\(\int (d+e x) (a+b x^2+c x^4) \, dx\) [1]

Optimal result

Integrand size = 18, antiderivative size = 50 \[ \int (d+e x) \left (a+b x^2+c x^4\right ) \, dx=a d x+\frac {1}{2} a e x^2+\frac {1}{3} b d x^3+\frac {1}{4} b e x^4+\frac {1}{5} c d x^5+\frac {1}{6} c e x^6 \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1685} \[ \int (d+e x) \left (a+b x^2+c x^4\right ) \, dx=a d x+\frac {1}{2} a e x^2+\frac {1}{3} b d x^3+\frac {1}{4} b e x^4+\frac {1}{5} c d x^5+\frac {1}{6} c e x^6 \]

[In]

Int[(d + e*x)*(a + b*x^2 + c*x^4),x]

[Out]

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

Rule 1685

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2 + c*x^4)^
p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, 0]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (a+b x^2+c x^4\right ) \, dx=a d x+\frac {1}{2} a e x^2+\frac {1}{3} b d x^3+\frac {1}{4} b e x^4+\frac {1}{5} c d x^5+\frac {1}{6} c e x^6 \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.82

method	result	size
gosper	\(a d x +\frac {1}{2} a e \,x^{2}+\frac {1}{3} x^{3} b d +\frac {1}{4} b e \,x^{4}+\frac {1}{5} c d \,x^{5}+\frac {1}{6} c e \,x^{6}\)	\(41\)
default	\(a d x +\frac {1}{2} a e \,x^{2}+\frac {1}{3} x^{3} b d +\frac {1}{4} b e \,x^{4}+\frac {1}{5} c d \,x^{5}+\frac {1}{6} c e \,x^{6}\)	\(41\)
norman	\(a d x +\frac {1}{2} a e \,x^{2}+\frac {1}{3} x^{3} b d +\frac {1}{4} b e \,x^{4}+\frac {1}{5} c d \,x^{5}+\frac {1}{6} c e \,x^{6}\)	\(41\)
risch	\(a d x +\frac {1}{2} a e \,x^{2}+\frac {1}{3} x^{3} b d +\frac {1}{4} b e \,x^{4}+\frac {1}{5} c d \,x^{5}+\frac {1}{6} c e \,x^{6}\)	\(41\)
parallelrisch	\(a d x +\frac {1}{2} a e \,x^{2}+\frac {1}{3} x^{3} b d +\frac {1}{4} b e \,x^{4}+\frac {1}{5} c d \,x^{5}+\frac {1}{6} c e \,x^{6}\)	\(41\)

[In]

int((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*x^3*b*d+1/4*b*e*x^4+1/5*c*d*x^5+1/6*c*e*x^6

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int (d+e x) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{6} \, c e x^{6} + \frac {1}{5} \, c d x^{5} + \frac {1}{4} \, b e x^{4} + \frac {1}{3} \, b d x^{3} + \frac {1}{2} \, a e x^{2} + a d x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92 \[ \int (d+e x) \left (a+b x^2+c x^4\right ) \, dx=a d x + \frac {a e x^{2}}{2} + \frac {b d x^{3}}{3} + \frac {b e x^{4}}{4} + \frac {c d x^{5}}{5} + \frac {c e x^{6}}{6} \]

[In]

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

[Out]

a*d*x + a*e*x**2/2 + b*d*x**3/3 + b*e*x**4/4 + c*d*x**5/5 + c*e*x**6/6

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int (d+e x) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{6} \, c e x^{6} + \frac {1}{5} \, c d x^{5} + \frac {1}{4} \, b e x^{4} + \frac {1}{3} \, b d x^{3} + \frac {1}{2} \, a e x^{2} + a d x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int (d+e x) \left (a+b x^2+c x^4\right ) \, dx=\frac {1}{6} \, c e x^{6} + \frac {1}{5} \, c d x^{5} + \frac {1}{4} \, b e x^{4} + \frac {1}{3} \, b d x^{3} + \frac {1}{2} \, a e x^{2} + a d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int (d+e x) \left (a+b x^2+c x^4\right ) \, dx=\frac {c\,e\,x^6}{6}+\frac {c\,d\,x^5}{5}+\frac {b\,e\,x^4}{4}+\frac {b\,d\,x^3}{3}+\frac {a\,e\,x^2}{2}+a\,d\,x \]

[In]

int((d + e*x)*(a + b*x^2 + c*x^4),x)

[Out]

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