3.2321 \(\int (A+B x) (d+e x) \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=180 \[ a^2 A d x+\frac{1}{5} x^5 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+\frac{1}{4} x^4 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+\frac{1}{3} x^3 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+\frac{1}{2} a x^2 (a A e+a B d+2 A b d)+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{7} B c^2 e x^7 \]

[Out]

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

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Rubi [A]  time = 0.611514, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ a^2 A d x+\frac{1}{5} x^5 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+\frac{1}{4} x^4 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+\frac{1}{3} x^3 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+\frac{1}{2} a x^2 (a A e+a B d+2 A b d)+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{7} B c^2 e x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c^{2} e x^{7}}{7} + a^{2} d \int A\, dx + a \left (A a e + 2 A b d + B a d\right ) \int x\, dx + \frac{c x^{6} \left (A c e + 2 B b e + B c d\right )}{6} + x^{5} \left (\frac{2 A b c e}{5} + \frac{A c^{2} d}{5} + \frac{2 B a c e}{5} + \frac{B b^{2} e}{5} + \frac{2 B b c d}{5}\right ) + x^{4} \left (\frac{A a c e}{2} + \frac{A b^{2} e}{4} + \frac{A b c d}{2} + \frac{B a b e}{2} + \frac{B a c d}{2} + \frac{B b^{2} d}{4}\right ) + x^{3} \left (\frac{2 A a b e}{3} + \frac{2 A a c d}{3} + \frac{A b^{2} d}{3} + \frac{B a^{2} e}{3} + \frac{2 B a b d}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*c**2*e*x**7/7 + a**2*d*Integral(A, x) + a*(A*a*e + 2*A*b*d + B*a*d)*Integral(x
, x) + c*x**6*(A*c*e + 2*B*b*e + B*c*d)/6 + x**5*(2*A*b*c*e/5 + A*c**2*d/5 + 2*B
*a*c*e/5 + B*b**2*e/5 + 2*B*b*c*d/5) + x**4*(A*a*c*e/2 + A*b**2*e/4 + A*b*c*d/2
+ B*a*b*e/2 + B*a*c*d/2 + B*b**2*d/4) + x**3*(2*A*a*b*e/3 + 2*A*a*c*d/3 + A*b**2
*d/3 + B*a**2*e/3 + 2*B*a*b*d/3)

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Mathematica [A]  time = 0.159784, size = 180, normalized size = 1. \[ a^2 A d x+\frac{1}{5} x^5 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+\frac{1}{4} x^4 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+\frac{1}{3} x^3 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+\frac{1}{2} a x^2 (a A e+a B d+2 A b d)+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{7} B c^2 e x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 167, normalized size = 0.9 \[{\frac{B{c}^{2}e{x}^{7}}{7}}+{\frac{ \left ( \left ( Ae+Bd \right ){c}^{2}+2\,Bebc \right ){x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}d+2\,bc \left ( Ae+Bd \right ) +Be \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,Abcd+ \left ( Ae+Bd \right ) \left ( 2\,ac+{b}^{2} \right ) +2\,Beab \right ){x}^{4}}{4}}+{\frac{ \left ( Ad \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( Ae+Bd \right ) ab+Be{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,Adab+ \left ( Ae+Bd \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{2}Adx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.69352, size = 248, normalized size = 1.38 \[ \frac{1}{7} \, B c^{2} e x^{7} + \frac{1}{6} \,{\left (B c^{2} d +{\left (2 \, B b c + A c^{2}\right )} e\right )} x^{6} + \frac{1}{5} \,{\left ({\left (2 \, B b c + A c^{2}\right )} d +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e\right )} x^{5} + A a^{2} d x + \frac{1}{4} \,{\left ({\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e\right )} x^{4} + \frac{1}{3} \,{\left ({\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d +{\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.235991, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e c^{2} B + \frac{1}{6} x^{6} d c^{2} B + \frac{1}{3} x^{6} e c b B + \frac{1}{6} x^{6} e c^{2} A + \frac{2}{5} x^{5} d c b B + \frac{1}{5} x^{5} e b^{2} B + \frac{2}{5} x^{5} e c a B + \frac{1}{5} x^{5} d c^{2} A + \frac{2}{5} x^{5} e c b A + \frac{1}{4} x^{4} d b^{2} B + \frac{1}{2} x^{4} d c a B + \frac{1}{2} x^{4} e b a B + \frac{1}{2} x^{4} d c b A + \frac{1}{4} x^{4} e b^{2} A + \frac{1}{2} x^{4} e c a A + \frac{2}{3} x^{3} d b a B + \frac{1}{3} x^{3} e a^{2} B + \frac{1}{3} x^{3} d b^{2} A + \frac{2}{3} x^{3} d c a A + \frac{2}{3} x^{3} e b a A + \frac{1}{2} x^{2} d a^{2} B + x^{2} d b a A + \frac{1}{2} x^{2} e a^{2} A + x d a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.201945, size = 231, normalized size = 1.28 \[ A a^{2} d x + \frac{B c^{2} e x^{7}}{7} + x^{6} \left (\frac{A c^{2} e}{6} + \frac{B b c e}{3} + \frac{B c^{2} d}{6}\right ) + x^{5} \left (\frac{2 A b c e}{5} + \frac{A c^{2} d}{5} + \frac{2 B a c e}{5} + \frac{B b^{2} e}{5} + \frac{2 B b c d}{5}\right ) + x^{4} \left (\frac{A a c e}{2} + \frac{A b^{2} e}{4} + \frac{A b c d}{2} + \frac{B a b e}{2} + \frac{B a c d}{2} + \frac{B b^{2} d}{4}\right ) + x^{3} \left (\frac{2 A a b e}{3} + \frac{2 A a c d}{3} + \frac{A b^{2} d}{3} + \frac{B a^{2} e}{3} + \frac{2 B a b d}{3}\right ) + x^{2} \left (\frac{A a^{2} e}{2} + A a b d + \frac{B a^{2} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.256183, size = 320, normalized size = 1.78 \[ \frac{1}{7} \, B c^{2} x^{7} e + \frac{1}{6} \, B c^{2} d x^{6} + \frac{1}{3} \, B b c x^{6} e + \frac{1}{6} \, A c^{2} x^{6} e + \frac{2}{5} \, B b c d x^{5} + \frac{1}{5} \, A c^{2} d x^{5} + \frac{1}{5} \, B b^{2} x^{5} e + \frac{2}{5} \, B a c x^{5} e + \frac{2}{5} \, A b c x^{5} e + \frac{1}{4} \, B b^{2} d x^{4} + \frac{1}{2} \, B a c d x^{4} + \frac{1}{2} \, A b c d x^{4} + \frac{1}{2} \, B a b x^{4} e + \frac{1}{4} \, A b^{2} x^{4} e + \frac{1}{2} \, A a c x^{4} e + \frac{2}{3} \, B a b d x^{3} + \frac{1}{3} \, A b^{2} d x^{3} + \frac{2}{3} \, A a c d x^{3} + \frac{1}{3} \, B a^{2} x^{3} e + \frac{2}{3} \, A a b x^{3} e + \frac{1}{2} \, B a^{2} d x^{2} + A a b d x^{2} + \frac{1}{2} \, A a^{2} x^{2} e + A a^{2} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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