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

Optimal. Leaf size=92 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.104594, antiderivative size = 92, 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 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00591937, size = 92, normalized size = 1. \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0., size = 77, normalized size = 0.8 \[ a{e}^{2}x+{\frac{b{e}^{2}{x}^{2}}{2}}+{\frac{c{e}^{2}{x}^{3}}{3}}+{\frac{2\,aef{x}^{5}}{5}}+{\frac{bef{x}^{6}}{3}}+{\frac{2\,cef{x}^{7}}{7}}+{\frac{a{f}^{2}{x}^{9}}{9}}+{\frac{b{f}^{2}{x}^{10}}{10}}+{\frac{c{f}^{2}{x}^{11}}{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.40846, size = 103, normalized size = 1.12 \[ \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{10} \, b f^{2} x^{10} + \frac{1}{9} \, a f^{2} x^{9} + \frac{2}{7} \, c e f x^{7} + \frac{1}{3} \, b e f x^{6} + \frac{2}{5} \, a e f x^{5} + \frac{1}{3} \, c e^{2} x^{3} + \frac{1}{2} \, b e^{2} x^{2} + a e^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.207, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} f^{2} c + \frac{1}{10} x^{10} f^{2} b + \frac{1}{9} x^{9} f^{2} a + \frac{2}{7} x^{7} f e c + \frac{1}{3} x^{6} f e b + \frac{2}{5} x^{5} f e a + \frac{1}{3} x^{3} e^{2} c + \frac{1}{2} x^{2} e^{2} b + x e^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.064328, size = 90, normalized size = 0.98 \[ a e^{2} x + \frac{2 a e f x^{5}}{5} + \frac{a f^{2} x^{9}}{9} + \frac{b e^{2} x^{2}}{2} + \frac{b e f x^{6}}{3} + \frac{b f^{2} x^{10}}{10} + \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210448, size = 103, normalized size = 1.12 \[ \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{10} \, b f^{2} x^{10} + \frac{1}{9} \, a f^{2} x^{9} + \frac{2}{7} \, c f x^{7} e + \frac{1}{3} \, b f x^{6} e + \frac{2}{5} \, a f x^{5} e + \frac{1}{3} \, c x^{3} e^{2} + \frac{1}{2} \, b x^{2} e^{2} + a x e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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