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

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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} \begin {gather*} a A x+\frac {B \left (a+c x^2\right )^2}{4 c}+\frac {1}{3} A c x^3 \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 32, normalized size = 1.03 \begin {gather*} a A x+\frac {1}{2} a B x^2+\frac {1}{3} A c x^3+\frac {1}{4} B c x^4 \end {gather*}

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)*(a + c*x^2),x]

[Out]

IntegrateAlgebraic[(A + B*x)*(a + c*x^2), x]

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fricas [A]  time = 0.37, size = 26, normalized size = 0.84 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.15, size = 26, normalized size = 0.84 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.04, size = 27, normalized size = 0.87 \begin {gather*} \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 {gather*}

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*B*a*x^2+A*a*x

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maxima [A]  time = 0.49, size = 26, normalized size = 0.84 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.04, size = 26, normalized size = 0.84 \begin {gather*} \frac {B\,c\,x^4}{4}+\frac {A\,c\,x^3}{3}+\frac {B\,a\,x^2}{2}+A\,a\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)*(A + B*x),x)

[Out]

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

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sympy [A]  time = 0.06, size = 29, normalized size = 0.94 \begin {gather*} A a x + \frac {A c x^{3}}{3} + \frac {B a x^{2}}{2} + \frac {B c x^{4}}{4} \end {gather*}

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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