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

Optimal. Leaf size=134 \[ a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+\frac{1}{2} a^2 b e x^6+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{3} a b^2 e x^9+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11}+\frac{1}{12} b^3 e x^{12} \]

[Out]

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

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Rubi [A]  time = 0.183741, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+\frac{1}{2} a^2 b e x^6+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{3} a b^2 e x^9+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11}+\frac{1}{12} b^3 e x^{12} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{3} d \int x\, dx + a^{3} \int c\, dx + \frac{3 a^{2} b c x^{4}}{4} + \frac{3 a^{2} b d x^{5}}{5} + \frac{3 a b^{2} c x^{7}}{7} + \frac{3 a b^{2} d x^{8}}{8} + \frac{b^{3} c x^{10}}{10} + \frac{b^{3} d x^{11}}{11} + \frac{e \left (a + b x^{3}\right )^{4}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*d*Integral(x, x) + a**3*Integral(c, x) + 3*a**2*b*c*x**4/4 + 3*a**2*b*d*x**
5/5 + 3*a*b**2*c*x**7/7 + 3*a*b**2*d*x**8/8 + b**3*c*x**10/10 + b**3*d*x**11/11
+ e*(a + b*x**3)**4/(12*b)

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Mathematica [A]  time = 0.00637086, size = 134, normalized size = 1. \[ a^3 c x+\frac{1}{2} a^3 d x^2+\frac{1}{3} a^3 e x^3+\frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+\frac{1}{2} a^2 b e x^6+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{3} a b^2 e x^9+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11}+\frac{1}{12} b^3 e x^{12} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 113, normalized size = 0.8 \[{a}^{3}cx+{\frac{{a}^{3}d{x}^{2}}{2}}+{\frac{{a}^{3}e{x}^{3}}{3}}+{\frac{3\,{a}^{2}bc{x}^{4}}{4}}+{\frac{3\,{a}^{2}bd{x}^{5}}{5}}+{\frac{{a}^{2}be{x}^{6}}{2}}+{\frac{3\,a{b}^{2}c{x}^{7}}{7}}+{\frac{3\,a{b}^{2}d{x}^{8}}{8}}+{\frac{a{b}^{2}e{x}^{9}}{3}}+{\frac{{b}^{3}c{x}^{10}}{10}}+{\frac{{b}^{3}d{x}^{11}}{11}}+{\frac{{b}^{3}e{x}^{12}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.40354, size = 151, normalized size = 1.13 \[ \frac{1}{12} \, b^{3} e x^{12} + \frac{1}{11} \, b^{3} d x^{11} + \frac{1}{10} \, b^{3} c x^{10} + \frac{1}{3} \, a b^{2} e x^{9} + \frac{3}{8} \, a b^{2} d x^{8} + \frac{3}{7} \, a b^{2} c x^{7} + \frac{1}{2} \, a^{2} b e x^{6} + \frac{3}{5} \, a^{2} b d x^{5} + \frac{3}{4} \, a^{2} b c x^{4} + \frac{1}{3} \, a^{3} e x^{3} + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.185684, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} e b^{3} + \frac{1}{11} x^{11} d b^{3} + \frac{1}{10} x^{10} c b^{3} + \frac{1}{3} x^{9} e b^{2} a + \frac{3}{8} x^{8} d b^{2} a + \frac{3}{7} x^{7} c b^{2} a + \frac{1}{2} x^{6} e b a^{2} + \frac{3}{5} x^{5} d b a^{2} + \frac{3}{4} x^{4} c b a^{2} + \frac{1}{3} x^{3} e a^{3} + \frac{1}{2} x^{2} d a^{3} + x c a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.075547, size = 134, normalized size = 1. \[ a^{3} c x + \frac{a^{3} d x^{2}}{2} + \frac{a^{3} e x^{3}}{3} + \frac{3 a^{2} b c x^{4}}{4} + \frac{3 a^{2} b d x^{5}}{5} + \frac{a^{2} b e x^{6}}{2} + \frac{3 a b^{2} c x^{7}}{7} + \frac{3 a b^{2} d x^{8}}{8} + \frac{a b^{2} e x^{9}}{3} + \frac{b^{3} c x^{10}}{10} + \frac{b^{3} d x^{11}}{11} + \frac{b^{3} e x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209451, size = 157, normalized size = 1.17 \[ \frac{1}{12} \, b^{3} x^{12} e + \frac{1}{11} \, b^{3} d x^{11} + \frac{1}{10} \, b^{3} c x^{10} + \frac{1}{3} \, a b^{2} x^{9} e + \frac{3}{8} \, a b^{2} d x^{8} + \frac{3}{7} \, a b^{2} c x^{7} + \frac{1}{2} \, a^{2} b x^{6} e + \frac{3}{5} \, a^{2} b d x^{5} + \frac{3}{4} \, a^{2} b c x^{4} + \frac{1}{3} \, a^{3} x^{3} e + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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