\(\int \frac {a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^4} \, dx\) [1836]

Optimal result

Integrand size = 33, antiderivative size = 35 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^4} \, dx=\frac {(a e+c d x)^2}{2 \left (c d^2-a e^2\right ) (d+e x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {24, 37} \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^4} \, dx=\frac {(a e+c d x)^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )} \]

[In]

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

[Out]

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

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^4} \, dx=-\frac {a e^2+c d (d+2 e x)}{2 e^2 (d+e x)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86

method	result	size
gosper	\(-\frac {2 x c d e +e^{2} a +c \,d^{2}}{2 \left (e x +d \right )^{2} e^{2}}\)	\(30\)
parallelrisch	\(\frac {-2 x c d e -e^{2} a -c \,d^{2}}{2 e^{2} \left (e x +d \right )^{2}}\)	\(32\)
risch	\(\frac {-\frac {c d x}{e}-\frac {e^{2} a +c \,d^{2}}{2 e^{2}}}{\left (e x +d \right )^{2}}\)	\(34\)
default	\(-\frac {c d}{e^{2} \left (e x +d \right )}-\frac {e^{2} a -c \,d^{2}}{2 e^{2} \left (e x +d \right )^{2}}\)	\(40\)
norman	\(\frac {-\frac {d \left (a \,e^{3}+d^{2} e c \right )}{2 e^{3}}-\frac {\left (a \,e^{3}+3 d^{2} e c \right ) x}{2 e^{2}}-c d \,x^{2}}{\left (e x +d \right )^{3}}\)	\(54\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^4} \, dx=-\frac {2 \, c d e x + c d^{2} + a e^{2}}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]

[In]

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

[Out]

-1/2*(2*c*d*e*x + c*d^2 + a*e^2)/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^4} \, dx=\frac {- a e^{2} - c d^{2} - 2 c d e x}{2 d^{2} e^{2} + 4 d e^{3} x + 2 e^{4} x^{2}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^4} \, dx=-\frac {2 \, c d e x + c d^{2} + a e^{2}}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]

[In]

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

[Out]

-1/2*(2*c*d*e*x + c*d^2 + a*e^2)/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^4} \, dx=-\frac {2 \, c d e x + c d^{2} + a e^{2}}{2 \, {\left (e x + d\right )}^{2} e^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^4} \, dx=-\frac {\frac {a}{2}-\frac {c\,x^2}{2}}{d^2+2\,d\,e\,x+e^2\,x^2} \]

[In]

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

[Out]

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