3.56 \(\int \frac{a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx\)

Optimal. Leaf size=12 \[ c x+\frac{d x^2}{2} \]

[Out]

c*x + (d*x^2)/2

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Rubi [A]  time = 0.0138278, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {1586} \[ c x+\frac{d x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3),x]

[Out]

c*x + (d*x^2)/2

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rubi steps

\begin{align*} \int \frac{a c+a d x+b c x^3+b d x^4}{a+b x^3} \, dx &=\int (c+d x) \, dx\\ &=c x+\frac{d x^2}{2}\\ \end{align*}

Mathematica [A]  time = 0.0007036, size = 12, normalized size = 1. \[ c x+\frac{d x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*c + a*d*x + b*c*x^3 + b*d*x^4)/(a + b*x^3),x]

[Out]

c*x + (d*x^2)/2

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Maple [A]  time = 0.002, size = 11, normalized size = 0.9 \begin{align*} cx+{\frac{d{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d*x^4+b*c*x^3+a*d*x+a*c)/(b*x^3+a),x)

[Out]

c*x+1/2*d*x^2

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Maxima [A]  time = 0.949419, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, d x^{2} + c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*d*x^4+b*c*x^3+a*d*x+a*c)/(b*x^3+a),x, algorithm="maxima")

[Out]

1/2*d*x^2 + c*x

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Fricas [A]  time = 1.25331, size = 23, normalized size = 1.92 \begin{align*} \frac{1}{2} \, d x^{2} + c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*d*x^4+b*c*x^3+a*d*x+a*c)/(b*x^3+a),x, algorithm="fricas")

[Out]

1/2*d*x^2 + c*x

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Sympy [A]  time = 0.069176, size = 8, normalized size = 0.67 \begin{align*} c x + \frac{d x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*d*x**4+b*c*x**3+a*d*x+a*c)/(b*x**3+a),x)

[Out]

c*x + d*x**2/2

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Giac [A]  time = 1.06273, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, d x^{2} + c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*d*x^4+b*c*x^3+a*d*x+a*c)/(b*x^3+a),x, algorithm="giac")

[Out]

1/2*d*x^2 + c*x