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

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

[Out]

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

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Rubi [A]  time = 0.213196, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{1}{2} a^3 c x^2+\frac{1}{3} a^3 d x^3+\frac{1}{4} a^3 e x^4+\frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+\frac{3}{8} a b^2 c x^8+\frac{1}{3} a b^2 d x^9+\frac{3}{10} a b^2 e x^{10}+\frac{1}{11} b^3 c x^{11}+\frac{1}{12} b^3 d x^{12}+\frac{1}{13} b^3 e x^{13} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{3} c \int x\, dx + \frac{a^{3} e x^{4}}{4} + \frac{3 a^{2} b c x^{5}}{5} + \frac{3 a^{2} b e x^{7}}{7} + \frac{3 a b^{2} c x^{8}}{8} + \frac{3 a b^{2} e x^{10}}{10} + \frac{b^{3} c x^{11}}{11} + \frac{b^{3} e x^{13}}{13} + \frac{d \left (a + b x^{3}\right )^{4}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00643486, size = 139, normalized size = 1. \[ \frac{1}{2} a^3 c x^2+\frac{1}{3} a^3 d x^3+\frac{1}{4} a^3 e x^4+\frac{3}{5} a^2 b c x^5+\frac{1}{2} a^2 b d x^6+\frac{3}{7} a^2 b e x^7+\frac{3}{8} a b^2 c x^8+\frac{1}{3} a b^2 d x^9+\frac{3}{10} a b^2 e x^{10}+\frac{1}{11} b^3 c x^{11}+\frac{1}{12} b^3 d x^{12}+\frac{1}{13} b^3 e x^{13} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.41346, size = 155, normalized size = 1.12 \[ \frac{1}{13} \, b^{3} e x^{13} + \frac{1}{12} \, b^{3} d x^{12} + \frac{1}{11} \, b^{3} c x^{11} + \frac{3}{10} \, a b^{2} e x^{10} + \frac{1}{3} \, a b^{2} d x^{9} + \frac{3}{8} \, a b^{2} c x^{8} + \frac{3}{7} \, a^{2} b e x^{7} + \frac{1}{2} \, a^{2} b d x^{6} + \frac{3}{5} \, a^{2} b c x^{5} + \frac{1}{4} \, a^{3} e x^{4} + \frac{1}{3} \, a^{3} d x^{3} + \frac{1}{2} \, a^{3} c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.186303, size = 1, normalized size = 0.01 \[ \frac{1}{13} x^{13} e b^{3} + \frac{1}{12} x^{12} d b^{3} + \frac{1}{11} x^{11} c b^{3} + \frac{3}{10} x^{10} e b^{2} a + \frac{1}{3} x^{9} d b^{2} a + \frac{3}{8} x^{8} c b^{2} a + \frac{3}{7} x^{7} e b a^{2} + \frac{1}{2} x^{6} d b a^{2} + \frac{3}{5} x^{5} c b a^{2} + \frac{1}{4} x^{4} e a^{3} + \frac{1}{3} x^{3} d a^{3} + \frac{1}{2} x^{2} c a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.077943, size = 138, normalized size = 0.99 \[ \frac{a^{3} c x^{2}}{2} + \frac{a^{3} d x^{3}}{3} + \frac{a^{3} e x^{4}}{4} + \frac{3 a^{2} b c x^{5}}{5} + \frac{a^{2} b d x^{6}}{2} + \frac{3 a^{2} b e x^{7}}{7} + \frac{3 a b^{2} c x^{8}}{8} + \frac{a b^{2} d x^{9}}{3} + \frac{3 a b^{2} e x^{10}}{10} + \frac{b^{3} c x^{11}}{11} + \frac{b^{3} d x^{12}}{12} + \frac{b^{3} e x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209888, size = 161, normalized size = 1.16 \[ \frac{1}{13} \, b^{3} x^{13} e + \frac{1}{12} \, b^{3} d x^{12} + \frac{1}{11} \, b^{3} c x^{11} + \frac{3}{10} \, a b^{2} x^{10} e + \frac{1}{3} \, a b^{2} d x^{9} + \frac{3}{8} \, a b^{2} c x^{8} + \frac{3}{7} \, a^{2} b x^{7} e + \frac{1}{2} \, a^{2} b d x^{6} + \frac{3}{5} \, a^{2} b c x^{5} + \frac{1}{4} \, a^{3} x^{4} e + \frac{1}{3} \, a^{3} d x^{3} + \frac{1}{2} \, a^{3} c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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