\(\int \frac {a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^6} \, dx\) [1838]

Optimal result

Integrand size = 33, antiderivative size = 39 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx=-\frac {a-\frac {c d^2}{e^2}}{4 (d+e x)^4}-\frac {c d}{3 e^2 (d+e x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {24, 45} \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx=-\frac {a-\frac {c d^2}{e^2}}{4 (d+e x)^4}-\frac {c d}{3 e^2 (d+e x)^3} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^6,x]

[Out]

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

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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {a e^3+c d e^2 x}{(d+e x)^5} \, dx}{e^2} \\ & = \frac {\int \left (\frac {-c d^2 e+a e^3}{(d+e x)^5}+\frac {c d e}{(d+e x)^4}\right ) \, dx}{e^2} \\ & = -\frac {a-\frac {c d^2}{e^2}}{4 (d+e x)^4}-\frac {c d}{3 e^2 (d+e x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx=-\frac {3 a e^2+c d (d+4 e x)}{12 e^2 (d+e x)^4} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^6,x]

[Out]

-1/12*(3*a*e^2 + c*d*(d + 4*e*x))/(e^2*(d + e*x)^4)

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79

method	result	size
gosper	\(-\frac {4 x c d e +3 e^{2} a +c \,d^{2}}{12 e^{2} \left (e x +d \right )^{4}}\)	\(31\)
risch	\(\frac {-\frac {c d x}{3 e}-\frac {3 e^{2} a +c \,d^{2}}{12 e^{2}}}{\left (e x +d \right )^{4}}\)	\(35\)
parallelrisch	\(\frac {-4 c d \,e^{3} x -3 e^{4} a -d^{2} e^{2} c}{12 e^{4} \left (e x +d \right )^{4}}\)	\(37\)
default	\(-\frac {c d}{3 e^{2} \left (e x +d \right )^{3}}-\frac {e^{2} a -c \,d^{2}}{4 e^{2} \left (e x +d \right )^{4}}\)	\(40\)
norman	\(\frac {-\frac {d \left (3 a \,e^{5}+d^{2} e^{3} c \right )}{12 e^{5}}-\frac {\left (3 a \,e^{5}+5 d^{2} e^{3} c \right ) x}{12 e^{4}}-\frac {c d \,x^{2}}{3}}{\left (e x +d \right )^{5}}\)	\(60\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

-1/12/e^2*(4*c*d*e*x+3*a*e^2+c*d^2)/(e*x+d)^4

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.69 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx=-\frac {4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \, {\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/12*(4*c*d*e*x + c*d^2 + 3*a*e^2)/(e^6*x^4 + 4*d*e^5*x^3 + 6*d^2*e^4*x^2 + 4*d^3*e^3*x + d^4*e^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (34) = 68\).

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.79 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx=\frac {- 3 a e^{2} - c d^{2} - 4 c d e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{4}} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d)**6,x)

[Out]

(-3*a*e**2 - c*d**2 - 4*c*d*e*x)/(12*d**4*e**2 + 48*d**3*e**3*x + 72*d**2*e**4*x**2 + 48*d*e**5*x**3 + 12*e**6
*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.69 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx=-\frac {4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \, {\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/12*(4*c*d*e*x + c*d^2 + 3*a*e^2)/(e^6*x^4 + 4*d*e^5*x^3 + 6*d^2*e^4*x^2 + 4*d^3*e^3*x + d^4*e^2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx=-\frac {4 \, c d e x + c d^{2} + 3 \, a e^{2}}{12 \, {\left (e x + d\right )}^{4} e^{2}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/12*(4*c*d*e*x + c*d^2 + 3*a*e^2)/((e*x + d)^4*e^2)

Mupad [B] (verification not implemented)

Time = 10.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.74 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx=-\frac {\frac {c\,d^2+3\,a\,e^2}{12\,e^2}+\frac {c\,d\,x}{3\,e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)/(d + e*x)^6,x)

[Out]

-((3*a*e^2 + c*d^2)/(12*e^2) + (c*d*x)/(3*e))/(d^4 + e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x)