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

Optimal. Leaf size=77 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{d \left (e+f x^4\right )^3}{12 f} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.116891, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{d \left (e+f x^4\right )^3}{12 f} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*e*f*x**5/5 + a*f**2*x**9/9 + c*e**2*x**3/3 + 2*c*e*f*x**7/7 + c*f**2*x**11/1
1 + d*(e + f*x**4)**3/(12*f) + e**2*Integral(a, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00620127, size = 92, normalized size = 1.19 \[ a e^2 x+\frac{2}{5} a e f x^5+\frac{1}{9} a f^2 x^9+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 77, normalized size = 1. \[{\frac{d{f}^{2}{x}^{12}}{12}}+{\frac{c{f}^{2}{x}^{11}}{11}}+{\frac{a{f}^{2}{x}^{9}}{9}}+{\frac{def{x}^{8}}{4}}+{\frac{2\,cef{x}^{7}}{7}}+{\frac{2\,aef{x}^{5}}{5}}+{\frac{d{e}^{2}{x}^{4}}{4}}+{\frac{c{e}^{2}{x}^{3}}{3}}+a{e}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.37371, size = 103, normalized size = 1.34 \[ \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{9} \, a f^{2} x^{9} + \frac{1}{4} \, d e f x^{8} + \frac{2}{7} \, c e f x^{7} + \frac{2}{5} \, a e f x^{5} + \frac{1}{4} \, d e^{2} x^{4} + \frac{1}{3} \, c e^{2} x^{3} + a e^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.202007, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} f^{2} d + \frac{1}{11} x^{11} f^{2} c + \frac{1}{9} x^{9} f^{2} a + \frac{1}{4} x^{8} f e d + \frac{2}{7} x^{7} f e c + \frac{2}{5} x^{5} f e a + \frac{1}{4} x^{4} e^{2} d + \frac{1}{3} x^{3} e^{2} c + x e^{2} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.066001, size = 90, normalized size = 1.17 \[ a e^{2} x + \frac{2 a e f x^{5}}{5} + \frac{a f^{2} x^{9}}{9} + \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} + \frac{d e^{2} x^{4}}{4} + \frac{d e f x^{8}}{4} + \frac{d f^{2} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**3+c*x**2+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 + c*e**2*x**3/3 + 2*c*e*f*x**7/7 + c*f
**2*x**11/11 + d*e**2*x**4/4 + d*e*f*x**8/4 + d*f**2*x**12/12

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207852, size = 103, normalized size = 1.34 \[ \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{9} \, a f^{2} x^{9} + \frac{1}{4} \, d f x^{8} e + \frac{2}{7} \, c f x^{7} e + \frac{2}{5} \, a f x^{5} e + \frac{1}{4} \, d x^{4} e^{2} + \frac{1}{3} \, c x^{3} e^{2} + a x e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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