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

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

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, 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} \begin {gather*} c d x+\frac {1}{2} c e x^2 \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} c \left (d x+\frac {e x^2}{2}\right ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 13, normalized size = 0.93 \begin {gather*} \frac {1}{2} c x (2 d+e x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(2*d + e*x))/2

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fricas [A]  time = 0.38, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, c e x^{2} + c d x \end {gather*}

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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giac [A]  time = 0.18, size = 19, normalized size = 1.36 \begin {gather*} \frac {1}{2} \, {\left (c x^{2} e^{3} + 2 \, c d x e^{2}\right )} e^{\left (-2\right )} \end {gather*}

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)

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maple [A]  time = 0.05, size = 13, normalized size = 0.93 \begin {gather*} \left (\frac {1}{2} e \,x^{2}+d x \right ) c \end {gather*}

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 = 1.33, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{2} \, c e x^{2} + c d x \end {gather*}

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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mupad [B]  time = 0.02, size = 11, normalized size = 0.79 \begin {gather*} \frac {c\,x\,\left (2\,d+e\,x\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x*(2*d + e*x))/2

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sympy [A]  time = 0.08, size = 12, normalized size = 0.86 \begin {gather*} c d x + \frac {c e x^{2}}{2} \end {gather*}

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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