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

Optimal. Leaf size=181 \[ \frac{1}{3} a^4 c x^3+\frac{1}{4} a^4 d x^4+\frac{1}{5} a^4 e x^5+\frac{2}{3} a^3 b c x^6+\frac{4}{7} a^3 b d x^7+\frac{1}{2} a^3 b e x^8+\frac{2}{3} a^2 b^2 c x^9+\frac{3}{5} a^2 b^2 d x^{10}+\frac{6}{11} a^2 b^2 e x^{11}+\frac{1}{3} a b^3 c x^{12}+\frac{4}{13} a b^3 d x^{13}+\frac{2}{7} a b^3 e x^{14}+\frac{1}{15} b^4 c x^{15}+\frac{1}{16} b^4 d x^{16}+\frac{1}{17} b^4 e x^{17} \]

[Out]

(a^4*c*x^3)/3 + (a^4*d*x^4)/4 + (a^4*e*x^5)/5 + (2*a^3*b*c*x^6)/3 + (4*a^3*b*d*x
^7)/7 + (a^3*b*e*x^8)/2 + (2*a^2*b^2*c*x^9)/3 + (3*a^2*b^2*d*x^10)/5 + (6*a^2*b^
2*e*x^11)/11 + (a*b^3*c*x^12)/3 + (4*a*b^3*d*x^13)/13 + (2*a*b^3*e*x^14)/7 + (b^
4*c*x^15)/15 + (b^4*d*x^16)/16 + (b^4*e*x^17)/17

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Rubi [A]  time = 0.333306, antiderivative size = 181, 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 \[ \frac{1}{3} a^4 c x^3+\frac{1}{4} a^4 d x^4+\frac{1}{5} a^4 e x^5+\frac{2}{3} a^3 b c x^6+\frac{4}{7} a^3 b d x^7+\frac{1}{2} a^3 b e x^8+\frac{2}{3} a^2 b^2 c x^9+\frac{3}{5} a^2 b^2 d x^{10}+\frac{6}{11} a^2 b^2 e x^{11}+\frac{1}{3} a b^3 c x^{12}+\frac{4}{13} a b^3 d x^{13}+\frac{2}{7} a b^3 e x^{14}+\frac{1}{15} b^4 c x^{15}+\frac{1}{16} b^4 d x^{16}+\frac{1}{17} b^4 e x^{17} \]

Antiderivative was successfully verified.

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

[Out]

(a^4*c*x^3)/3 + (a^4*d*x^4)/4 + (a^4*e*x^5)/5 + (2*a^3*b*c*x^6)/3 + (4*a^3*b*d*x
^7)/7 + (a^3*b*e*x^8)/2 + (2*a^2*b^2*c*x^9)/3 + (3*a^2*b^2*d*x^10)/5 + (6*a^2*b^
2*e*x^11)/11 + (a*b^3*c*x^12)/3 + (4*a*b^3*d*x^13)/13 + (2*a*b^3*e*x^14)/7 + (b^
4*c*x^15)/15 + (b^4*d*x^16)/16 + (b^4*e*x^17)/17

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Rubi in Sympy [A]  time = 42.2621, size = 136, normalized size = 0.75 \[ \frac{a^{4} d x^{4}}{4} + \frac{a^{4} e x^{5}}{5} + \frac{4 a^{3} b d x^{7}}{7} + \frac{a^{3} b e x^{8}}{2} + \frac{3 a^{2} b^{2} d x^{10}}{5} + \frac{6 a^{2} b^{2} e x^{11}}{11} + \frac{4 a b^{3} d x^{13}}{13} + \frac{2 a b^{3} e x^{14}}{7} + \frac{b^{4} d x^{16}}{16} + \frac{b^{4} e x^{17}}{17} + \frac{c \left (a + b x^{3}\right )^{5}}{15 b} \]

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)**4,x)

[Out]

a**4*d*x**4/4 + a**4*e*x**5/5 + 4*a**3*b*d*x**7/7 + a**3*b*e*x**8/2 + 3*a**2*b**
2*d*x**10/5 + 6*a**2*b**2*e*x**11/11 + 4*a*b**3*d*x**13/13 + 2*a*b**3*e*x**14/7
+ b**4*d*x**16/16 + b**4*e*x**17/17 + c*(a + b*x**3)**5/(15*b)

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Mathematica [A]  time = 0.00786134, size = 181, normalized size = 1. \[ \frac{1}{3} a^4 c x^3+\frac{1}{4} a^4 d x^4+\frac{1}{5} a^4 e x^5+\frac{2}{3} a^3 b c x^6+\frac{4}{7} a^3 b d x^7+\frac{1}{2} a^3 b e x^8+\frac{2}{3} a^2 b^2 c x^9+\frac{3}{5} a^2 b^2 d x^{10}+\frac{6}{11} a^2 b^2 e x^{11}+\frac{1}{3} a b^3 c x^{12}+\frac{4}{13} a b^3 d x^{13}+\frac{2}{7} a b^3 e x^{14}+\frac{1}{15} b^4 c x^{15}+\frac{1}{16} b^4 d x^{16}+\frac{1}{17} b^4 e x^{17} \]

Antiderivative was successfully verified.

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

[Out]

(a^4*c*x^3)/3 + (a^4*d*x^4)/4 + (a^4*e*x^5)/5 + (2*a^3*b*c*x^6)/3 + (4*a^3*b*d*x
^7)/7 + (a^3*b*e*x^8)/2 + (2*a^2*b^2*c*x^9)/3 + (3*a^2*b^2*d*x^10)/5 + (6*a^2*b^
2*e*x^11)/11 + (a*b^3*c*x^12)/3 + (4*a*b^3*d*x^13)/13 + (2*a*b^3*e*x^14)/7 + (b^
4*c*x^15)/15 + (b^4*d*x^16)/16 + (b^4*e*x^17)/17

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Maple [A]  time = 0.002, size = 152, normalized size = 0.8 \[{\frac{{a}^{4}c{x}^{3}}{3}}+{\frac{{a}^{4}d{x}^{4}}{4}}+{\frac{{a}^{4}e{x}^{5}}{5}}+{\frac{2\,{a}^{3}bc{x}^{6}}{3}}+{\frac{4\,{a}^{3}bd{x}^{7}}{7}}+{\frac{{a}^{3}be{x}^{8}}{2}}+{\frac{2\,{a}^{2}{b}^{2}c{x}^{9}}{3}}+{\frac{3\,{a}^{2}{b}^{2}d{x}^{10}}{5}}+{\frac{6\,{a}^{2}{b}^{2}e{x}^{11}}{11}}+{\frac{a{b}^{3}c{x}^{12}}{3}}+{\frac{4\,a{b}^{3}d{x}^{13}}{13}}+{\frac{2\,a{b}^{3}e{x}^{14}}{7}}+{\frac{{b}^{4}c{x}^{15}}{15}}+{\frac{{b}^{4}d{x}^{16}}{16}}+{\frac{{b}^{4}e{x}^{17}}{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^4*c*x^3+1/4*a^4*d*x^4+1/5*a^4*e*x^5+2/3*a^3*b*c*x^6+4/7*a^3*b*d*x^7+1/2*a^
3*b*e*x^8+2/3*a^2*b^2*c*x^9+3/5*a^2*b^2*d*x^10+6/11*a^2*b^2*e*x^11+1/3*a*b^3*c*x
^12+4/13*a*b^3*d*x^13+2/7*a*b^3*e*x^14+1/15*b^4*c*x^15+1/16*b^4*d*x^16+1/17*b^4*
e*x^17

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Maxima [A]  time = 1.37662, size = 204, normalized size = 1.13 \[ \frac{1}{17} \, b^{4} e x^{17} + \frac{1}{16} \, b^{4} d x^{16} + \frac{1}{15} \, b^{4} c x^{15} + \frac{2}{7} \, a b^{3} e x^{14} + \frac{4}{13} \, a b^{3} d x^{13} + \frac{1}{3} \, a b^{3} c x^{12} + \frac{6}{11} \, a^{2} b^{2} e x^{11} + \frac{3}{5} \, a^{2} b^{2} d x^{10} + \frac{2}{3} \, a^{2} b^{2} c x^{9} + \frac{1}{2} \, a^{3} b e x^{8} + \frac{4}{7} \, a^{3} b d x^{7} + \frac{2}{3} \, a^{3} b c x^{6} + \frac{1}{5} \, a^{4} e x^{5} + \frac{1}{4} \, a^{4} d x^{4} + \frac{1}{3} \, a^{4} c x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/17*b^4*e*x^17 + 1/16*b^4*d*x^16 + 1/15*b^4*c*x^15 + 2/7*a*b^3*e*x^14 + 4/13*a*
b^3*d*x^13 + 1/3*a*b^3*c*x^12 + 6/11*a^2*b^2*e*x^11 + 3/5*a^2*b^2*d*x^10 + 2/3*a
^2*b^2*c*x^9 + 1/2*a^3*b*e*x^8 + 4/7*a^3*b*d*x^7 + 2/3*a^3*b*c*x^6 + 1/5*a^4*e*x
^5 + 1/4*a^4*d*x^4 + 1/3*a^4*c*x^3

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Fricas [A]  time = 0.18471, size = 1, normalized size = 0.01 \[ \frac{1}{17} x^{17} e b^{4} + \frac{1}{16} x^{16} d b^{4} + \frac{1}{15} x^{15} c b^{4} + \frac{2}{7} x^{14} e b^{3} a + \frac{4}{13} x^{13} d b^{3} a + \frac{1}{3} x^{12} c b^{3} a + \frac{6}{11} x^{11} e b^{2} a^{2} + \frac{3}{5} x^{10} d b^{2} a^{2} + \frac{2}{3} x^{9} c b^{2} a^{2} + \frac{1}{2} x^{8} e b a^{3} + \frac{4}{7} x^{7} d b a^{3} + \frac{2}{3} x^{6} c b a^{3} + \frac{1}{5} x^{5} e a^{4} + \frac{1}{4} x^{4} d a^{4} + \frac{1}{3} x^{3} c a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/17*x^17*e*b^4 + 1/16*x^16*d*b^4 + 1/15*x^15*c*b^4 + 2/7*x^14*e*b^3*a + 4/13*x^
13*d*b^3*a + 1/3*x^12*c*b^3*a + 6/11*x^11*e*b^2*a^2 + 3/5*x^10*d*b^2*a^2 + 2/3*x
^9*c*b^2*a^2 + 1/2*x^8*e*b*a^3 + 4/7*x^7*d*b*a^3 + 2/3*x^6*c*b*a^3 + 1/5*x^5*e*a
^4 + 1/4*x^4*d*a^4 + 1/3*x^3*c*a^4

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Sympy [A]  time = 0.088768, size = 184, normalized size = 1.02 \[ \frac{a^{4} c x^{3}}{3} + \frac{a^{4} d x^{4}}{4} + \frac{a^{4} e x^{5}}{5} + \frac{2 a^{3} b c x^{6}}{3} + \frac{4 a^{3} b d x^{7}}{7} + \frac{a^{3} b e x^{8}}{2} + \frac{2 a^{2} b^{2} c x^{9}}{3} + \frac{3 a^{2} b^{2} d x^{10}}{5} + \frac{6 a^{2} b^{2} e x^{11}}{11} + \frac{a b^{3} c x^{12}}{3} + \frac{4 a b^{3} d x^{13}}{13} + \frac{2 a b^{3} e x^{14}}{7} + \frac{b^{4} c x^{15}}{15} + \frac{b^{4} d x^{16}}{16} + \frac{b^{4} e x^{17}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**4*c*x**3/3 + a**4*d*x**4/4 + a**4*e*x**5/5 + 2*a**3*b*c*x**6/3 + 4*a**3*b*d*x
**7/7 + a**3*b*e*x**8/2 + 2*a**2*b**2*c*x**9/3 + 3*a**2*b**2*d*x**10/5 + 6*a**2*
b**2*e*x**11/11 + a*b**3*c*x**12/3 + 4*a*b**3*d*x**13/13 + 2*a*b**3*e*x**14/7 +
b**4*c*x**15/15 + b**4*d*x**16/16 + b**4*e*x**17/17

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GIAC/XCAS [A]  time = 0.209389, size = 211, normalized size = 1.17 \[ \frac{1}{17} \, b^{4} x^{17} e + \frac{1}{16} \, b^{4} d x^{16} + \frac{1}{15} \, b^{4} c x^{15} + \frac{2}{7} \, a b^{3} x^{14} e + \frac{4}{13} \, a b^{3} d x^{13} + \frac{1}{3} \, a b^{3} c x^{12} + \frac{6}{11} \, a^{2} b^{2} x^{11} e + \frac{3}{5} \, a^{2} b^{2} d x^{10} + \frac{2}{3} \, a^{2} b^{2} c x^{9} + \frac{1}{2} \, a^{3} b x^{8} e + \frac{4}{7} \, a^{3} b d x^{7} + \frac{2}{3} \, a^{3} b c x^{6} + \frac{1}{5} \, a^{4} x^{5} e + \frac{1}{4} \, a^{4} d x^{4} + \frac{1}{3} \, a^{4} c x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/17*b^4*x^17*e + 1/16*b^4*d*x^16 + 1/15*b^4*c*x^15 + 2/7*a*b^3*x^14*e + 4/13*a*
b^3*d*x^13 + 1/3*a*b^3*c*x^12 + 6/11*a^2*b^2*x^11*e + 3/5*a^2*b^2*d*x^10 + 2/3*a
^2*b^2*c*x^9 + 1/2*a^3*b*x^8*e + 4/7*a^3*b*d*x^7 + 2/3*a^3*b*c*x^6 + 1/5*a^4*x^5
*e + 1/4*a^4*d*x^4 + 1/3*a^4*c*x^3