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

Optimal. Leaf size=109 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{f \left (a+b x^4\right )^3}{12 b}+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]

[Out]

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

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Rubi [A]  time = 0.144531, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{f \left (a+b x^4\right )^3}{12 b}+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} d \int x\, dx + \frac{a^{2} e x^{3}}{3} + a^{2} \int c\, dx + \frac{2 a b c x^{5}}{5} + \frac{a b d x^{6}}{3} + \frac{2 a b e x^{7}}{7} + \frac{b^{2} c x^{9}}{9} + \frac{b^{2} d x^{10}}{10} + \frac{b^{2} e x^{11}}{11} + \frac{f \left (a + b x^{4}\right )^{3}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00625311, size = 124, normalized size = 1.14 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{1}{4} a^2 f x^4+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{1}{4} a b f x^8+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}+\frac{1}{12} b^2 f x^{12} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 103, normalized size = 0.9 \[{\frac{{b}^{2}f{x}^{12}}{12}}+{\frac{{b}^{2}e{x}^{11}}{11}}+{\frac{{b}^{2}d{x}^{10}}{10}}+{\frac{{b}^{2}c{x}^{9}}{9}}+{\frac{fab{x}^{8}}{4}}+{\frac{2\,abe{x}^{7}}{7}}+{\frac{abd{x}^{6}}{3}}+{\frac{2\,abc{x}^{5}}{5}}+{\frac{f{a}^{2}{x}^{4}}{4}}+{\frac{{a}^{2}e{x}^{3}}{3}}+{\frac{{a}^{2}d{x}^{2}}{2}}+{a}^{2}cx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36978, size = 138, normalized size = 1.27 \[ \frac{1}{12} \, b^{2} f x^{12} + \frac{1}{11} \, b^{2} e x^{11} + \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{9} \, b^{2} c x^{9} + \frac{1}{4} \, a b f x^{8} + \frac{2}{7} \, a b e x^{7} + \frac{1}{3} \, a b d x^{6} + \frac{2}{5} \, a b c x^{5} + \frac{1}{4} \, a^{2} f x^{4} + \frac{1}{3} \, a^{2} e x^{3} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \]

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, algorithm="maxima")

[Out]

1/12*b^2*f*x^12 + 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*f*
x^8 + 2/7*a*b*e*x^7 + 1/3*a*b*d*x^6 + 2/5*a*b*c*x^5 + 1/4*a^2*f*x^4 + 1/3*a^2*e*
x^3 + 1/2*a^2*d*x^2 + a^2*c*x

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Fricas [A]  time = 0.187411, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} f b^{2} + \frac{1}{11} x^{11} e b^{2} + \frac{1}{10} x^{10} d b^{2} + \frac{1}{9} x^{9} c b^{2} + \frac{1}{4} x^{8} f b a + \frac{2}{7} x^{7} e b a + \frac{1}{3} x^{6} d b a + \frac{2}{5} x^{5} c b a + \frac{1}{4} x^{4} f a^{2} + \frac{1}{3} x^{3} e a^{2} + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \]

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, algorithm="fricas")

[Out]

1/12*x^12*f*b^2 + 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*f*
b*a + 2/7*x^7*e*b*a + 1/3*x^6*d*b*a + 2/5*x^5*c*b*a + 1/4*x^4*f*a^2 + 1/3*x^3*e*
a^2 + 1/2*x^2*d*a^2 + x*c*a^2

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Sympy [A]  time = 0.078685, size = 121, normalized size = 1.11 \[ a^{2} c x + \frac{a^{2} d x^{2}}{2} + \frac{a^{2} e x^{3}}{3} + \frac{a^{2} f x^{4}}{4} + \frac{2 a b c x^{5}}{5} + \frac{a b d x^{6}}{3} + \frac{2 a b e x^{7}}{7} + \frac{a b f x^{8}}{4} + \frac{b^{2} c x^{9}}{9} + \frac{b^{2} d x^{10}}{10} + \frac{b^{2} e x^{11}}{11} + \frac{b^{2} f x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.228603, size = 142, normalized size = 1.3 \[ \frac{1}{12} \, b^{2} f x^{12} + \frac{1}{11} \, b^{2} x^{11} e + \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{9} \, b^{2} c x^{9} + \frac{1}{4} \, a b f x^{8} + \frac{2}{7} \, a b x^{7} e + \frac{1}{3} \, a b d x^{6} + \frac{2}{5} \, a b c x^{5} + \frac{1}{4} \, a^{2} f x^{4} + \frac{1}{3} \, a^{2} x^{3} e + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \]

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, algorithm="giac")

[Out]

1/12*b^2*f*x^12 + 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*f*
x^8 + 2/7*a*b*x^7*e + 1/3*a*b*d*x^6 + 2/5*a*b*c*x^5 + 1/4*a^2*f*x^4 + 1/3*a^2*x^
3*e + 1/2*a^2*d*x^2 + a^2*c*x