3.9.5 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=5 \[ c^2 x \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 5, normalized size of antiderivative = 1.00, 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, 8} \begin {gather*} c^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^4,x]

[Out]

c^2*x

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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]

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)^4} \, dx &=\int c^2 \, dx\\ &=c^2 x\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 5, normalized size = 1.00 \begin {gather*} c^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^4,x]

[Out]

c^2*x

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 0.04, size = 12, normalized size = 2.40 \begin {gather*} \frac {c^2 (d+e x)}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^4,x]

[Out]

(c^2*(d + e*x))/e

________________________________________________________________________________________

fricas [A]  time = 0.38, size = 5, normalized size = 1.00 \begin {gather*} c^{2} 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)^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

c^2*x

________________________________________________________________________________________

giac [A]  time = 0.16, size = 5, normalized size = 1.00 \begin {gather*} c^{2} 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)^2/(e*x+d)^4,x, algorithm="giac")

[Out]

c^2*x

________________________________________________________________________________________

maple [A]  time = 0.04, size = 6, normalized size = 1.20 \begin {gather*} c^{2} x \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)^2/(e*x+d)^4,x)

[Out]

c^2*x

________________________________________________________________________________________

maxima [A]  time = 1.35, size = 5, normalized size = 1.00 \begin {gather*} c^{2} 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)^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

c^2*x

________________________________________________________________________________________

mupad [B]  time = 0.01, size = 5, normalized size = 1.00 \begin {gather*} c^2\,x \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)^2/(d + e*x)^4,x)

[Out]

c^2*x

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 3, normalized size = 0.60 \begin {gather*} c^{2} 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)**2/(e*x+d)**4,x)

[Out]

c**2*x

________________________________________________________________________________________