3.1131 \(\int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=139 \[ \frac{1}{4} A b^3 d x^4+\frac{1}{7} c x^7 \left (3 b c (A e+B d)+A c^2 d+3 b^2 B e\right )+\frac{1}{6} b x^6 \left (3 b c (A e+B d)+3 A c^2 d+b^2 B e\right )+\frac{1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]

[Out]

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

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Rubi [A]  time = 0.403917, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{4} A b^3 d x^4+\frac{1}{7} c x^7 \left (3 b c (A e+B d)+A c^2 d+3 b^2 B e\right )+\frac{1}{6} b x^6 \left (3 b c (A e+B d)+3 A c^2 d+b^2 B e\right )+\frac{1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 51.0924, size = 151, normalized size = 1.09 \[ \frac{A b^{3} d x^{4}}{4} + \frac{B c^{3} e x^{9}}{9} + \frac{b^{2} x^{5} \left (A b e + 3 A c d + B b d\right )}{5} + \frac{b x^{6} \left (3 A b c e + 3 A c^{2} d + B b^{2} e + 3 B b c d\right )}{6} + \frac{c^{2} x^{8} \left (A c e + 3 B b e + B c d\right )}{8} + \frac{c x^{7} \left (3 A b c e + A c^{2} d + 3 B b^{2} e + 3 B b c d\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0770839, size = 141, normalized size = 1.01 \[ \frac{1}{4} A b^3 d x^4+\frac{1}{7} c x^7 \left (3 A b c e+A c^2 d+3 b^2 B e+3 b B c d\right )+\frac{1}{6} b x^6 \left (3 A b c e+3 A c^2 d+b^2 B e+3 b B c d\right )+\frac{1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\frac{1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac{1}{9} B c^3 e x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 138, normalized size = 1. \[{\frac{B{c}^{3}e{x}^{9}}{9}}+{\frac{ \left ( \left ( Ae+Bd \right ){c}^{3}+3\,Beb{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( Ad{c}^{3}+3\, \left ( Ae+Bd \right ) b{c}^{2}+3\,Be{b}^{2}c \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,Adb{c}^{2}+3\,{b}^{2}c \left ( Ae+Bd \right ) +{b}^{3}Be \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,Ad{b}^{2}c+{b}^{3} \left ( Ae+Bd \right ) \right ){x}^{5}}{5}}+{\frac{A{b}^{3}d{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.710359, size = 201, normalized size = 1.45 \[ \frac{1}{9} \, B c^{3} e x^{9} + \frac{1}{4} \, A b^{3} d x^{4} + \frac{1}{8} \,{\left (B c^{3} d +{\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac{1}{7} \,{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \,{\left (B b^{2} c + A b c^{2}\right )} e\right )} x^{7} + \frac{1}{6} \,{\left (3 \,{\left (B b^{2} c + A b c^{2}\right )} d +{\left (B b^{3} + 3 \, A b^{2} c\right )} e\right )} x^{6} + \frac{1}{5} \,{\left (A b^{3} e +{\left (B b^{3} + 3 \, A b^{2} c\right )} d\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.26875, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e c^{3} B + \frac{1}{8} x^{8} d c^{3} B + \frac{3}{8} x^{8} e c^{2} b B + \frac{1}{8} x^{8} e c^{3} A + \frac{3}{7} x^{7} d c^{2} b B + \frac{3}{7} x^{7} e c b^{2} B + \frac{1}{7} x^{7} d c^{3} A + \frac{3}{7} x^{7} e c^{2} b A + \frac{1}{2} x^{6} d c b^{2} B + \frac{1}{6} x^{6} e b^{3} B + \frac{1}{2} x^{6} d c^{2} b A + \frac{1}{2} x^{6} e c b^{2} A + \frac{1}{5} x^{5} d b^{3} B + \frac{3}{5} x^{5} d c b^{2} A + \frac{1}{5} x^{5} e b^{3} A + \frac{1}{4} x^{4} d b^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.176282, size = 177, normalized size = 1.27 \[ \frac{A b^{3} d x^{4}}{4} + \frac{B c^{3} e x^{9}}{9} + x^{8} \left (\frac{A c^{3} e}{8} + \frac{3 B b c^{2} e}{8} + \frac{B c^{3} d}{8}\right ) + x^{7} \left (\frac{3 A b c^{2} e}{7} + \frac{A c^{3} d}{7} + \frac{3 B b^{2} c e}{7} + \frac{3 B b c^{2} d}{7}\right ) + x^{6} \left (\frac{A b^{2} c e}{2} + \frac{A b c^{2} d}{2} + \frac{B b^{3} e}{6} + \frac{B b^{2} c d}{2}\right ) + x^{5} \left (\frac{A b^{3} e}{5} + \frac{3 A b^{2} c d}{5} + \frac{B b^{3} d}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.277885, size = 239, normalized size = 1.72 \[ \frac{1}{9} \, B c^{3} x^{9} e + \frac{1}{8} \, B c^{3} d x^{8} + \frac{3}{8} \, B b c^{2} x^{8} e + \frac{1}{8} \, A c^{3} x^{8} e + \frac{3}{7} \, B b c^{2} d x^{7} + \frac{1}{7} \, A c^{3} d x^{7} + \frac{3}{7} \, B b^{2} c x^{7} e + \frac{3}{7} \, A b c^{2} x^{7} e + \frac{1}{2} \, B b^{2} c d x^{6} + \frac{1}{2} \, A b c^{2} d x^{6} + \frac{1}{6} \, B b^{3} x^{6} e + \frac{1}{2} \, A b^{2} c x^{6} e + \frac{1}{5} \, B b^{3} d x^{5} + \frac{3}{5} \, A b^{2} c d x^{5} + \frac{1}{5} \, A b^{3} x^{5} e + \frac{1}{4} \, A b^{3} d x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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