\(\int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx\) [985]

Optimal result

Integrand size = 28, antiderivative size = 15 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx=-\frac {c}{2 e (d+e x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, 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} \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx=-\frac {c}{2 e (d+e x)^2} \]

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^5,x]

[Out]

-1/2*c/(e*(d + e*x)^2)

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)^4} \, dx}{e^2} \\ & = c \int \frac {1}{(d+e x)^3} \, dx \\ & = -\frac {c}{2 e (d+e x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx=-\frac {c}{2 e (d+e x)^2} \]

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)/(d + e*x)^5,x]

[Out]

-1/2*c/(e*(d + e*x)^2)

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {c}{2 e \left (e x +d \right )^{2}}\)	\(14\)
default	\(-\frac {c}{2 e \left (e x +d \right )^{2}}\)	\(14\)
risch	\(-\frac {c}{2 e \left (e x +d \right )^{2}}\)	\(14\)
parallelrisch	\(-\frac {c}{2 e \left (e x +d \right )^{2}}\)	\(14\)
norman	\(\frac {-\frac {d^{2} c}{2 e}-\frac {c e \,x^{2}}{2}-c d x}{\left (e x +d \right )^{4}}\)	\(31\)

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx=-\frac {c}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

-1/2*c/(e^3*x^2 + 2*d*e^2*x + d^2*e)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx=- \frac {c}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \]

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)/(e*x+d)**5,x)

[Out]

-c/(2*d**2*e + 4*d*e**2*x + 2*e**3*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx=-\frac {c}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/2*c/(e^3*x^2 + 2*d*e^2*x + d^2*e)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx=-\frac {c}{2 \, {\left (e x + d\right )}^{2} e} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)/(e*x+d)^5,x, algorithm="giac")

[Out]

-1/2*c/((e*x + d)^2*e)

Mupad [B] (verification not implemented)

Time = 9.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx=-\frac {c}{2\,e\,\left (d^2+2\,d\,e\,x+e^2\,x^2\right )} \]

[In]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)/(d + e*x)^5,x)

[Out]

-c/(2*e*(d^2 + e^2*x^2 + 2*d*e*x))