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

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

[Out]

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

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Rubi [A]  time = 0.306498, antiderivative size = 181, 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 \[ \frac{1}{2} a^4 c x^2+\frac{1}{3} a^4 d x^3+\frac{1}{4} a^4 e x^4+\frac{4}{5} a^3 b c x^5+\frac{2}{3} a^3 b d x^6+\frac{4}{7} a^3 b e x^7+\frac{3}{4} a^2 b^2 c x^8+\frac{2}{3} a^2 b^2 d x^9+\frac{3}{5} a^2 b^2 e x^{10}+\frac{4}{11} a b^3 c x^{11}+\frac{1}{3} a b^3 d x^{12}+\frac{4}{13} a b^3 e x^{13}+\frac{1}{14} b^4 c x^{14}+\frac{1}{15} b^4 d x^{15}+\frac{1}{16} b^4 e x^{16} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{4} c \int x\, dx + \frac{a^{4} e x^{4}}{4} + \frac{4 a^{3} b c x^{5}}{5} + \frac{4 a^{3} b e x^{7}}{7} + \frac{3 a^{2} b^{2} c x^{8}}{4} + \frac{3 a^{2} b^{2} e x^{10}}{5} + \frac{4 a b^{3} c x^{11}}{11} + \frac{4 a b^{3} e x^{13}}{13} + \frac{b^{4} c x^{14}}{14} + \frac{b^{4} e x^{16}}{16} + \frac{d \left (a + b x^{3}\right )^{5}}{15 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00765655, size = 181, normalized size = 1. \[ \frac{1}{2} a^4 c x^2+\frac{1}{3} a^4 d x^3+\frac{1}{4} a^4 e x^4+\frac{4}{5} a^3 b c x^5+\frac{2}{3} a^3 b d x^6+\frac{4}{7} a^3 b e x^7+\frac{3}{4} a^2 b^2 c x^8+\frac{2}{3} a^2 b^2 d x^9+\frac{3}{5} a^2 b^2 e x^{10}+\frac{4}{11} a b^3 c x^{11}+\frac{1}{3} a b^3 d x^{12}+\frac{4}{13} a b^3 e x^{13}+\frac{1}{14} b^4 c x^{14}+\frac{1}{15} b^4 d x^{15}+\frac{1}{16} b^4 e x^{16} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 152, normalized size = 0.8 \[{\frac{{a}^{4}c{x}^{2}}{2}}+{\frac{{a}^{4}d{x}^{3}}{3}}+{\frac{{a}^{4}e{x}^{4}}{4}}+{\frac{4\,{a}^{3}bc{x}^{5}}{5}}+{\frac{2\,{a}^{3}bd{x}^{6}}{3}}+{\frac{4\,{a}^{3}be{x}^{7}}{7}}+{\frac{3\,{a}^{2}{b}^{2}c{x}^{8}}{4}}+{\frac{2\,{a}^{2}{b}^{2}d{x}^{9}}{3}}+{\frac{3\,{a}^{2}{b}^{2}e{x}^{10}}{5}}+{\frac{4\,a{b}^{3}c{x}^{11}}{11}}+{\frac{a{b}^{3}d{x}^{12}}{3}}+{\frac{4\,a{b}^{3}e{x}^{13}}{13}}+{\frac{{b}^{4}c{x}^{14}}{14}}+{\frac{{b}^{4}d{x}^{15}}{15}}+{\frac{{b}^{4}e{x}^{16}}{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.42459, size = 204, normalized size = 1.13 \[ \frac{1}{16} \, b^{4} e x^{16} + \frac{1}{15} \, b^{4} d x^{15} + \frac{1}{14} \, b^{4} c x^{14} + \frac{4}{13} \, a b^{3} e x^{13} + \frac{1}{3} \, a b^{3} d x^{12} + \frac{4}{11} \, a b^{3} c x^{11} + \frac{3}{5} \, a^{2} b^{2} e x^{10} + \frac{2}{3} \, a^{2} b^{2} d x^{9} + \frac{3}{4} \, a^{2} b^{2} c x^{8} + \frac{4}{7} \, a^{3} b e x^{7} + \frac{2}{3} \, a^{3} b d x^{6} + \frac{4}{5} \, a^{3} b c x^{5} + \frac{1}{4} \, a^{4} e x^{4} + \frac{1}{3} \, a^{4} d x^{3} + \frac{1}{2} \, a^{4} c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.181537, size = 1, normalized size = 0.01 \[ \frac{1}{16} x^{16} e b^{4} + \frac{1}{15} x^{15} d b^{4} + \frac{1}{14} x^{14} c b^{4} + \frac{4}{13} x^{13} e b^{3} a + \frac{1}{3} x^{12} d b^{3} a + \frac{4}{11} x^{11} c b^{3} a + \frac{3}{5} x^{10} e b^{2} a^{2} + \frac{2}{3} x^{9} d b^{2} a^{2} + \frac{3}{4} x^{8} c b^{2} a^{2} + \frac{4}{7} x^{7} e b a^{3} + \frac{2}{3} x^{6} d b a^{3} + \frac{4}{5} x^{5} c b a^{3} + \frac{1}{4} x^{4} e a^{4} + \frac{1}{3} x^{3} d a^{4} + \frac{1}{2} x^{2} c a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.089343, size = 185, normalized size = 1.02 \[ \frac{a^{4} c x^{2}}{2} + \frac{a^{4} d x^{3}}{3} + \frac{a^{4} e x^{4}}{4} + \frac{4 a^{3} b c x^{5}}{5} + \frac{2 a^{3} b d x^{6}}{3} + \frac{4 a^{3} b e x^{7}}{7} + \frac{3 a^{2} b^{2} c x^{8}}{4} + \frac{2 a^{2} b^{2} d x^{9}}{3} + \frac{3 a^{2} b^{2} e x^{10}}{5} + \frac{4 a b^{3} c x^{11}}{11} + \frac{a b^{3} d x^{12}}{3} + \frac{4 a b^{3} e x^{13}}{13} + \frac{b^{4} c x^{14}}{14} + \frac{b^{4} d x^{15}}{15} + \frac{b^{4} e x^{16}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210272, size = 211, normalized size = 1.17 \[ \frac{1}{16} \, b^{4} x^{16} e + \frac{1}{15} \, b^{4} d x^{15} + \frac{1}{14} \, b^{4} c x^{14} + \frac{4}{13} \, a b^{3} x^{13} e + \frac{1}{3} \, a b^{3} d x^{12} + \frac{4}{11} \, a b^{3} c x^{11} + \frac{3}{5} \, a^{2} b^{2} x^{10} e + \frac{2}{3} \, a^{2} b^{2} d x^{9} + \frac{3}{4} \, a^{2} b^{2} c x^{8} + \frac{4}{7} \, a^{3} b x^{7} e + \frac{2}{3} \, a^{3} b d x^{6} + \frac{4}{5} \, a^{3} b c x^{5} + \frac{1}{4} \, a^{4} x^{4} e + \frac{1}{3} \, a^{4} d x^{3} + \frac{1}{2} \, a^{4} c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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