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

Optimal. Leaf size=139 \[ \frac{1}{3} a^3 c x^3+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{1}{2} a^2 b c x^6+\frac{3}{7} a^2 b d x^7+\frac{3}{8} a^2 b e x^8+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{1}{12} b^3 c x^{12}+\frac{1}{13} b^3 d x^{13}+\frac{1}{14} b^3 e x^{14} \]

[Out]

(a^3*c*x^3)/3 + (a^3*d*x^4)/4 + (a^3*e*x^5)/5 + (a^2*b*c*x^6)/2 + (3*a^2*b*d*x^7
)/7 + (3*a^2*b*e*x^8)/8 + (a*b^2*c*x^9)/3 + (3*a*b^2*d*x^10)/10 + (3*a*b^2*e*x^1
1)/11 + (b^3*c*x^12)/12 + (b^3*d*x^13)/13 + (b^3*e*x^14)/14

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Rubi [A]  time = 0.251333, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{1}{3} a^3 c x^3+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{1}{2} a^2 b c x^6+\frac{3}{7} a^2 b d x^7+\frac{3}{8} a^2 b e x^8+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{1}{12} b^3 c x^{12}+\frac{1}{13} b^3 d x^{13}+\frac{1}{14} b^3 e x^{14} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*c*x^3)/3 + (a^3*d*x^4)/4 + (a^3*e*x^5)/5 + (a^2*b*c*x^6)/2 + (3*a^2*b*d*x^7
)/7 + (3*a^2*b*e*x^8)/8 + (a*b^2*c*x^9)/3 + (3*a*b^2*d*x^10)/10 + (3*a*b^2*e*x^1
1)/11 + (b^3*c*x^12)/12 + (b^3*d*x^13)/13 + (b^3*e*x^14)/14

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Rubi in Sympy [A]  time = 34.6413, size = 107, normalized size = 0.77 \[ \frac{a^{3} d x^{4}}{4} + \frac{a^{3} e x^{5}}{5} + \frac{3 a^{2} b d x^{7}}{7} + \frac{3 a^{2} b e x^{8}}{8} + \frac{3 a b^{2} d x^{10}}{10} + \frac{3 a b^{2} e x^{11}}{11} + \frac{b^{3} d x^{13}}{13} + \frac{b^{3} e x^{14}}{14} + \frac{c \left (a + b x^{3}\right )^{4}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*d*x**4/4 + a**3*e*x**5/5 + 3*a**2*b*d*x**7/7 + 3*a**2*b*e*x**8/8 + 3*a*b**2
*d*x**10/10 + 3*a*b**2*e*x**11/11 + b**3*d*x**13/13 + b**3*e*x**14/14 + c*(a + b
*x**3)**4/(12*b)

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Mathematica [A]  time = 0.00765015, size = 139, normalized size = 1. \[ \frac{1}{3} a^3 c x^3+\frac{1}{4} a^3 d x^4+\frac{1}{5} a^3 e x^5+\frac{1}{2} a^2 b c x^6+\frac{3}{7} a^2 b d x^7+\frac{3}{8} a^2 b e x^8+\frac{1}{3} a b^2 c x^9+\frac{3}{10} a b^2 d x^{10}+\frac{3}{11} a b^2 e x^{11}+\frac{1}{12} b^3 c x^{12}+\frac{1}{13} b^3 d x^{13}+\frac{1}{14} b^3 e x^{14} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*c*x^3)/3 + (a^3*d*x^4)/4 + (a^3*e*x^5)/5 + (a^2*b*c*x^6)/2 + (3*a^2*b*d*x^7
)/7 + (3*a^2*b*e*x^8)/8 + (a*b^2*c*x^9)/3 + (3*a*b^2*d*x^10)/10 + (3*a*b^2*e*x^1
1)/11 + (b^3*c*x^12)/12 + (b^3*d*x^13)/13 + (b^3*e*x^14)/14

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Maple [A]  time = 0.001, size = 116, normalized size = 0.8 \[{\frac{{a}^{3}c{x}^{3}}{3}}+{\frac{{a}^{3}d{x}^{4}}{4}}+{\frac{{a}^{3}e{x}^{5}}{5}}+{\frac{{a}^{2}bc{x}^{6}}{2}}+{\frac{3\,{a}^{2}bd{x}^{7}}{7}}+{\frac{3\,{a}^{2}be{x}^{8}}{8}}+{\frac{a{b}^{2}c{x}^{9}}{3}}+{\frac{3\,a{b}^{2}d{x}^{10}}{10}}+{\frac{3\,a{b}^{2}e{x}^{11}}{11}}+{\frac{{b}^{3}c{x}^{12}}{12}}+{\frac{{b}^{3}d{x}^{13}}{13}}+{\frac{{b}^{3}e{x}^{14}}{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^3*c*x^3+1/4*a^3*d*x^4+1/5*a^3*e*x^5+1/2*a^2*b*c*x^6+3/7*a^2*b*d*x^7+3/8*a^
2*b*e*x^8+1/3*a*b^2*c*x^9+3/10*a*b^2*d*x^10+3/11*a*b^2*e*x^11+1/12*b^3*c*x^12+1/
13*b^3*d*x^13+1/14*b^3*e*x^14

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Maxima [A]  time = 1.37136, size = 155, normalized size = 1.12 \[ \frac{1}{14} \, b^{3} e x^{14} + \frac{1}{13} \, b^{3} d x^{13} + \frac{1}{12} \, b^{3} c x^{12} + \frac{3}{11} \, a b^{2} e x^{11} + \frac{3}{10} \, a b^{2} d x^{10} + \frac{1}{3} \, a b^{2} c x^{9} + \frac{3}{8} \, a^{2} b e x^{8} + \frac{3}{7} \, a^{2} b d x^{7} + \frac{1}{2} \, a^{2} b c x^{6} + \frac{1}{5} \, a^{3} e x^{5} + \frac{1}{4} \, a^{3} d x^{4} + \frac{1}{3} \, a^{3} c x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*b^3*e*x^14 + 1/13*b^3*d*x^13 + 1/12*b^3*c*x^12 + 3/11*a*b^2*e*x^11 + 3/10*a
*b^2*d*x^10 + 1/3*a*b^2*c*x^9 + 3/8*a^2*b*e*x^8 + 3/7*a^2*b*d*x^7 + 1/2*a^2*b*c*
x^6 + 1/5*a^3*e*x^5 + 1/4*a^3*d*x^4 + 1/3*a^3*c*x^3

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Fricas [A]  time = 0.185472, size = 1, normalized size = 0.01 \[ \frac{1}{14} x^{14} e b^{3} + \frac{1}{13} x^{13} d b^{3} + \frac{1}{12} x^{12} c b^{3} + \frac{3}{11} x^{11} e b^{2} a + \frac{3}{10} x^{10} d b^{2} a + \frac{1}{3} x^{9} c b^{2} a + \frac{3}{8} x^{8} e b a^{2} + \frac{3}{7} x^{7} d b a^{2} + \frac{1}{2} x^{6} c b a^{2} + \frac{1}{5} x^{5} e a^{3} + \frac{1}{4} x^{4} d a^{3} + \frac{1}{3} x^{3} c a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*x^14*e*b^3 + 1/13*x^13*d*b^3 + 1/12*x^12*c*b^3 + 3/11*x^11*e*b^2*a + 3/10*x
^10*d*b^2*a + 1/3*x^9*c*b^2*a + 3/8*x^8*e*b*a^2 + 3/7*x^7*d*b*a^2 + 1/2*x^6*c*b*
a^2 + 1/5*x^5*e*a^3 + 1/4*x^4*d*a^3 + 1/3*x^3*c*a^3

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Sympy [A]  time = 0.07861, size = 138, normalized size = 0.99 \[ \frac{a^{3} c x^{3}}{3} + \frac{a^{3} d x^{4}}{4} + \frac{a^{3} e x^{5}}{5} + \frac{a^{2} b c x^{6}}{2} + \frac{3 a^{2} b d x^{7}}{7} + \frac{3 a^{2} b e x^{8}}{8} + \frac{a b^{2} c x^{9}}{3} + \frac{3 a b^{2} d x^{10}}{10} + \frac{3 a b^{2} e x^{11}}{11} + \frac{b^{3} c x^{12}}{12} + \frac{b^{3} d x^{13}}{13} + \frac{b^{3} e x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*c*x**3/3 + a**3*d*x**4/4 + a**3*e*x**5/5 + a**2*b*c*x**6/2 + 3*a**2*b*d*x**
7/7 + 3*a**2*b*e*x**8/8 + a*b**2*c*x**9/3 + 3*a*b**2*d*x**10/10 + 3*a*b**2*e*x**
11/11 + b**3*c*x**12/12 + b**3*d*x**13/13 + b**3*e*x**14/14

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GIAC/XCAS [A]  time = 0.208852, size = 161, normalized size = 1.16 \[ \frac{1}{14} \, b^{3} x^{14} e + \frac{1}{13} \, b^{3} d x^{13} + \frac{1}{12} \, b^{3} c x^{12} + \frac{3}{11} \, a b^{2} x^{11} e + \frac{3}{10} \, a b^{2} d x^{10} + \frac{1}{3} \, a b^{2} c x^{9} + \frac{3}{8} \, a^{2} b x^{8} e + \frac{3}{7} \, a^{2} b d x^{7} + \frac{1}{2} \, a^{2} b c x^{6} + \frac{1}{5} \, a^{3} x^{5} e + \frac{1}{4} \, a^{3} d x^{4} + \frac{1}{3} \, a^{3} c x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*b^3*x^14*e + 1/13*b^3*d*x^13 + 1/12*b^3*c*x^12 + 3/11*a*b^2*x^11*e + 3/10*a
*b^2*d*x^10 + 1/3*a*b^2*c*x^9 + 3/8*a^2*b*x^8*e + 3/7*a^2*b*d*x^7 + 1/2*a^2*b*c*
x^6 + 1/5*a^3*x^5*e + 1/4*a^3*d*x^4 + 1/3*a^3*c*x^3