} e^{2} c b^{2} + \frac{2}{3} x^{6} e^{2} c^{2} a + x^{5} d^{2} c^{2} b + \frac{8}{5} x^{5} e d c b^{2} + \frac{1}{5} x^{5} e^{2} b^{3} + \frac{8}{5} x^{5} e d c^{2} a + \frac{6}{5} x^{5} e^{2} c b a + x^{4} d^{2} c b^{2} + \frac{1}{2} x^{4} e d b^{3} + x^{4} d^{2} c^{2} a + 3 x^{4} e d c b a + \frac{1}{2} x^{4} e^{2} b^{2} a + \frac{1}{2} x^{4} e^{2} c a^{2} + \frac{1}{3} x^{3} d^{2} b^{3} + 2 x^{3} d^{2} c b a + \frac{4}{3} x^{3} e d b^{2} a + \frac{4}{3} x^{3} e d c a^{2} + \frac{1}{3} x^{3} e^{2} b a^{2} + x^{2} d^{2} b^{2} a + x^{2} d^{2} c a^{2} + x^{2} e d b a^{2} + x d^{2} b a^{2} \]

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.253206, size = 294, normalized size = 1.22 \[ a^{2} b d^{2} x + \frac{c^{3} e^{2} x^{8}}{4} + x^{7} \left (\frac{5 b c^{2} e^{2}}{7} + \frac{4 c^{3} d e}{7}\right ) + x^{6} \left (\frac{2 a c^{2} e^{2}}{3} + \frac{2 b^{2} c e^{2}}{3} + \frac{5 b c^{2} d e}{3} + \frac{c^{3} d^{2}}{3}\right ) + x^{5} \left (\frac{6 a b c e^{2}}{5} + \frac{8 a c^{2} d e}{5} + \frac{b^{3} e^{2}}{5} + \frac{8 b^{2} c d e}{5} + b c^{2} d^{2}\right ) + x^{4} \left (\frac{a^{2} c e^{2}}{2} + \frac{a b^{2} e^{2}}{2} + 3 a b c d e + a c^{2} d^{2} + \frac{b^{3} d e}{2} + b^{2} c d^{2}\right ) + x^{3} \left (\frac{a^{2} b e^{2}}{3} + \frac{4 a^{2} c d e}{3} + \frac{4 a b^{2} d e}{3} + 2 a b c d^{2} + \frac{b^{3} d^{2}}{3}\right ) + x^{2} \left (a^{2} b d e + a^{2} c d^{2} + a b^{2} d^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.267811, size = 402, normalized size = 1.68 \[ \frac{1}{4} \, c^{3} x^{8} e^{2} + \frac{4}{7} \, c^{3} d x^{7} e + \frac{1}{3} \, c^{3} d^{2} x^{6} + \frac{5}{7} \, b c^{2} x^{7} e^{2} + \frac{5}{3} \, b c^{2} d x^{6} e + b c^{2} d^{2} x^{5} + \frac{2}{3} \, b^{2} c x^{6} e^{2} + \frac{2}{3} \, a c^{2} x^{6} e^{2} + \frac{8}{5} \, b^{2} c d x^{5} e + \frac{8}{5} \, a c^{2} d x^{5} e + b^{2} c d^{2} x^{4} + a c^{2} d^{2} x^{4} + \frac{1}{5} \, b^{3} x^{5} e^{2} + \frac{6}{5} \, a b c x^{5} e^{2} + \frac{1}{2} \, b^{3} d x^{4} e + 3 \, a b c d x^{4} e + \frac{1}{3} \, b^{3} d^{2} x^{3} + 2 \, a b c d^{2} x^{3} + \frac{1}{2} \, a b^{2} x^{4} e^{2} + \frac{1}{2} \, a^{2} c x^{4} e^{2} + \frac{4}{3} \, a b^{2} d x^{3} e + \frac{4}{3} \, a^{2} c d x^{3} e + a b^{2} d^{2} x^{2} + a^{2} c d^{2} x^{2} + \frac{1}{3} \, a^{2} b x^{3} e^{2} + a^{2} b d x^{2} e + a^{2} b d^{2} x \]

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)^2,x, algorithm="giac")

[Out]

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