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

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

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Rubi [A]  time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, 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} \begin {gather*} 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 {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 50, normalized size = 1.00 \begin {gather*} 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 {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (a+b x^2+c x^4\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.85, size = 40, normalized size = 0.80 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.32, size = 43, normalized size = 0.86 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.00, size = 41, normalized size = 0.82 \begin {gather*} \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 {gather*}

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 = 1.25, size = 40, normalized size = 0.80 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.03, size = 40, normalized size = 0.80 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.06, size = 46, normalized size = 0.92 \begin {gather*} 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 {gather*}

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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