3.1289 \(\int (A+B x) (a+c x^2) \, dx\)

Optimal. Leaf size=31 \[ a A x+\frac{B \left (a+c x^2\right )^2}{4 c}+\frac{1}{3} A c x^3 \]

[Out]

a*A*x + (A*c*x^3)/3 + (B*(a + c*x^2)^2)/(4*c)

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Rubi [A]  time = 0.0067756, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {641} \[ a A x+\frac{B \left (a+c x^2\right )^2}{4 c}+\frac{1}{3} A c x^3 \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(a + c*x^2),x]

[Out]

a*A*x + (A*c*x^3)/3 + (B*(a + c*x^2)^2)/(4*c)

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

\begin{align*} \int (A+B x) \left (a+c x^2\right ) \, dx &=\frac{B \left (a+c x^2\right )^2}{4 c}+A \int \left (a+c x^2\right ) \, dx\\ &=a A x+\frac{1}{3} A c x^3+\frac{B \left (a+c x^2\right )^2}{4 c}\\ \end{align*}

Mathematica [A]  time = 0.0017835, size = 32, normalized size = 1.03 \[ a A x+\frac{1}{2} a B x^2+\frac{1}{3} A c x^3+\frac{1}{4} B c x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(a + c*x^2),x]

[Out]

a*A*x + (a*B*x^2)/2 + (A*c*x^3)/3 + (B*c*x^4)/4

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Maple [A]  time = 0.001, size = 27, normalized size = 0.9 \begin{align*}{\frac{Bc{x}^{4}}{4}}+{\frac{Ac{x}^{3}}{3}}+{\frac{aB{x}^{2}}{2}}+aAx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a),x)

[Out]

1/4*B*c*x^4+1/3*A*c*x^3+1/2*a*B*x^2+a*A*x

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Maxima [A]  time = 1.06404, size = 35, normalized size = 1.13 \begin{align*} \frac{1}{4} \, B c x^{4} + \frac{1}{3} \, A c x^{3} + \frac{1}{2} \, B a x^{2} + A a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a),x, algorithm="maxima")

[Out]

1/4*B*c*x^4 + 1/3*A*c*x^3 + 1/2*B*a*x^2 + A*a*x

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Fricas [A]  time = 1.50926, size = 66, normalized size = 2.13 \begin{align*} \frac{1}{4} x^{4} c B + \frac{1}{3} x^{3} c A + \frac{1}{2} x^{2} a B + x a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a),x, algorithm="fricas")

[Out]

1/4*x^4*c*B + 1/3*x^3*c*A + 1/2*x^2*a*B + x*a*A

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Sympy [A]  time = 0.126366, size = 29, normalized size = 0.94 \begin{align*} A a x + \frac{A c x^{3}}{3} + \frac{B a x^{2}}{2} + \frac{B c x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a),x)

[Out]

A*a*x + A*c*x**3/3 + B*a*x**2/2 + B*c*x**4/4

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Giac [A]  time = 1.28257, size = 35, normalized size = 1.13 \begin{align*} \frac{1}{4} \, B c x^{4} + \frac{1}{3} \, A c x^{3} + \frac{1}{2} \, B a x^{2} + A a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a),x, algorithm="giac")

[Out]

1/4*B*c*x^4 + 1/3*A*c*x^3 + 1/2*B*a*x^2 + A*a*x