3.992 \(\int \frac{(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=17 \[ \frac{c^2 (d+e x)^2}{2 e} \]

[Out]

(c^2*(d + e*x)^2)/(2*e)

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Rubi [A]  time = 0.0035307, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {27, 9} \[ \frac{c^2 (d+e x)^2}{2 e} \]

Antiderivative was successfully verified.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^3,x]

[Out]

(c^2*(d + e*x)^2)/(2*e)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, x]

Rubi steps

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

Mathematica [A]  time = 0.000673, size = 16, normalized size = 0.94 \[ c^2 \left (d x+\frac{e x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^3,x]

[Out]

c^2*(d*x + (e*x^2)/2)

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Maple [A]  time = 0.038, size = 15, normalized size = 0.9 \begin{align*}{c}^{2} \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)^2/(e*x+d)^3,x)

[Out]

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

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Maxima [A]  time = 1.16155, size = 22, normalized size = 1.29 \begin{align*} \frac{1}{2} \, c^{2} e x^{2} + c^{2} 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)^2/(e*x+d)^3,x, algorithm="maxima")

[Out]

1/2*c^2*e*x^2 + c^2*d*x

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Fricas [A]  time = 2.05921, size = 34, normalized size = 2. \begin{align*} \frac{1}{2} \, c^{2} e x^{2} + c^{2} 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)^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/2*c^2*e*x^2 + c^2*d*x

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Sympy [A]  time = 0.116893, size = 15, normalized size = 0.88 \begin{align*} c^{2} d x + \frac{c^{2} 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)**2/(e*x+d)**3,x)

[Out]

c**2*d*x + c**2*e*x**2/2

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Giac [A]  time = 1.14977, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{2} \,{\left (c^{2} x^{2} e^{7} + 2 \, c^{2} d x e^{6}\right )} e^{\left (-6\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)^2/(e*x+d)^3,x, algorithm="giac")

[Out]

1/2*(c^2*x^2*e^7 + 2*c^2*d*x*e^6)*e^(-6)