\(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^7} \, dx\) [996]

Optimal result

Integrand size = 30, antiderivative size = 17 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {c^2}{2 e (d+e x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {c^2}{2 e (d+e x)^2} \]

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^7,x]

[Out]

-1/2*c^2/(e*(d + e*x)^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {c^2}{2 e (d+e x)^2} \]

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^7,x]

[Out]

-1/2*c^2/(e*(d + e*x)^2)

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {c^{2}}{2 e \left (e x +d \right )^{2}}\)	\(16\)
default	\(-\frac {c^{2}}{2 e \left (e x +d \right )^{2}}\)	\(16\)
risch	\(-\frac {c^{2}}{2 e \left (e x +d \right )^{2}}\)	\(16\)
parallelrisch	\(-\frac {c^{2}}{2 e \left (e x +d \right )^{2}}\)	\(16\)
norman	\(\frac {-\frac {c^{2} d^{4}}{2 e}-\frac {c^{2} x^{4} e^{3}}{2}-2 d \,e^{2} c^{2} x^{3}-3 c^{2} d^{2} e \,x^{2}-2 c^{2} d^{3} x}{\left (e x +d \right )^{6}}\)	\(65\)

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {c^{2}}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/2*c^2/(e^3*x^2 + 2*d*e^2*x + d^2*e)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx=- \frac {c^{2}}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \]

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**2/(e*x+d)**7,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {c^{2}}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^7,x, algorithm="maxima")

[Out]

-1/2*c^2/(e^3*x^2 + 2*d*e^2*x + d^2*e)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {c^{2}}{2 \, {\left (e x + d\right )}^{2} e} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/2*c^2/((e*x + d)^2*e)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {c^2}{2\,e\,\left (d^2+2\,d\,e\,x+e^2\,x^2\right )} \]

[In]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^2/(d + e*x)^7,x)

[Out]

-c^2/(2*e*(d^2 + e^2*x^2 + 2*d*e*x))