3.1833 \(\int \frac {a d e+(c d^2+a e^2) x+c d e x^2}{d+e x} \, dx\)

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

[Out]

a*e*x+1/2*c*d*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 = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {24} \[ a e x+\frac {1}{2} c d x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ a e x+\frac {1}{2} c d x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*e*x + (c*d*x^2)/2

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fricas [A]  time = 1.14, size = 12, normalized size = 0.86 \[ \frac {1}{2} \, c d x^{2} + a e x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*d*x^2 + a*e*x

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giac [A]  time = 0.26, size = 19, normalized size = 1.36 \[ \frac {1}{2} \, {\left (c d x^{2} e^{2} + 2 \, a x e^{3}\right )} e^{\left (-2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*d*x^2*e^2 + 2*a*x*e^3)*e^(-2)

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maple [A]  time = 0.04, size = 13, normalized size = 0.93 \[ \frac {1}{2} c d \,x^{2}+a e x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.03, size = 12, normalized size = 0.86 \[ \frac {1}{2} \, c d x^{2} + a e x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*d*x^2 + a*e*x

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mupad [B]  time = 0.02, size = 12, normalized size = 0.86 \[ \frac {c\,d\,x^2}{2}+a\,e\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*e*x + (c*d*x^2)/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*e*x + c*d*x**2/2

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