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

Optimal. Leaf size=65 \[ \frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]

[Out]

(c*e^2*x^3)/3 + (d*e^2*x^4)/4 + (2*c*e*f*x^7)/7 + (d*e*f*x^8)/4 + (c*f^2*x^11)/1
1 + (d*f^2*x^12)/12

_______________________________________________________________________________________

Rubi [A]  time = 0.1929, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]

Antiderivative was successfully verified.

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

[Out]

(c*e^2*x^3)/3 + (d*e^2*x^4)/4 + (2*c*e*f*x^7)/7 + (d*e*f*x^8)/4 + (c*f^2*x^11)/1
1 + (d*f^2*x^12)/12

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.3557, size = 44, normalized size = 0.68 \[ \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} + \frac{d \left (e + f x^{4}\right )^{3}}{12 f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00467111, size = 65, normalized size = 1. \[ \frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]

Antiderivative was successfully verified.

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

[Out]

(c*e^2*x^3)/3 + (d*e^2*x^4)/4 + (2*c*e*f*x^7)/7 + (d*e*f*x^8)/4 + (c*f^2*x^11)/1
1 + (d*f^2*x^12)/12

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 54, normalized size = 0.8 \[{\frac{c{e}^{2}{x}^{3}}{3}}+{\frac{d{e}^{2}{x}^{4}}{4}}+{\frac{2\,cef{x}^{7}}{7}}+{\frac{def{x}^{8}}{4}}+{\frac{c{f}^{2}{x}^{11}}{11}}+{\frac{d{f}^{2}{x}^{12}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.38366, size = 72, normalized size = 1.11 \[ \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{4} \, d e f x^{8} + \frac{2}{7} \, c e f x^{7} + \frac{1}{4} \, d e^{2} x^{4} + \frac{1}{3} \, c e^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.20125, size = 1, normalized size = 0.02 \[ \frac{1}{12} x^{12} f^{2} d + \frac{1}{11} x^{11} f^{2} c + \frac{1}{4} x^{8} f e d + \frac{2}{7} x^{7} f e c + \frac{1}{4} x^{4} e^{2} d + \frac{1}{3} x^{3} e^{2} c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.060133, size = 61, normalized size = 0.94 \[ \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} + \frac{d e^{2} x^{4}}{4} + \frac{d e f x^{8}}{4} + \frac{d f^{2} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 = 0.210158, size = 72, normalized size = 1.11 \[ \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{4} \, d f x^{8} e + \frac{2}{7} \, c f x^{7} e + \frac{1}{4} \, d x^{4} e^{2} + \frac{1}{3} \, c x^{3} e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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