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

Optimal. Leaf size=173 \[ a^4 c x+\frac{1}{2} a^4 d x^2+\frac{1}{3} a^4 e x^3+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{2}{3} a^3 b e x^6+\frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{3} a^2 b^2 e x^9+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{1}{3} a b^3 e x^{12}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14}+\frac{1}{15} b^4 e x^{15} \]

[Out]

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

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Rubi [A]  time = 0.259694, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ a^4 c x+\frac{1}{2} a^4 d x^2+\frac{1}{3} a^4 e x^3+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{2}{3} a^3 b e x^6+\frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{3} a^2 b^2 e x^9+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{1}{3} a b^3 e x^{12}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14}+\frac{1}{15} b^4 e x^{15} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{4} d \int x\, dx + a^{4} \int c\, dx + a^{3} b c x^{4} + \frac{4 a^{3} b d x^{5}}{5} + \frac{6 a^{2} b^{2} c x^{7}}{7} + \frac{3 a^{2} b^{2} d x^{8}}{4} + \frac{2 a b^{3} c x^{10}}{5} + \frac{4 a b^{3} d x^{11}}{11} + \frac{b^{4} c x^{13}}{13} + \frac{b^{4} d x^{14}}{14} + \frac{e \left (a + b x^{3}\right )^{5}}{15 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00801525, size = 173, normalized size = 1. \[ a^4 c x+\frac{1}{2} a^4 d x^2+\frac{1}{3} a^4 e x^3+a^3 b c x^4+\frac{4}{5} a^3 b d x^5+\frac{2}{3} a^3 b e x^6+\frac{6}{7} a^2 b^2 c x^7+\frac{3}{4} a^2 b^2 d x^8+\frac{2}{3} a^2 b^2 e x^9+\frac{2}{5} a b^3 c x^{10}+\frac{4}{11} a b^3 d x^{11}+\frac{1}{3} a b^3 e x^{12}+\frac{1}{13} b^4 c x^{13}+\frac{1}{14} b^4 d x^{14}+\frac{1}{15} b^4 e x^{15} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 148, normalized size = 0.9 \[{a}^{4}cx+{\frac{{a}^{4}d{x}^{2}}{2}}+{\frac{{a}^{4}e{x}^{3}}{3}}+{a}^{3}bc{x}^{4}+{\frac{4\,{a}^{3}bd{x}^{5}}{5}}+{\frac{2\,{a}^{3}be{x}^{6}}{3}}+{\frac{6\,{a}^{2}{b}^{2}c{x}^{7}}{7}}+{\frac{3\,{a}^{2}{b}^{2}d{x}^{8}}{4}}+{\frac{2\,{a}^{2}{b}^{2}e{x}^{9}}{3}}+{\frac{2\,a{b}^{3}c{x}^{10}}{5}}+{\frac{4\,a{b}^{3}d{x}^{11}}{11}}+{\frac{a{b}^{3}e{x}^{12}}{3}}+{\frac{{b}^{4}c{x}^{13}}{13}}+{\frac{{b}^{4}d{x}^{14}}{14}}+{\frac{{b}^{4}e{x}^{15}}{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.41452, size = 198, normalized size = 1.14 \[ \frac{1}{15} \, b^{4} e x^{15} + \frac{1}{14} \, b^{4} d x^{14} + \frac{1}{13} \, b^{4} c x^{13} + \frac{1}{3} \, a b^{3} e x^{12} + \frac{4}{11} \, a b^{3} d x^{11} + \frac{2}{5} \, a b^{3} c x^{10} + \frac{2}{3} \, a^{2} b^{2} e x^{9} + \frac{3}{4} \, a^{2} b^{2} d x^{8} + \frac{6}{7} \, a^{2} b^{2} c x^{7} + \frac{2}{3} \, a^{3} b e x^{6} + \frac{4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac{1}{3} \, a^{4} e x^{3} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.185243, size = 1, normalized size = 0.01 \[ \frac{1}{15} x^{15} e b^{4} + \frac{1}{14} x^{14} d b^{4} + \frac{1}{13} x^{13} c b^{4} + \frac{1}{3} x^{12} e b^{3} a + \frac{4}{11} x^{11} d b^{3} a + \frac{2}{5} x^{10} c b^{3} a + \frac{2}{3} x^{9} e b^{2} a^{2} + \frac{3}{4} x^{8} d b^{2} a^{2} + \frac{6}{7} x^{7} c b^{2} a^{2} + \frac{2}{3} x^{6} e b a^{3} + \frac{4}{5} x^{5} d b a^{3} + x^{4} c b a^{3} + \frac{1}{3} x^{3} e a^{4} + \frac{1}{2} x^{2} d a^{4} + x c a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.088664, size = 178, normalized size = 1.03 \[ a^{4} c x + \frac{a^{4} d x^{2}}{2} + \frac{a^{4} e x^{3}}{3} + a^{3} b c x^{4} + \frac{4 a^{3} b d x^{5}}{5} + \frac{2 a^{3} b e x^{6}}{3} + \frac{6 a^{2} b^{2} c x^{7}}{7} + \frac{3 a^{2} b^{2} d x^{8}}{4} + \frac{2 a^{2} b^{2} e x^{9}}{3} + \frac{2 a b^{3} c x^{10}}{5} + \frac{4 a b^{3} d x^{11}}{11} + \frac{a b^{3} e x^{12}}{3} + \frac{b^{4} c x^{13}}{13} + \frac{b^{4} d x^{14}}{14} + \frac{b^{4} e x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21047, size = 205, normalized size = 1.18 \[ \frac{1}{15} \, b^{4} x^{15} e + \frac{1}{14} \, b^{4} d x^{14} + \frac{1}{13} \, b^{4} c x^{13} + \frac{1}{3} \, a b^{3} x^{12} e + \frac{4}{11} \, a b^{3} d x^{11} + \frac{2}{5} \, a b^{3} c x^{10} + \frac{2}{3} \, a^{2} b^{2} x^{9} e + \frac{3}{4} \, a^{2} b^{2} d x^{8} + \frac{6}{7} \, a^{2} b^{2} c x^{7} + \frac{2}{3} \, a^{3} b x^{6} e + \frac{4}{5} \, a^{3} b d x^{5} + a^{3} b c x^{4} + \frac{1}{3} \, a^{4} x^{3} e + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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