3.244 \(\int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=225 \[ \frac{1}{4} b^3 d^4 x^4+\frac{1}{3} c e^2 x^9 \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+\frac{1}{2} b d^2 x^6 \left (2 b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac{1}{8} e x^8 \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )+\frac{1}{7} d x^7 \left (4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right )+\frac{1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac{1}{11} c^3 e^4 x^{11} \]

[Out]

(b^3*d^4*x^4)/4 + (b^2*d^3*(3*c*d + 4*b*e)*x^5)/5 + (b*d^2*(c^2*d^2 + 4*b*c*d*e
+ 2*b^2*e^2)*x^6)/2 + (d*(c^3*d^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3)
*x^7)/7 + (e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3)*x^8)/8 + (c
*e^2*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^9)/3 + (c^2*e^3*(4*c*d + 3*b*e)*x^10)/1
0 + (c^3*e^4*x^11)/11

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Rubi [A]  time = 0.521784, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{1}{4} b^3 d^4 x^4+\frac{1}{3} c e^2 x^9 \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+\frac{1}{2} b d^2 x^6 \left (2 b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac{1}{8} e x^8 \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )+\frac{1}{7} d x^7 \left (4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right )+\frac{1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac{1}{11} c^3 e^4 x^{11} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4*(b*x + c*x^2)^3,x]

[Out]

(b^3*d^4*x^4)/4 + (b^2*d^3*(3*c*d + 4*b*e)*x^5)/5 + (b*d^2*(c^2*d^2 + 4*b*c*d*e
+ 2*b^2*e^2)*x^6)/2 + (d*(c^3*d^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3)
*x^7)/7 + (e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3)*x^8)/8 + (c
*e^2*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^9)/3 + (c^2*e^3*(4*c*d + 3*b*e)*x^10)/1
0 + (c^3*e^4*x^11)/11

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Rubi in Sympy [A]  time = 74.6656, size = 226, normalized size = 1. \[ \frac{b^{3} d^{4} x^{4}}{4} + \frac{b^{2} d^{3} x^{5} \left (4 b e + 3 c d\right )}{5} + \frac{b d^{2} x^{6} \left (2 b^{2} e^{2} + 4 b c d e + c^{2} d^{2}\right )}{2} + \frac{c^{3} e^{4} x^{11}}{11} + \frac{c^{2} e^{3} x^{10} \left (3 b e + 4 c d\right )}{10} + \frac{c e^{2} x^{9} \left (b^{2} e^{2} + 4 b c d e + 2 c^{2} d^{2}\right )}{3} + \frac{d x^{7} \left (4 b^{3} e^{3} + 18 b^{2} c d e^{2} + 12 b c^{2} d^{2} e + c^{3} d^{3}\right )}{7} + \frac{e x^{8} \left (b^{3} e^{3} + 12 b^{2} c d e^{2} + 18 b c^{2} d^{2} e + 4 c^{3} d^{3}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4*(c*x**2+b*x)**3,x)

[Out]

b**3*d**4*x**4/4 + b**2*d**3*x**5*(4*b*e + 3*c*d)/5 + b*d**2*x**6*(2*b**2*e**2 +
4*b*c*d*e + c**2*d**2)/2 + c**3*e**4*x**11/11 + c**2*e**3*x**10*(3*b*e + 4*c*d)
/10 + c*e**2*x**9*(b**2*e**2 + 4*b*c*d*e + 2*c**2*d**2)/3 + d*x**7*(4*b**3*e**3
+ 18*b**2*c*d*e**2 + 12*b*c**2*d**2*e + c**3*d**3)/7 + e*x**8*(b**3*e**3 + 12*b*
*2*c*d*e**2 + 18*b*c**2*d**2*e + 4*c**3*d**3)/8

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Mathematica [A]  time = 0.0604998, size = 225, normalized size = 1. \[ \frac{1}{4} b^3 d^4 x^4+\frac{1}{3} c e^2 x^9 \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+\frac{1}{2} b d^2 x^6 \left (2 b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac{1}{8} e x^8 \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )+\frac{1}{7} d x^7 \left (4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right )+\frac{1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac{1}{11} c^3 e^4 x^{11} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4*(b*x + c*x^2)^3,x]

[Out]

(b^3*d^4*x^4)/4 + (b^2*d^3*(3*c*d + 4*b*e)*x^5)/5 + (b*d^2*(c^2*d^2 + 4*b*c*d*e
+ 2*b^2*e^2)*x^6)/2 + (d*(c^3*d^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3)
*x^7)/7 + (e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 + b^3*e^3)*x^8)/8 + (c
*e^2*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^9)/3 + (c^2*e^3*(4*c*d + 3*b*e)*x^10)/1
0 + (c^3*e^4*x^11)/11

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Maple [A]  time = 0.002, size = 232, normalized size = 1. \[{\frac{{c}^{3}{e}^{4}{x}^{11}}{11}}+{\frac{ \left ( 3\,{e}^{4}b{c}^{2}+4\,d{e}^{3}{c}^{3} \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,{e}^{4}{b}^{2}c+12\,d{e}^{3}b{c}^{2}+6\,{d}^{2}{e}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ({e}^{4}{b}^{3}+12\,d{e}^{3}{b}^{2}c+18\,{d}^{2}{e}^{2}b{c}^{2}+4\,{d}^{3}e{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 4\,d{e}^{3}{b}^{3}+18\,{d}^{2}{e}^{2}{b}^{2}c+12\,{d}^{3}eb{c}^{2}+{d}^{4}{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{b}^{3}+12\,{d}^{3}e{b}^{2}c+3\,{d}^{4}b{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,{d}^{3}e{b}^{3}+3\,{d}^{4}{b}^{2}c \right ){x}^{5}}{5}}+{\frac{{b}^{3}{d}^{4}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4*(c*x^2+b*x)^3,x)

[Out]

1/11*c^3*e^4*x^11+1/10*(3*b*c^2*e^4+4*c^3*d*e^3)*x^10+1/9*(3*b^2*c*e^4+12*b*c^2*
d*e^3+6*c^3*d^2*e^2)*x^9+1/8*(b^3*e^4+12*b^2*c*d*e^3+18*b*c^2*d^2*e^2+4*c^3*d^3*
e)*x^8+1/7*(4*b^3*d*e^3+18*b^2*c*d^2*e^2+12*b*c^2*d^3*e+c^3*d^4)*x^7+1/6*(6*b^3*
d^2*e^2+12*b^2*c*d^3*e+3*b*c^2*d^4)*x^6+1/5*(4*b^3*d^3*e+3*b^2*c*d^4)*x^5+1/4*b^
3*d^4*x^4

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Maxima [A]  time = 0.703481, size = 309, normalized size = 1.37 \[ \frac{1}{11} \, c^{3} e^{4} x^{11} + \frac{1}{4} \, b^{3} d^{4} x^{4} + \frac{1}{10} \,{\left (4 \, c^{3} d e^{3} + 3 \, b c^{2} e^{4}\right )} x^{10} + \frac{1}{3} \,{\left (2 \, c^{3} d^{2} e^{2} + 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} x^{9} + \frac{1}{8} \,{\left (4 \, c^{3} d^{3} e + 18 \, b c^{2} d^{2} e^{2} + 12 \, b^{2} c d e^{3} + b^{3} e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (c^{3} d^{4} + 12 \, b c^{2} d^{3} e + 18 \, b^{2} c d^{2} e^{2} + 4 \, b^{3} d e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (b c^{2} d^{4} + 4 \, b^{2} c d^{3} e + 2 \, b^{3} d^{2} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (3 \, b^{2} c d^{4} + 4 \, b^{3} d^{3} e\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(e*x + d)^4,x, algorithm="maxima")

[Out]

1/11*c^3*e^4*x^11 + 1/4*b^3*d^4*x^4 + 1/10*(4*c^3*d*e^3 + 3*b*c^2*e^4)*x^10 + 1/
3*(2*c^3*d^2*e^2 + 4*b*c^2*d*e^3 + b^2*c*e^4)*x^9 + 1/8*(4*c^3*d^3*e + 18*b*c^2*
d^2*e^2 + 12*b^2*c*d*e^3 + b^3*e^4)*x^8 + 1/7*(c^3*d^4 + 12*b*c^2*d^3*e + 18*b^2
*c*d^2*e^2 + 4*b^3*d*e^3)*x^7 + 1/2*(b*c^2*d^4 + 4*b^2*c*d^3*e + 2*b^3*d^2*e^2)*
x^6 + 1/5*(3*b^2*c*d^4 + 4*b^3*d^3*e)*x^5

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Fricas [A]  time = 0.203249, size = 1, normalized size = 0. \[ \frac{1}{11} x^{11} e^{4} c^{3} + \frac{2}{5} x^{10} e^{3} d c^{3} + \frac{3}{10} x^{10} e^{4} c^{2} b + \frac{2}{3} x^{9} e^{2} d^{2} c^{3} + \frac{4}{3} x^{9} e^{3} d c^{2} b + \frac{1}{3} x^{9} e^{4} c b^{2} + \frac{1}{2} x^{8} e d^{3} c^{3} + \frac{9}{4} x^{8} e^{2} d^{2} c^{2} b + \frac{3}{2} x^{8} e^{3} d c b^{2} + \frac{1}{8} x^{8} e^{4} b^{3} + \frac{1}{7} x^{7} d^{4} c^{3} + \frac{12}{7} x^{7} e d^{3} c^{2} b + \frac{18}{7} x^{7} e^{2} d^{2} c b^{2} + \frac{4}{7} x^{7} e^{3} d b^{3} + \frac{1}{2} x^{6} d^{4} c^{2} b + 2 x^{6} e d^{3} c b^{2} + x^{6} e^{2} d^{2} b^{3} + \frac{3}{5} x^{5} d^{4} c b^{2} + \frac{4}{5} x^{5} e d^{3} b^{3} + \frac{1}{4} x^{4} d^{4} b^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(e*x + d)^4,x, algorithm="fricas")

[Out]

1/11*x^11*e^4*c^3 + 2/5*x^10*e^3*d*c^3 + 3/10*x^10*e^4*c^2*b + 2/3*x^9*e^2*d^2*c
^3 + 4/3*x^9*e^3*d*c^2*b + 1/3*x^9*e^4*c*b^2 + 1/2*x^8*e*d^3*c^3 + 9/4*x^8*e^2*d
^2*c^2*b + 3/2*x^8*e^3*d*c*b^2 + 1/8*x^8*e^4*b^3 + 1/7*x^7*d^4*c^3 + 12/7*x^7*e*
d^3*c^2*b + 18/7*x^7*e^2*d^2*c*b^2 + 4/7*x^7*e^3*d*b^3 + 1/2*x^6*d^4*c^2*b + 2*x
^6*e*d^3*c*b^2 + x^6*e^2*d^2*b^3 + 3/5*x^5*d^4*c*b^2 + 4/5*x^5*e*d^3*b^3 + 1/4*x
^4*d^4*b^3

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Sympy [A]  time = 0.236438, size = 257, normalized size = 1.14 \[ \frac{b^{3} d^{4} x^{4}}{4} + \frac{c^{3} e^{4} x^{11}}{11} + x^{10} \left (\frac{3 b c^{2} e^{4}}{10} + \frac{2 c^{3} d e^{3}}{5}\right ) + x^{9} \left (\frac{b^{2} c e^{4}}{3} + \frac{4 b c^{2} d e^{3}}{3} + \frac{2 c^{3} d^{2} e^{2}}{3}\right ) + x^{8} \left (\frac{b^{3} e^{4}}{8} + \frac{3 b^{2} c d e^{3}}{2} + \frac{9 b c^{2} d^{2} e^{2}}{4} + \frac{c^{3} d^{3} e}{2}\right ) + x^{7} \left (\frac{4 b^{3} d e^{3}}{7} + \frac{18 b^{2} c d^{2} e^{2}}{7} + \frac{12 b c^{2} d^{3} e}{7} + \frac{c^{3} d^{4}}{7}\right ) + x^{6} \left (b^{3} d^{2} e^{2} + 2 b^{2} c d^{3} e + \frac{b c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac{4 b^{3} d^{3} e}{5} + \frac{3 b^{2} c d^{4}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4*(c*x**2+b*x)**3,x)

[Out]

b**3*d**4*x**4/4 + c**3*e**4*x**11/11 + x**10*(3*b*c**2*e**4/10 + 2*c**3*d*e**3/
5) + x**9*(b**2*c*e**4/3 + 4*b*c**2*d*e**3/3 + 2*c**3*d**2*e**2/3) + x**8*(b**3*
e**4/8 + 3*b**2*c*d*e**3/2 + 9*b*c**2*d**2*e**2/4 + c**3*d**3*e/2) + x**7*(4*b**
3*d*e**3/7 + 18*b**2*c*d**2*e**2/7 + 12*b*c**2*d**3*e/7 + c**3*d**4/7) + x**6*(b
**3*d**2*e**2 + 2*b**2*c*d**3*e + b*c**2*d**4/2) + x**5*(4*b**3*d**3*e/5 + 3*b**
2*c*d**4/5)

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GIAC/XCAS [A]  time = 0.21032, size = 327, normalized size = 1.45 \[ \frac{1}{11} \, c^{3} x^{11} e^{4} + \frac{2}{5} \, c^{3} d x^{10} e^{3} + \frac{2}{3} \, c^{3} d^{2} x^{9} e^{2} + \frac{1}{2} \, c^{3} d^{3} x^{8} e + \frac{1}{7} \, c^{3} d^{4} x^{7} + \frac{3}{10} \, b c^{2} x^{10} e^{4} + \frac{4}{3} \, b c^{2} d x^{9} e^{3} + \frac{9}{4} \, b c^{2} d^{2} x^{8} e^{2} + \frac{12}{7} \, b c^{2} d^{3} x^{7} e + \frac{1}{2} \, b c^{2} d^{4} x^{6} + \frac{1}{3} \, b^{2} c x^{9} e^{4} + \frac{3}{2} \, b^{2} c d x^{8} e^{3} + \frac{18}{7} \, b^{2} c d^{2} x^{7} e^{2} + 2 \, b^{2} c d^{3} x^{6} e + \frac{3}{5} \, b^{2} c d^{4} x^{5} + \frac{1}{8} \, b^{3} x^{8} e^{4} + \frac{4}{7} \, b^{3} d x^{7} e^{3} + b^{3} d^{2} x^{6} e^{2} + \frac{4}{5} \, b^{3} d^{3} x^{5} e + \frac{1}{4} \, b^{3} d^{4} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3*(e*x + d)^4,x, algorithm="giac")

[Out]

1/11*c^3*x^11*e^4 + 2/5*c^3*d*x^10*e^3 + 2/3*c^3*d^2*x^9*e^2 + 1/2*c^3*d^3*x^8*e
+ 1/7*c^3*d^4*x^7 + 3/10*b*c^2*x^10*e^4 + 4/3*b*c^2*d*x^9*e^3 + 9/4*b*c^2*d^2*x
^8*e^2 + 12/7*b*c^2*d^3*x^7*e + 1/2*b*c^2*d^4*x^6 + 1/3*b^2*c*x^9*e^4 + 3/2*b^2*
c*d*x^8*e^3 + 18/7*b^2*c*d^2*x^7*e^2 + 2*b^2*c*d^3*x^6*e + 3/5*b^2*c*d^4*x^5 + 1
/8*b^3*x^8*e^4 + 4/7*b^3*d*x^7*e^3 + b^3*d^2*x^6*e^2 + 4/5*b^3*d^3*x^5*e + 1/4*b
^3*d^4*x^4