3.1130 \(\int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=225 \[ \frac{1}{4} A b^3 d^2 x^4+\frac{1}{8} c x^8 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac{1}{6} b x^6 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 A b e+3 A c d+b B d)+\frac{1}{7} x^7 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+\frac{1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac{1}{10} B c^3 e^2 x^{10} \]

[Out]

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

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Rubi [A]  time = 0.803501, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{1}{4} A b^3 d^2 x^4+\frac{1}{8} c x^8 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac{1}{6} b x^6 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 A b e+3 A c d+b B d)+\frac{1}{7} x^7 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+\frac{1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac{1}{10} B c^3 e^2 x^{10} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 117.207, size = 274, normalized size = 1.22 \[ \frac{A b^{3} d^{2} x^{4}}{4} + \frac{B c^{3} e^{2} x^{10}}{10} + \frac{b^{2} d x^{5} \left (2 A b e + 3 A c d + B b d\right )}{5} + \frac{b x^{6} \left (A b^{2} e^{2} + 6 A b c d e + 3 A c^{2} d^{2} + 2 B b^{2} d e + 3 B b c d^{2}\right )}{6} + \frac{c^{2} e x^{9} \left (A c e + 3 B b e + 2 B c d\right )}{9} + \frac{c x^{8} \left (3 A b c e^{2} + 2 A c^{2} d e + 3 B b^{2} e^{2} + 6 B b c d e + B c^{2} d^{2}\right )}{8} + x^{7} \left (\frac{3 A b^{2} c e^{2}}{7} + \frac{6 A b c^{2} d e}{7} + \frac{A c^{3} d^{2}}{7} + \frac{B b^{3} e^{2}}{7} + \frac{6 B b^{2} c d e}{7} + \frac{3 B b c^{2} d^{2}}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.149242, size = 225, normalized size = 1. \[ \frac{1}{4} A b^3 d^2 x^4+\frac{1}{8} c x^8 \left (A c e (3 b e+2 c d)+B \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac{1}{6} b x^6 \left (b^2 e (A e+2 B d)+3 b c d (2 A e+B d)+3 A c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 A b e+3 A c d+b B d)+\frac{1}{7} x^7 \left (3 b^2 c e (A e+2 B d)+3 b c^2 d (2 A e+B d)+A c^3 d^2+b^3 B e^2\right )+\frac{1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac{1}{10} B c^3 e^2 x^{10} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 240, normalized size = 1.1 \[{\frac{B{c}^{3}{e}^{2}{x}^{10}}{10}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ){c}^{3}+3\,B{e}^{2}b{c}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){c}^{3}+3\, \left ( A{e}^{2}+2\,Bde \right ) b{c}^{2}+3\,B{e}^{2}{b}^{2}c \right ){x}^{8}}{8}}+{\frac{ \left ( A{c}^{3}{d}^{2}+3\, \left ( 2\,Ade+B{d}^{2} \right ) b{c}^{2}+3\, \left ( A{e}^{2}+2\,Bde \right ){b}^{2}c+{b}^{3}B{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,A{d}^{2}b{c}^{2}+3\, \left ( 2\,Ade+B{d}^{2} \right ){b}^{2}c+ \left ( A{e}^{2}+2\,Bde \right ){b}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{d}^{2}{b}^{2}c+ \left ( 2\,Ade+B{d}^{2} \right ){b}^{3} \right ){x}^{5}}{5}}+{\frac{A{b}^{3}{d}^{2}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*B*c^3*e^2*x^10+1/9*((A*e^2+2*B*d*e)*c^3+3*B*e^2*b*c^2)*x^9+1/8*((2*A*d*e+B*
d^2)*c^3+3*(A*e^2+2*B*d*e)*b*c^2+3*B*e^2*b^2*c)*x^8+1/7*(A*c^3*d^2+3*(2*A*d*e+B*
d^2)*b*c^2+3*(A*e^2+2*B*d*e)*b^2*c+b^3*B*e^2)*x^7+1/6*(3*A*d^2*b*c^2+3*(2*A*d*e+
B*d^2)*b^2*c+(A*e^2+2*B*d*e)*b^3)*x^6+1/5*(3*A*d^2*b^2*c+(2*A*d*e+B*d^2)*b^3)*x^
5+1/4*A*b^3*d^2*x^4

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Maxima [A]  time = 0.704241, size = 327, normalized size = 1.45 \[ \frac{1}{10} \, B c^{3} e^{2} x^{10} + \frac{1}{4} \, A b^{3} d^{2} x^{4} + \frac{1}{9} \,{\left (2 \, B c^{3} d e +{\left (3 \, B b c^{2} + A c^{3}\right )} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d^{2} + 2 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d e + 3 \,{\left (B b^{2} c + A b c^{2}\right )} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} + 6 \,{\left (B b^{2} c + A b c^{2}\right )} d e +{\left (B b^{3} + 3 \, A b^{2} c\right )} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (A b^{3} e^{2} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{2} + 2 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d e\right )} x^{6} + \frac{1}{5} \,{\left (2 \, A b^{3} d e +{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2}\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*B*c^3*e^2*x^10 + 1/4*A*b^3*d^2*x^4 + 1/9*(2*B*c^3*d*e + (3*B*b*c^2 + A*c^3)
*e^2)*x^9 + 1/8*(B*c^3*d^2 + 2*(3*B*b*c^2 + A*c^3)*d*e + 3*(B*b^2*c + A*b*c^2)*e
^2)*x^8 + 1/7*((3*B*b*c^2 + A*c^3)*d^2 + 6*(B*b^2*c + A*b*c^2)*d*e + (B*b^3 + 3*
A*b^2*c)*e^2)*x^7 + 1/6*(A*b^3*e^2 + 3*(B*b^2*c + A*b*c^2)*d^2 + 2*(B*b^3 + 3*A*
b^2*c)*d*e)*x^6 + 1/5*(2*A*b^3*d*e + (B*b^3 + 3*A*b^2*c)*d^2)*x^5

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Fricas [A]  time = 0.262929, size = 1, normalized size = 0. \[ \frac{1}{10} x^{10} e^{2} c^{3} B + \frac{2}{9} x^{9} e d c^{3} B + \frac{1}{3} x^{9} e^{2} c^{2} b B + \frac{1}{9} x^{9} e^{2} c^{3} A + \frac{1}{8} x^{8} d^{2} c^{3} B + \frac{3}{4} x^{8} e d c^{2} b B + \frac{3}{8} x^{8} e^{2} c b^{2} B + \frac{1}{4} x^{8} e d c^{3} A + \frac{3}{8} x^{8} e^{2} c^{2} b A + \frac{3}{7} x^{7} d^{2} c^{2} b B + \frac{6}{7} x^{7} e d c b^{2} B + \frac{1}{7} x^{7} e^{2} b^{3} B + \frac{1}{7} x^{7} d^{2} c^{3} A + \frac{6}{7} x^{7} e d c^{2} b A + \frac{3}{7} x^{7} e^{2} c b^{2} A + \frac{1}{2} x^{6} d^{2} c b^{2} B + \frac{1}{3} x^{6} e d b^{3} B + \frac{1}{2} x^{6} d^{2} c^{2} b A + x^{6} e d c b^{2} A + \frac{1}{6} x^{6} e^{2} b^{3} A + \frac{1}{5} x^{5} d^{2} b^{3} B + \frac{3}{5} x^{5} d^{2} c b^{2} A + \frac{2}{5} x^{5} e d b^{3} A + \frac{1}{4} x^{4} d^{2} b^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*x^10*e^2*c^3*B + 2/9*x^9*e*d*c^3*B + 1/3*x^9*e^2*c^2*b*B + 1/9*x^9*e^2*c^3*
A + 1/8*x^8*d^2*c^3*B + 3/4*x^8*e*d*c^2*b*B + 3/8*x^8*e^2*c*b^2*B + 1/4*x^8*e*d*
c^3*A + 3/8*x^8*e^2*c^2*b*A + 3/7*x^7*d^2*c^2*b*B + 6/7*x^7*e*d*c*b^2*B + 1/7*x^
7*e^2*b^3*B + 1/7*x^7*d^2*c^3*A + 6/7*x^7*e*d*c^2*b*A + 3/7*x^7*e^2*c*b^2*A + 1/
2*x^6*d^2*c*b^2*B + 1/3*x^6*e*d*b^3*B + 1/2*x^6*d^2*c^2*b*A + x^6*e*d*c*b^2*A +
1/6*x^6*e^2*b^3*A + 1/5*x^5*d^2*b^3*B + 3/5*x^5*d^2*c*b^2*A + 2/5*x^5*e*d*b^3*A
+ 1/4*x^4*d^2*b^3*A

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Sympy [A]  time = 0.236124, size = 303, normalized size = 1.35 \[ \frac{A b^{3} d^{2} x^{4}}{4} + \frac{B c^{3} e^{2} x^{10}}{10} + x^{9} \left (\frac{A c^{3} e^{2}}{9} + \frac{B b c^{2} e^{2}}{3} + \frac{2 B c^{3} d e}{9}\right ) + x^{8} \left (\frac{3 A b c^{2} e^{2}}{8} + \frac{A c^{3} d e}{4} + \frac{3 B b^{2} c e^{2}}{8} + \frac{3 B b c^{2} d e}{4} + \frac{B c^{3} d^{2}}{8}\right ) + x^{7} \left (\frac{3 A b^{2} c e^{2}}{7} + \frac{6 A b c^{2} d e}{7} + \frac{A c^{3} d^{2}}{7} + \frac{B b^{3} e^{2}}{7} + \frac{6 B b^{2} c d e}{7} + \frac{3 B b c^{2} d^{2}}{7}\right ) + x^{6} \left (\frac{A b^{3} e^{2}}{6} + A b^{2} c d e + \frac{A b c^{2} d^{2}}{2} + \frac{B b^{3} d e}{3} + \frac{B b^{2} c d^{2}}{2}\right ) + x^{5} \left (\frac{2 A b^{3} d e}{5} + \frac{3 A b^{2} c d^{2}}{5} + \frac{B b^{3} d^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.275254, size = 394, normalized size = 1.75 \[ \frac{1}{10} \, B c^{3} x^{10} e^{2} + \frac{2}{9} \, B c^{3} d x^{9} e + \frac{1}{8} \, B c^{3} d^{2} x^{8} + \frac{1}{3} \, B b c^{2} x^{9} e^{2} + \frac{1}{9} \, A c^{3} x^{9} e^{2} + \frac{3}{4} \, B b c^{2} d x^{8} e + \frac{1}{4} \, A c^{3} d x^{8} e + \frac{3}{7} \, B b c^{2} d^{2} x^{7} + \frac{1}{7} \, A c^{3} d^{2} x^{7} + \frac{3}{8} \, B b^{2} c x^{8} e^{2} + \frac{3}{8} \, A b c^{2} x^{8} e^{2} + \frac{6}{7} \, B b^{2} c d x^{7} e + \frac{6}{7} \, A b c^{2} d x^{7} e + \frac{1}{2} \, B b^{2} c d^{2} x^{6} + \frac{1}{2} \, A b c^{2} d^{2} x^{6} + \frac{1}{7} \, B b^{3} x^{7} e^{2} + \frac{3}{7} \, A b^{2} c x^{7} e^{2} + \frac{1}{3} \, B b^{3} d x^{6} e + A b^{2} c d x^{6} e + \frac{1}{5} \, B b^{3} d^{2} x^{5} + \frac{3}{5} \, A b^{2} c d^{2} x^{5} + \frac{1}{6} \, A b^{3} x^{6} e^{2} + \frac{2}{5} \, A b^{3} d x^{5} e + \frac{1}{4} \, A b^{3} d^{2} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*B*c^3*x^10*e^2 + 2/9*B*c^3*d*x^9*e + 1/8*B*c^3*d^2*x^8 + 1/3*B*b*c^2*x^9*e^
2 + 1/9*A*c^3*x^9*e^2 + 3/4*B*b*c^2*d*x^8*e + 1/4*A*c^3*d*x^8*e + 3/7*B*b*c^2*d^
2*x^7 + 1/7*A*c^3*d^2*x^7 + 3/8*B*b^2*c*x^8*e^2 + 3/8*A*b*c^2*x^8*e^2 + 6/7*B*b^
2*c*d*x^7*e + 6/7*A*b*c^2*d*x^7*e + 1/2*B*b^2*c*d^2*x^6 + 1/2*A*b*c^2*d^2*x^6 +
1/7*B*b^3*x^7*e^2 + 3/7*A*b^2*c*x^7*e^2 + 1/3*B*b^3*d*x^6*e + A*b^2*c*d*x^6*e +
1/5*B*b^3*d^2*x^5 + 3/5*A*b^2*c*d^2*x^5 + 1/6*A*b^3*x^6*e^2 + 2/5*A*b^3*d*x^5*e
+ 1/4*A*b^3*d^2*x^4