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

Optimal. Leaf size=114 \[ \frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{7} a^2 f x^7+\frac{c \left (a+b x^4\right )^3}{12 b}+\frac{2}{9} a b d x^9+\frac{1}{5} a b e x^{10}+\frac{2}{11} a b f x^{11}+\frac{1}{13} b^2 d x^{13}+\frac{1}{14} b^2 e x^{14}+\frac{1}{15} b^2 f x^{15} \]

[Out]

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

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Rubi [A]  time = 0.318637, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{7} a^2 f x^7+\frac{c \left (a+b x^4\right )^3}{12 b}+\frac{2}{9} a b d x^9+\frac{1}{5} a b e x^{10}+\frac{2}{11} a b f x^{11}+\frac{1}{13} b^2 d x^{13}+\frac{1}{14} b^2 e x^{14}+\frac{1}{15} b^2 f x^{15} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.3562, size = 107, normalized size = 0.94 \[ \frac{a^{2} d x^{5}}{5} + \frac{a^{2} e x^{6}}{6} + \frac{a^{2} f x^{7}}{7} + \frac{2 a b d x^{9}}{9} + \frac{a b e x^{10}}{5} + \frac{2 a b f x^{11}}{11} + \frac{b^{2} d x^{13}}{13} + \frac{b^{2} e x^{14}}{14} + \frac{b^{2} f x^{15}}{15} + \frac{c \left (a + b x^{4}\right )^{3}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00889073, size = 129, normalized size = 1.13 \[ \frac{1}{4} a^2 c x^4+\frac{1}{5} a^2 d x^5+\frac{1}{6} a^2 e x^6+\frac{1}{7} a^2 f x^7+\frac{1}{4} a b c x^8+\frac{2}{9} a b d x^9+\frac{1}{5} a b e x^{10}+\frac{2}{11} a b f x^{11}+\frac{1}{12} b^2 c x^{12}+\frac{1}{13} b^2 d x^{13}+\frac{1}{14} b^2 e x^{14}+\frac{1}{15} b^2 f x^{15} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 106, normalized size = 0.9 \[{\frac{{b}^{2}f{x}^{15}}{15}}+{\frac{{b}^{2}e{x}^{14}}{14}}+{\frac{{b}^{2}d{x}^{13}}{13}}+{\frac{{b}^{2}c{x}^{12}}{12}}+{\frac{2\,abf{x}^{11}}{11}}+{\frac{abe{x}^{10}}{5}}+{\frac{2\,abd{x}^{9}}{9}}+{\frac{abc{x}^{8}}{4}}+{\frac{{a}^{2}f{x}^{7}}{7}}+{\frac{{a}^{2}e{x}^{6}}{6}}+{\frac{{a}^{2}d{x}^{5}}{5}}+{\frac{{a}^{2}c{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37863, size = 142, normalized size = 1.25 \[ \frac{1}{15} \, b^{2} f x^{15} + \frac{1}{14} \, b^{2} e x^{14} + \frac{1}{13} \, b^{2} d x^{13} + \frac{1}{12} \, b^{2} c x^{12} + \frac{2}{11} \, a b f x^{11} + \frac{1}{5} \, a b e x^{10} + \frac{2}{9} \, a b d x^{9} + \frac{1}{4} \, a b c x^{8} + \frac{1}{7} \, a^{2} f x^{7} + \frac{1}{6} \, a^{2} e x^{6} + \frac{1}{5} \, a^{2} d x^{5} + \frac{1}{4} \, a^{2} c x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.189671, size = 1, normalized size = 0.01 \[ \frac{1}{15} x^{15} f b^{2} + \frac{1}{14} x^{14} e b^{2} + \frac{1}{13} x^{13} d b^{2} + \frac{1}{12} x^{12} c b^{2} + \frac{2}{11} x^{11} f b a + \frac{1}{5} x^{10} e b a + \frac{2}{9} x^{9} d b a + \frac{1}{4} x^{8} c b a + \frac{1}{7} x^{7} f a^{2} + \frac{1}{6} x^{6} e a^{2} + \frac{1}{5} x^{5} d a^{2} + \frac{1}{4} x^{4} c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.080523, size = 124, normalized size = 1.09 \[ \frac{a^{2} c x^{4}}{4} + \frac{a^{2} d x^{5}}{5} + \frac{a^{2} e x^{6}}{6} + \frac{a^{2} f x^{7}}{7} + \frac{a b c x^{8}}{4} + \frac{2 a b d x^{9}}{9} + \frac{a b e x^{10}}{5} + \frac{2 a b f x^{11}}{11} + \frac{b^{2} c x^{12}}{12} + \frac{b^{2} d x^{13}}{13} + \frac{b^{2} e x^{14}}{14} + \frac{b^{2} f x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222893, size = 146, normalized size = 1.28 \[ \frac{1}{15} \, b^{2} f x^{15} + \frac{1}{14} \, b^{2} x^{14} e + \frac{1}{13} \, b^{2} d x^{13} + \frac{1}{12} \, b^{2} c x^{12} + \frac{2}{11} \, a b f x^{11} + \frac{1}{5} \, a b x^{10} e + \frac{2}{9} \, a b d x^{9} + \frac{1}{4} \, a b c x^{8} + \frac{1}{7} \, a^{2} f x^{7} + \frac{1}{6} \, a^{2} x^{6} e + \frac{1}{5} \, a^{2} d x^{5} + \frac{1}{4} \, a^{2} c x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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