\(\int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx\) [984]

Optimal result

Integrand size = 28, antiderivative size = 13 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx=-\frac {c}{e (d+e x)} \]

[Out]

-c/e/(e*x+d)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {24, 21, 32} \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx=-\frac {c}{e (d+e x)} \]

[In]

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

[Out]

-(c/(e*(d + e*x)))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {c d e^2+c e^3 x}{(d+e x)^3} \, dx}{e^2} \\ & = c \int \frac {1}{(d+e x)^2} \, dx \\ & = -\frac {c}{e (d+e x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx=-\frac {c}{e (d+e x)} \]

[In]

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

[Out]

-(c/(e*(d + e*x)))

Maple [A] (verified)

Time = 2.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
gosper	\(-\frac {c}{e \left (e x +d \right )}\)	\(14\)
default	\(-\frac {c}{e \left (e x +d \right )}\)	\(14\)
risch	\(-\frac {c}{e \left (e x +d \right )}\)	\(14\)
parallelrisch	\(-\frac {c}{e \left (e x +d \right )}\)	\(14\)
norman	\(\frac {c d x +2 c e \,x^{2}+\frac {e^{2} c \,x^{3}}{d}}{\left (e x +d \right )^{3}}\)	\(32\)

[In]

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

[Out]

-c/e/(e*x+d)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx=-\frac {c}{e^{2} x + d e} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

-c/(e^2*x + d*e)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx=- \frac {c}{d e + e^{2} x} \]

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)/(e*x+d)**4,x)

[Out]

-c/(d*e + e**2*x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx=-\frac {c}{e^{2} x + d e} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

-c/(e^2*x + d*e)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx=-\frac {c}{{\left (e x + d\right )} e} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^4,x, algorithm="giac")

[Out]

-c/((e*x + d)*e)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx=-\frac {c}{e\,\left (d+e\,x\right )} \]

[In]

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

[Out]

-c/(e*(d + e*x))