3.1678 \(\int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=118 \[ \frac{e (a+b x)^7 (-3 a B e+A b e+2 b B d)}{7 b^4}+\frac{(a+b x)^6 (b d-a e) (-3 a B e+2 A b e+b B d)}{6 b^4}+\frac{(a+b x)^5 (A b-a B) (b d-a e)^2}{5 b^4}+\frac{B e^2 (a+b x)^8}{8 b^4} \]

[Out]

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

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Rubi [A]  time = 0.554686, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{e (a+b x)^7 (-3 a B e+A b e+2 b B d)}{7 b^4}+\frac{(a+b x)^6 (b d-a e) (-3 a B e+2 A b e+b B d)}{6 b^4}+\frac{(a+b x)^5 (A b-a B) (b d-a e)^2}{5 b^4}+\frac{B e^2 (a+b x)^8}{8 b^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 74.1414, size = 112, normalized size = 0.95 \[ \frac{B e^{2} \left (a + b x\right )^{8}}{8 b^{4}} + \frac{e \left (a + b x\right )^{7} \left (A b e - 3 B a e + 2 B b d\right )}{7 b^{4}} - \frac{\left (a + b x\right )^{6} \left (a e - b d\right ) \left (2 A b e - 3 B a e + B b d\right )}{6 b^{4}} + \frac{\left (a + b x\right )^{5} \left (A b - B a\right ) \left (a e - b d\right )^{2}}{5 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.174061, size = 288, normalized size = 2.44 \[ a^4 A d^2 x+\frac{1}{2} a^3 d x^2 (2 a A e+a B d+4 A b d)+\frac{1}{5} b x^5 \left (A b \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+4 a B \left (a^2 e^2+3 a b d e+b^2 d^2\right )\right )+\frac{1}{4} a x^4 \left (4 A b \left (a^2 e^2+3 a b d e+b^2 d^2\right )+a B \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )\right )+\frac{1}{3} a^2 x^3 \left (A \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+2 a B d (a e+2 b d)\right )+\frac{1}{6} b^2 x^6 \left (6 a^2 B e^2+4 a b e (A e+2 B d)+b^2 d (2 A e+B d)\right )+\frac{1}{7} b^3 e x^7 (4 a B e+A b e+2 b B d)+\frac{1}{8} b^4 B e^2 x^8 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.001, size = 305, normalized size = 2.6 \[{\frac{B{e}^{2}{b}^{4}{x}^{8}}{8}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ){b}^{4}+4\,B{e}^{2}a{b}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){b}^{4}+4\, \left ( A{e}^{2}+2\,Bde \right ) a{b}^{3}+6\,B{e}^{2}{a}^{2}{b}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{2}{b}^{4}+4\, \left ( 2\,Ade+B{d}^{2} \right ) a{b}^{3}+6\, \left ( A{e}^{2}+2\,Bde \right ){a}^{2}{b}^{2}+4\,B{e}^{2}{a}^{3}b \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,A{d}^{2}a{b}^{3}+6\, \left ( 2\,Ade+B{d}^{2} \right ){a}^{2}{b}^{2}+4\, \left ( A{e}^{2}+2\,Bde \right ){a}^{3}b+B{e}^{2}{a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,A{d}^{2}{a}^{2}{b}^{2}+4\, \left ( 2\,Ade+B{d}^{2} \right ){a}^{3}b+ \left ( A{e}^{2}+2\,Bde \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{2}{a}^{3}b+ \left ( 2\,Ade+B{d}^{2} \right ){a}^{4} \right ){x}^{2}}{2}}+A{d}^{2}{a}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.698034, size = 435, normalized size = 3.69 \[ \frac{1}{8} \, B b^{4} e^{2} x^{8} + A a^{4} d^{2} x + \frac{1}{7} \,{\left (2 \, B b^{4} d e +{\left (4 \, B a b^{3} + A b^{4}\right )} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B b^{4} d^{2} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} + 4 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (A a^{4} e^{2} + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} + 2 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a^{4} d e +{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2}\right )} x^{2} \]

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.248825, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} e^{2} b^{4} B + \frac{2}{7} x^{7} e d b^{4} B + \frac{4}{7} x^{7} e^{2} b^{3} a B + \frac{1}{7} x^{7} e^{2} b^{4} A + \frac{1}{6} x^{6} d^{2} b^{4} B + \frac{4}{3} x^{6} e d b^{3} a B + x^{6} e^{2} b^{2} a^{2} B + \frac{1}{3} x^{6} e d b^{4} A + \frac{2}{3} x^{6} e^{2} b^{3} a A + \frac{4}{5} x^{5} d^{2} b^{3} a B + \frac{12}{5} x^{5} e d b^{2} a^{2} B + \frac{4}{5} x^{5} e^{2} b a^{3} B + \frac{1}{5} x^{5} d^{2} b^{4} A + \frac{8}{5} x^{5} e d b^{3} a A + \frac{6}{5} x^{5} e^{2} b^{2} a^{2} A + \frac{3}{2} x^{4} d^{2} b^{2} a^{2} B + 2 x^{4} e d b a^{3} B + \frac{1}{4} x^{4} e^{2} a^{4} B + x^{4} d^{2} b^{3} a A + 3 x^{4} e d b^{2} a^{2} A + x^{4} e^{2} b a^{3} A + \frac{4}{3} x^{3} d^{2} b a^{3} B + \frac{2}{3} x^{3} e d a^{4} B + 2 x^{3} d^{2} b^{2} a^{2} A + \frac{8}{3} x^{3} e d b a^{3} A + \frac{1}{3} x^{3} e^{2} a^{4} A + \frac{1}{2} x^{2} d^{2} a^{4} B + 2 x^{2} d^{2} b a^{3} A + x^{2} e d a^{4} A + x d^{2} a^{4} A \]

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)^2,x, algorithm="fricas")

[Out]

1/8*x^8*e^2*b^4*B + 2/7*x^7*e*d*b^4*B + 4/7*x^7*e^2*b^3*a*B + 1/7*x^7*e^2*b^4*A
+ 1/6*x^6*d^2*b^4*B + 4/3*x^6*e*d*b^3*a*B + x^6*e^2*b^2*a^2*B + 1/3*x^6*e*d*b^4*
A + 2/3*x^6*e^2*b^3*a*A + 4/5*x^5*d^2*b^3*a*B + 12/5*x^5*e*d*b^2*a^2*B + 4/5*x^5
*e^2*b*a^3*B + 1/5*x^5*d^2*b^4*A + 8/5*x^5*e*d*b^3*a*A + 6/5*x^5*e^2*b^2*a^2*A +
3/2*x^4*d^2*b^2*a^2*B + 2*x^4*e*d*b*a^3*B + 1/4*x^4*e^2*a^4*B + x^4*d^2*b^3*a*A
+ 3*x^4*e*d*b^2*a^2*A + x^4*e^2*b*a^3*A + 4/3*x^3*d^2*b*a^3*B + 2/3*x^3*e*d*a^4
*B + 2*x^3*d^2*b^2*a^2*A + 8/3*x^3*e*d*b*a^3*A + 1/3*x^3*e^2*a^4*A + 1/2*x^2*d^2
*a^4*B + 2*x^2*d^2*b*a^3*A + x^2*e*d*a^4*A + x*d^2*a^4*A

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Sympy [A]  time = 0.267599, size = 384, normalized size = 3.25 \[ A a^{4} d^{2} x + \frac{B b^{4} e^{2} x^{8}}{8} + x^{7} \left (\frac{A b^{4} e^{2}}{7} + \frac{4 B a b^{3} e^{2}}{7} + \frac{2 B b^{4} d e}{7}\right ) + x^{6} \left (\frac{2 A a b^{3} e^{2}}{3} + \frac{A b^{4} d e}{3} + B a^{2} b^{2} e^{2} + \frac{4 B a b^{3} d e}{3} + \frac{B b^{4} d^{2}}{6}\right ) + x^{5} \left (\frac{6 A a^{2} b^{2} e^{2}}{5} + \frac{8 A a b^{3} d e}{5} + \frac{A b^{4} d^{2}}{5} + \frac{4 B a^{3} b e^{2}}{5} + \frac{12 B a^{2} b^{2} d e}{5} + \frac{4 B a b^{3} d^{2}}{5}\right ) + x^{4} \left (A a^{3} b e^{2} + 3 A a^{2} b^{2} d e + A a b^{3} d^{2} + \frac{B a^{4} e^{2}}{4} + 2 B a^{3} b d e + \frac{3 B a^{2} b^{2} d^{2}}{2}\right ) + x^{3} \left (\frac{A a^{4} e^{2}}{3} + \frac{8 A a^{3} b d e}{3} + 2 A a^{2} b^{2} d^{2} + \frac{2 B a^{4} d e}{3} + \frac{4 B a^{3} b d^{2}}{3}\right ) + x^{2} \left (A a^{4} d e + 2 A a^{3} b d^{2} + \frac{B a^{4} d^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**4*d**2*x + B*b**4*e**2*x**8/8 + x**7*(A*b**4*e**2/7 + 4*B*a*b**3*e**2/7 + 2
*B*b**4*d*e/7) + x**6*(2*A*a*b**3*e**2/3 + A*b**4*d*e/3 + B*a**2*b**2*e**2 + 4*B
*a*b**3*d*e/3 + B*b**4*d**2/6) + x**5*(6*A*a**2*b**2*e**2/5 + 8*A*a*b**3*d*e/5 +
A*b**4*d**2/5 + 4*B*a**3*b*e**2/5 + 12*B*a**2*b**2*d*e/5 + 4*B*a*b**3*d**2/5) +
x**4*(A*a**3*b*e**2 + 3*A*a**2*b**2*d*e + A*a*b**3*d**2 + B*a**4*e**2/4 + 2*B*a
**3*b*d*e + 3*B*a**2*b**2*d**2/2) + x**3*(A*a**4*e**2/3 + 8*A*a**3*b*d*e/3 + 2*A
*a**2*b**2*d**2 + 2*B*a**4*d*e/3 + 4*B*a**3*b*d**2/3) + x**2*(A*a**4*d*e + 2*A*a
**3*b*d**2 + B*a**4*d**2/2)

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GIAC/XCAS [A]  time = 0.301755, size = 505, normalized size = 4.28 \[ \frac{1}{8} \, B b^{4} x^{8} e^{2} + \frac{2}{7} \, B b^{4} d x^{7} e + \frac{1}{6} \, B b^{4} d^{2} x^{6} + \frac{4}{7} \, B a b^{3} x^{7} e^{2} + \frac{1}{7} \, A b^{4} x^{7} e^{2} + \frac{4}{3} \, B a b^{3} d x^{6} e + \frac{1}{3} \, A b^{4} d x^{6} e + \frac{4}{5} \, B a b^{3} d^{2} x^{5} + \frac{1}{5} \, A b^{4} d^{2} x^{5} + B a^{2} b^{2} x^{6} e^{2} + \frac{2}{3} \, A a b^{3} x^{6} e^{2} + \frac{12}{5} \, B a^{2} b^{2} d x^{5} e + \frac{8}{5} \, A a b^{3} d x^{5} e + \frac{3}{2} \, B a^{2} b^{2} d^{2} x^{4} + A a b^{3} d^{2} x^{4} + \frac{4}{5} \, B a^{3} b x^{5} e^{2} + \frac{6}{5} \, A a^{2} b^{2} x^{5} e^{2} + 2 \, B a^{3} b d x^{4} e + 3 \, A a^{2} b^{2} d x^{4} e + \frac{4}{3} \, B a^{3} b d^{2} x^{3} + 2 \, A a^{2} b^{2} d^{2} x^{3} + \frac{1}{4} \, B a^{4} x^{4} e^{2} + A a^{3} b x^{4} e^{2} + \frac{2}{3} \, B a^{4} d x^{3} e + \frac{8}{3} \, A a^{3} b d x^{3} e + \frac{1}{2} \, B a^{4} d^{2} x^{2} + 2 \, A a^{3} b d^{2} x^{2} + \frac{1}{3} \, A a^{4} x^{3} e^{2} + A a^{4} d x^{2} e + A a^{4} d^{2} x \]

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)^2,x, algorithm="giac")

[Out]

1/8*B*b^4*x^8*e^2 + 2/7*B*b^4*d*x^7*e + 1/6*B*b^4*d^2*x^6 + 4/7*B*a*b^3*x^7*e^2
+ 1/7*A*b^4*x^7*e^2 + 4/3*B*a*b^3*d*x^6*e + 1/3*A*b^4*d*x^6*e + 4/5*B*a*b^3*d^2*
x^5 + 1/5*A*b^4*d^2*x^5 + B*a^2*b^2*x^6*e^2 + 2/3*A*a*b^3*x^6*e^2 + 12/5*B*a^2*b
^2*d*x^5*e + 8/5*A*a*b^3*d*x^5*e + 3/2*B*a^2*b^2*d^2*x^4 + A*a*b^3*d^2*x^4 + 4/5
*B*a^3*b*x^5*e^2 + 6/5*A*a^2*b^2*x^5*e^2 + 2*B*a^3*b*d*x^4*e + 3*A*a^2*b^2*d*x^4
*e + 4/3*B*a^3*b*d^2*x^3 + 2*A*a^2*b^2*d^2*x^3 + 1/4*B*a^4*x^4*e^2 + A*a^3*b*x^4
*e^2 + 2/3*B*a^4*d*x^3*e + 8/3*A*a^3*b*d*x^3*e + 1/2*B*a^4*d^2*x^2 + 2*A*a^3*b*d
^2*x^2 + 1/3*A*a^4*x^3*e^2 + A*a^4*d*x^2*e + A*a^4*d^2*x