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

Optimal. Leaf size=97 \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.165651, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 + d*(e + f*x**4)**3/(12*f)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00674172, size = 97, normalized size = 1. \[ \frac{1}{2} b e^2 x^2+\frac{1}{3} b e f x^6+\frac{1}{10} b f^2 x^{10}+\frac{1}{3} c e^2 x^3+\frac{2}{7} c e f x^7+\frac{1}{11} c f^2 x^{11}+\frac{1}{4} d e^2 x^4+\frac{1}{4} d e f x^8+\frac{1}{12} d f^2 x^{12} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 80, normalized size = 0.8 \[{\frac{b{e}^{2}{x}^{2}}{2}}+{\frac{c{e}^{2}{x}^{3}}{3}}+{\frac{d{e}^{2}{x}^{4}}{4}}+{\frac{bef{x}^{6}}{3}}+{\frac{2\,cef{x}^{7}}{7}}+{\frac{def{x}^{8}}{4}}+{\frac{b{f}^{2}{x}^{10}}{10}}+{\frac{c{f}^{2}{x}^{11}}{11}}+{\frac{d{f}^{2}{x}^{12}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 6.91078, size = 107, normalized size = 1.1 \[ \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{10} \, b f^{2} x^{10} + \frac{1}{4} \, d e f x^{8} + \frac{2}{7} \, c e f x^{7} + \frac{1}{3} \, b e f x^{6} + \frac{1}{4} \, d e^{2} x^{4} + \frac{1}{3} \, c e^{2} x^{3} + \frac{1}{2} \, b e^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.202662, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} f^{2} d + \frac{1}{11} x^{11} f^{2} c + \frac{1}{10} x^{10} f^{2} b + \frac{1}{4} x^{8} f e d + \frac{2}{7} x^{7} f e c + \frac{1}{3} x^{6} f e b + \frac{1}{4} x^{4} e^{2} d + \frac{1}{3} x^{3} e^{2} c + \frac{1}{2} x^{2} e^{2} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.066996, size = 92, normalized size = 0.95 \[ \frac{b e^{2} x^{2}}{2} + \frac{b e f x^{6}}{3} + \frac{b f^{2} x^{10}}{10} + \frac{c e^{2} x^{3}}{3} + \frac{2 c e f x^{7}}{7} + \frac{c f^{2} x^{11}}{11} + \frac{d e^{2} x^{4}}{4} + \frac{d e f x^{8}}{4} + \frac{d f^{2} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 + d*e**2*x**4/4 + d*e*f*x**8/4 + d*f**2*x**12/12

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.209136, size = 107, normalized size = 1.1 \[ \frac{1}{12} \, d f^{2} x^{12} + \frac{1}{11} \, c f^{2} x^{11} + \frac{1}{10} \, b f^{2} x^{10} + \frac{1}{4} \, d f x^{8} e + \frac{2}{7} \, c f x^{7} e + \frac{1}{3} \, b f x^{6} e + \frac{1}{4} \, d x^{4} e^{2} + \frac{1}{3} \, c x^{3} e^{2} + \frac{1}{2} \, b x^{2} e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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