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

Optimal. Leaf size=50 \[ a d x+\frac{1}{2} a e x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4+\frac{1}{5} c d x^5+\frac{1}{6} c e x^6 \]

[Out]

a*d*x + (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

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Rubi [A]  time = 0.0412663, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {1671} \[ a d x+\frac{1}{2} a e x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4+\frac{1}{5} c d x^5+\frac{1}{6} c e x^6 \]

Antiderivative was successfully verified.

[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 1671

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*} \int (d+e x) \left (a+b x^2+c x^4\right ) \, dx &=\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]  time = 0.0017897, size = 50, normalized size = 1. \[ a d x+\frac{1}{2} a e x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4+\frac{1}{5} c d x^5+\frac{1}{6} c e x^6 \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.007, size = 41, normalized size = 0.8 \begin{align*} adx+{\frac{ae{x}^{2}}{2}}+{\frac{bd{x}^{3}}{3}}+{\frac{be{x}^{4}}{4}}+{\frac{cd{x}^{5}}{5}}+{\frac{ce{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 1.32987, size = 54, normalized size = 1.08 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.46244, size = 104, normalized size = 2.08 \begin{align*} \frac{1}{6} x^{6} e c + \frac{1}{5} x^{5} d c + \frac{1}{4} x^{4} e b + \frac{1}{3} x^{3} d b + \frac{1}{2} x^{2} e a + x d a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.064181, size = 46, normalized size = 0.92 \begin{align*} 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.07894, size = 58, normalized size = 1.16 \begin{align*} \frac{1}{6} \, c x^{6} e + \frac{1}{5} \, c d x^{5} + \frac{1}{4} \, b x^{4} e + \frac{1}{3} \, b d x^{3} + \frac{1}{2} \, a x^{2} e + a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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