\(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^4} \, dx\) [993]

Optimal result

Integrand size = 30, antiderivative size = 5 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^4} \, dx=c^2 x \]

[Out]

c^2*x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 8} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^4} \, dx=c^2 x \]

[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*} \text {integral}& = \int c^2 \, dx \\ & = c^2 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^4} \, dx=c^2 x \]

[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

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20

method	result	size
default	\(c^{2} x\)	\(6\)
risch	\(c^{2} x\)	\(6\)
norman	\(\frac {c^{2} x^{4} e^{3}-\frac {3 c^{2} d^{4}}{e}-6 c^{2} d^{2} e \,x^{2}-8 c^{2} d^{3} x}{\left (e x +d \right )^{3}}\)	\(52\)

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

c^2*x

Fricas [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^4} \, dx=c^{2} x \]

[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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.60 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^4} \, dx=c^{2} x \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^4} \, dx=c^{2} x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^4} \, dx=c^{2} x \]

[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

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^4} \, dx=c^2\,x \]

[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