3.1679 \(\int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=75 \[ \frac{(a+b x)^6 (-2 a B e+A b e+b B d)}{6 b^3}+\frac{(a+b x)^5 (A b-a B) (b d-a e)}{5 b^3}+\frac{B e (a+b x)^7}{7 b^3} \]

[Out]

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

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Rubi [A]  time = 0.288773, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(a+b x)^6 (-2 a B e+A b e+b B d)}{6 b^3}+\frac{(a+b x)^5 (A b-a B) (b d-a e)}{5 b^3}+\frac{B e (a+b x)^7}{7 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.5023, size = 68, normalized size = 0.91 \[ \frac{B e \left (a + b x\right )^{7}}{7 b^{3}} + \frac{\left (a + b x\right )^{6} \left (A b e - 2 B a e + B b d\right )}{6 b^{3}} - \frac{\left (a + b x\right )^{5} \left (A b - B a\right ) \left (a e - b d\right )}{5 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.0918434, size = 172, normalized size = 2.29 \[ a^4 A d x+\frac{1}{2} a^3 x^2 (a A e+a B d+4 A b d)+\frac{1}{3} a^2 x^3 (2 A b (2 a e+3 b d)+a B (a e+4 b d))+\frac{1}{6} b^3 x^6 (4 a B e+A b e+b B d)+\frac{1}{5} b^2 x^5 (A b (4 a e+b d)+2 a B (3 a e+2 b d))+\frac{1}{2} a b x^4 (A b (3 a e+2 b d)+a B (2 a e+3 b d))+\frac{1}{7} b^4 B e x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.002, size = 176, normalized size = 2.4 \[{\frac{Be{b}^{4}{x}^{7}}{7}}+{\frac{ \left ( \left ( Ae+Bd \right ){b}^{4}+4\,Bea{b}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( Ad{b}^{4}+4\, \left ( Ae+Bd \right ) a{b}^{3}+6\,Be{a}^{2}{b}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,Ada{b}^{3}+6\, \left ( Ae+Bd \right ){a}^{2}{b}^{2}+4\,Be{a}^{3}b \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,Ad{a}^{2}{b}^{2}+4\, \left ( Ae+Bd \right ){a}^{3}b+Be{a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,Ad{a}^{3}b+ \left ( Ae+Bd \right ){a}^{4} \right ){x}^{2}}{2}}+Ad{a}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.689752, size = 267, normalized size = 3.56 \[ \frac{1}{7} \, B b^{4} e x^{7} + A a^{4} d x + \frac{1}{6} \,{\left (B b^{4} d +{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} x^{6} + \frac{1}{5} \,{\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e\right )} x^{5} + \frac{1}{2} \,{\left ({\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d +{\left (B a^{4} + 4 \, A a^{3} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{4} e +{\left (B a^{4} + 4 \, A a^{3} b\right )} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.251807, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e b^{4} B + \frac{1}{6} x^{6} d b^{4} B + \frac{2}{3} x^{6} e b^{3} a B + \frac{1}{6} x^{6} e b^{4} A + \frac{4}{5} x^{5} d b^{3} a B + \frac{6}{5} x^{5} e b^{2} a^{2} B + \frac{1}{5} x^{5} d b^{4} A + \frac{4}{5} x^{5} e b^{3} a A + \frac{3}{2} x^{4} d b^{2} a^{2} B + x^{4} e b a^{3} B + x^{4} d b^{3} a A + \frac{3}{2} x^{4} e b^{2} a^{2} A + \frac{4}{3} x^{3} d b a^{3} B + \frac{1}{3} x^{3} e a^{4} B + 2 x^{3} d b^{2} a^{2} A + \frac{4}{3} x^{3} e b a^{3} A + \frac{1}{2} x^{2} d a^{4} B + 2 x^{2} d b a^{3} A + \frac{1}{2} x^{2} e a^{4} A + x d a^{4} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.207065, size = 226, normalized size = 3.01 \[ A a^{4} d x + \frac{B b^{4} e x^{7}}{7} + x^{6} \left (\frac{A b^{4} e}{6} + \frac{2 B a b^{3} e}{3} + \frac{B b^{4} d}{6}\right ) + x^{5} \left (\frac{4 A a b^{3} e}{5} + \frac{A b^{4} d}{5} + \frac{6 B a^{2} b^{2} e}{5} + \frac{4 B a b^{3} d}{5}\right ) + x^{4} \left (\frac{3 A a^{2} b^{2} e}{2} + A a b^{3} d + B a^{3} b e + \frac{3 B a^{2} b^{2} d}{2}\right ) + x^{3} \left (\frac{4 A a^{3} b e}{3} + 2 A a^{2} b^{2} d + \frac{B a^{4} e}{3} + \frac{4 B a^{3} b d}{3}\right ) + x^{2} \left (\frac{A a^{4} e}{2} + 2 A a^{3} b d + \frac{B a^{4} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.310364, size = 305, normalized size = 4.07 \[ \frac{1}{7} \, B b^{4} x^{7} e + \frac{1}{6} \, B b^{4} d x^{6} + \frac{2}{3} \, B a b^{3} x^{6} e + \frac{1}{6} \, A b^{4} x^{6} e + \frac{4}{5} \, B a b^{3} d x^{5} + \frac{1}{5} \, A b^{4} d x^{5} + \frac{6}{5} \, B a^{2} b^{2} x^{5} e + \frac{4}{5} \, A a b^{3} x^{5} e + \frac{3}{2} \, B a^{2} b^{2} d x^{4} + A a b^{3} d x^{4} + B a^{3} b x^{4} e + \frac{3}{2} \, A a^{2} b^{2} x^{4} e + \frac{4}{3} \, B a^{3} b d x^{3} + 2 \, A a^{2} b^{2} d x^{3} + \frac{1}{3} \, B a^{4} x^{3} e + \frac{4}{3} \, A a^{3} b x^{3} e + \frac{1}{2} \, B a^{4} d x^{2} + 2 \, A a^{3} b d x^{2} + \frac{1}{2} \, A a^{4} x^{2} e + A a^{4} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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