3.981 \(\int \frac{c d^2+2 c d e x+c e^2 x^2}{d+e x} \, dx\)

Optimal. Leaf size=14 \[ c d x+\frac{1}{2} c e x^2 \]

[Out]

c*d*x + (c*e*x^2)/2

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Rubi [A]  time = 0.0083603, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {24} \[ c d x+\frac{1}{2} c e x^2 \]

Antiderivative was successfully verified.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x),x]

[Out]

c*d*x + (c*e*x^2)/2

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0005952, size = 14, normalized size = 1. \[ c \left (d x+\frac{e x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x),x]

[Out]

c*(d*x + (e*x^2)/2)

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Maple [A]  time = 0.038, size = 13, normalized size = 0.9 \begin{align*} c \left ({\frac{e{x}^{2}}{2}}+dx \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d),x)

[Out]

c*(1/2*e*x^2+d*x)

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Maxima [A]  time = 1.11703, size = 16, normalized size = 1.14 \begin{align*} \frac{1}{2} \, c e x^{2} + c d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*e*x^2 + c*d*x

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Fricas [A]  time = 1.93557, size = 28, normalized size = 2. \begin{align*} \frac{1}{2} \, c e x^{2} + c d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*e*x^2 + c*d*x

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Sympy [A]  time = 0.085971, size = 12, normalized size = 0.86 \begin{align*} c d x + \frac{c e x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)/(e*x+d),x)

[Out]

c*d*x + c*e*x**2/2

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Giac [A]  time = 1.13159, size = 26, normalized size = 1.86 \begin{align*} \frac{1}{2} \,{\left (c x^{2} e^{3} + 2 \, c d x e^{2}\right )} e^{\left (-2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^2*e^3 + 2*c*d*x*e^2)*e^(-2)