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

Optimal. Leaf size=97 \[ \frac{1}{3} a^2 c x^3+\frac{1}{4} a^2 d x^4+\frac{1}{5} a^2 e x^5+\frac{1}{3} a b c x^6+\frac{2}{7} a b d x^7+\frac{1}{4} a b e x^8+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]

[Out]

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

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Rubi [A]  time = 0.161161, antiderivative size = 97, 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 \[ \frac{1}{3} a^2 c x^3+\frac{1}{4} a^2 d x^4+\frac{1}{5} a^2 e x^5+\frac{1}{3} a b c x^6+\frac{2}{7} a b d x^7+\frac{1}{4} a b e x^8+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.1027, size = 75, normalized size = 0.77 \[ \frac{a^{2} d x^{4}}{4} + \frac{a^{2} e x^{5}}{5} + \frac{2 a b d x^{7}}{7} + \frac{a b e x^{8}}{4} + \frac{b^{2} d x^{10}}{10} + \frac{b^{2} e x^{11}}{11} + \frac{c \left (a + b x^{3}\right )^{3}}{9 b} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.00599648, size = 97, normalized size = 1. \[ \frac{1}{3} a^2 c x^3+\frac{1}{4} a^2 d x^4+\frac{1}{5} a^2 e x^5+\frac{1}{3} a b c x^6+\frac{2}{7} a b d x^7+\frac{1}{4} a b e x^8+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 80, normalized size = 0.8 \[{\frac{{a}^{2}c{x}^{3}}{3}}+{\frac{{a}^{2}d{x}^{4}}{4}}+{\frac{{a}^{2}e{x}^{5}}{5}}+{\frac{abc{x}^{6}}{3}}+{\frac{2\,abd{x}^{7}}{7}}+{\frac{abe{x}^{8}}{4}}+{\frac{{b}^{2}c{x}^{9}}{9}}+{\frac{{b}^{2}d{x}^{10}}{10}}+{\frac{{b}^{2}e{x}^{11}}{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37591, size = 107, normalized size = 1.1 \[ \frac{1}{11} \, b^{2} e x^{11} + \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{9} \, b^{2} c x^{9} + \frac{1}{4} \, a b e x^{8} + \frac{2}{7} \, a b d x^{7} + \frac{1}{3} \, a b c x^{6} + \frac{1}{5} \, a^{2} e x^{5} + \frac{1}{4} \, a^{2} d x^{4} + \frac{1}{3} \, a^{2} c x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.183414, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} e b^{2} + \frac{1}{10} x^{10} d b^{2} + \frac{1}{9} x^{9} c b^{2} + \frac{1}{4} x^{8} e b a + \frac{2}{7} x^{7} d b a + \frac{1}{3} x^{6} c b a + \frac{1}{5} x^{5} e a^{2} + \frac{1}{4} x^{4} d a^{2} + \frac{1}{3} x^{3} c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.065967, size = 92, normalized size = 0.95 \[ \frac{a^{2} c x^{3}}{3} + \frac{a^{2} d x^{4}}{4} + \frac{a^{2} e x^{5}}{5} + \frac{a b c x^{6}}{3} + \frac{2 a b d x^{7}}{7} + \frac{a b e x^{8}}{4} + \frac{b^{2} c x^{9}}{9} + \frac{b^{2} d x^{10}}{10} + \frac{b^{2} e x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209191, size = 111, normalized size = 1.14 \[ \frac{1}{11} \, b^{2} x^{11} e + \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{9} \, b^{2} c x^{9} + \frac{1}{4} \, a b x^{8} e + \frac{2}{7} \, a b d x^{7} + \frac{1}{3} \, a b c x^{6} + \frac{1}{5} \, a^{2} x^{5} e + \frac{1}{4} \, a^{2} d x^{4} + \frac{1}{3} \, a^{2} c x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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