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

Optimal result

Integrand size = 23, antiderivative size = 69 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=a d x+\frac {1}{2} a e x^2+\frac {1}{3} (b d+a f) x^3+\frac {1}{4} b e x^4+\frac {1}{5} (c d+b f) x^5+\frac {1}{6} c e x^6+\frac {1}{7} c f x^7 \]

[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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1671} \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{3} x^3 (a f+b d)+a d x+\frac {1}{2} a e x^2+\frac {1}{5} x^5 (b f+c d)+\frac {1}{4} b e x^4+\frac {1}{6} c e x^6+\frac {1}{7} c f x^7 \]

[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

Rule 1671

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps \begin{align*} \text {integral}& = \int \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx \\ & = a d x+\frac {1}{2} a e x^2+\frac {1}{3} (b d+a f) x^3+\frac {1}{4} b e x^4+\frac {1}{5} (c d+b f) x^5+\frac {1}{6} c e x^6+\frac {1}{7} c f x^7 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=a d x+\frac {1}{2} a e x^2+\frac {1}{3} (b d+a f) x^3+\frac {1}{4} b e x^4+\frac {1}{5} (c d+b f) x^5+\frac {1}{6} c e x^6+\frac {1}{7} c f x^7 \]

[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] (verified)

Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84

method	result	size
default	\(a d x +\frac {a e \,x^{2}}{2}+\frac {\left (a f +b d \right ) x^{3}}{3}+\frac {b e \,x^{4}}{4}+\frac {\left (b f +c d \right ) x^{5}}{5}+\frac {c e \,x^{6}}{6}+\frac {c f \,x^{7}}{7}\)	\(58\)
norman	\(\frac {c f \,x^{7}}{7}+\frac {c e \,x^{6}}{6}+\left (\frac {b f}{5}+\frac {c d}{5}\right ) x^{5}+\frac {b e \,x^{4}}{4}+\left (\frac {a f}{3}+\frac {b d}{3}\right ) x^{3}+\frac {a e \,x^{2}}{2}+a d x\)	\(60\)
gosper	\(\frac {1}{7} c f \,x^{7}+\frac {1}{6} c e \,x^{6}+\frac {1}{5} x^{5} b f +\frac {1}{5} c d \,x^{5}+\frac {1}{4} b e \,x^{4}+\frac {1}{3} x^{3} a f +\frac {1}{3} x^{3} b d +\frac {1}{2} a e \,x^{2}+a d x\)	\(62\)
risch	\(\frac {1}{7} c f \,x^{7}+\frac {1}{6} c e \,x^{6}+\frac {1}{5} x^{5} b f +\frac {1}{5} c d \,x^{5}+\frac {1}{4} b e \,x^{4}+\frac {1}{3} x^{3} a f +\frac {1}{3} x^{3} b d +\frac {1}{2} a e \,x^{2}+a d x\)	\(62\)
parallelrisch	\(\frac {1}{7} c f \,x^{7}+\frac {1}{6} c e \,x^{6}+\frac {1}{5} x^{5} b f +\frac {1}{5} c d \,x^{5}+\frac {1}{4} b e \,x^{4}+\frac {1}{3} x^{3} a f +\frac {1}{3} x^{3} b d +\frac {1}{2} a e \,x^{2}+a d x\)	\(62\)

[In]

int((f*x^2+e*x+d)*(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, c f x^{7} + \frac {1}{6} \, c e x^{6} + \frac {1}{4} \, b e x^{4} + \frac {1}{5} \, {\left (c d + b f\right )} x^{5} + \frac {1}{2} \, a e x^{2} + \frac {1}{3} \, {\left (b d + a f\right )} x^{3} + a d x \]

[In]

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

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=a d x + \frac {a e x^{2}}{2} + \frac {b e x^{4}}{4} + \frac {c e x^{6}}{6} + \frac {c f x^{7}}{7} + x^{5} \left (\frac {b f}{5} + \frac {c d}{5}\right ) + x^{3} \left (\frac {a f}{3} + \frac {b d}{3}\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, c f x^{7} + \frac {1}{6} \, c e x^{6} + \frac {1}{4} \, b e x^{4} + \frac {1}{5} \, {\left (c d + b f\right )} x^{5} + \frac {1}{2} \, a e x^{2} + \frac {1}{3} \, {\left (b d + a f\right )} x^{3} + a d x \]

[In]

integrate((f*x^2+e*x+d)*(c*x^4+b*x^2+a),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

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{7} \, c f x^{7} + \frac {1}{6} \, c e x^{6} + \frac {1}{5} \, c d x^{5} + \frac {1}{5} \, b f x^{5} + \frac {1}{4} \, b e x^{4} + \frac {1}{3} \, b d x^{3} + \frac {1}{3} \, a f x^{3} + \frac {1}{2} \, a e x^{2} + a d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \left (d+e x+f x^2\right ) \left (a+b x^2+c x^4\right ) \, dx=\frac {c\,f\,x^7}{7}+\frac {c\,e\,x^6}{6}+\left (\frac {c\,d}{5}+\frac {b\,f}{5}\right )\,x^5+\frac {b\,e\,x^4}{4}+\left (\frac {b\,d}{3}+\frac {a\,f}{3}\right )\,x^3+\frac {a\,e\,x^2}{2}+a\,d\,x \]

[In]

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

[Out]

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