3.315 \(\int (c+d x+e x^2) (a+b x^3) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0245263, 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 = {1657} \[ a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{4} b c x^4+\frac{1}{5} b d x^5+\frac{1}{6} b e x^6 \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2)*(a + b*x^3),x]

[Out]

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

Rule 1657

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

Mathematica [A]  time = 0.0018268, size = 50, normalized size = 1. \[ a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{4} b c x^4+\frac{1}{5} b d x^5+\frac{1}{6} b e x^6 \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2)*(a + b*x^3),x]

[Out]

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

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Maple [A]  time = 0.002, size = 41, normalized size = 0.8 \begin{align*} acx+{\frac{ad{x}^{2}}{2}}+{\frac{ae{x}^{3}}{3}}+{\frac{bc{x}^{4}}{4}}+{\frac{bd{x}^{5}}{5}}+{\frac{be{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d*x+c)*(b*x^3+a),x)

[Out]

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

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Maxima [A]  time = 0.950306, size = 54, normalized size = 1.08 \begin{align*} \frac{1}{6} \, b e x^{6} + \frac{1}{5} \, b d x^{5} + \frac{1}{4} \, b c x^{4} + \frac{1}{3} \, a e x^{3} + \frac{1}{2} \, a d x^{2} + a c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)*(b*x^3+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.28266, size = 104, normalized size = 2.08 \begin{align*} \frac{1}{6} x^{6} e b + \frac{1}{5} x^{5} d b + \frac{1}{4} x^{4} c b + \frac{1}{3} x^{3} e a + \frac{1}{2} x^{2} d a + x c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)*(b*x^3+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.064803, size = 46, normalized size = 0.92 \begin{align*} a c x + \frac{a d x^{2}}{2} + \frac{a e x^{3}}{3} + \frac{b c x^{4}}{4} + \frac{b d x^{5}}{5} + \frac{b e x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d*x+c)*(b*x**3+a),x)

[Out]

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

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Giac [A]  time = 1.04892, size = 57, normalized size = 1.14 \begin{align*} \frac{1}{6} \, b x^{6} e + \frac{1}{5} \, b d x^{5} + \frac{1}{4} \, b c x^{4} + \frac{1}{3} \, a x^{3} e + \frac{1}{2} \, a d x^{2} + a c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)*(b*x^3+a),x, algorithm="giac")

[Out]

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