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

Optimal. Leaf size=69 \[ \frac{1}{3} x^3 (a f+b d)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{4} b e x^4+\frac{1}{6} c e x^6+\frac{1}{7} c f x^7 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0913705, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{1}{3} x^3 (a f+b d)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{4} b e x^4+\frac{1}{6} c e x^6+\frac{1}{7} c f x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*e*Integral(x, x) + b*e*x**4/4 + c*e*x**6/6 + c*f*x**7/7 + d*Integral(a, x) + x
**5*(b*f/5 + c*d/5) + x**3*(a*f/3 + b*d/3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0374614, size = 69, normalized size = 1. \[ \frac{1}{3} x^3 (a f+b d)+a d x+\frac{1}{2} a e x^2+\frac{1}{5} x^5 (b f+c d)+\frac{1}{4} b e x^4+\frac{1}{6} c e x^6+\frac{1}{7} c f x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 58, normalized size = 0.8 \[ adx+{\frac{ae{x}^{2}}{2}}+{\frac{ \left ( fa+bd \right ){x}^{3}}{3}}+{\frac{be{x}^{4}}{4}}+{\frac{ \left ( bf+cd \right ){x}^{5}}{5}}+{\frac{ce{x}^{6}}{6}}+{\frac{cf{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.697054, size = 77, normalized size = 1.12 \[ \frac{1}{7} \, c f x^{7} + \frac{1}{6} \, c e x^{6} + \frac{1}{4} \, b e x^{4} + \frac{1}{5} \,{\left (c d + b f\right )} x^{5} + \frac{1}{2} \, a e x^{2} + \frac{1}{3} \,{\left (b d + a f\right )} x^{3} + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.241331, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} f c + \frac{1}{6} x^{6} e c + \frac{1}{5} x^{5} d c + \frac{1}{5} x^{5} f b + \frac{1}{4} x^{4} e b + \frac{1}{3} x^{3} d b + \frac{1}{3} x^{3} f a + \frac{1}{2} x^{2} e a + x d a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.102619, size = 65, normalized size = 0.94 \[ a d x + \frac{a e x^{2}}{2} + \frac{b e x^{4}}{4} + \frac{c e x^{6}}{6} + \frac{c f x^{7}}{7} + x^{5} \left (\frac{b f}{5} + \frac{c d}{5}\right ) + x^{3} \left (\frac{a f}{3} + \frac{b d}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d*x + a*e*x**2/2 + b*e*x**4/4 + c*e*x**6/6 + c*f*x**7/7 + x**5*(b*f/5 + c*d/5)
+ x**3*(a*f/3 + b*d/3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.285878, size = 86, normalized size = 1.25 \[ \frac{1}{7} \, c f x^{7} + \frac{1}{6} \, c x^{6} e + \frac{1}{5} \, c d x^{5} + \frac{1}{5} \, b f x^{5} + \frac{1}{4} \, b x^{4} e + \frac{1}{3} \, b d x^{3} + \frac{1}{3} \, a f x^{3} + \frac{1}{2} \, a x^{2} e + a d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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