\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^6} \, dx\) [1847]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^6} \, dx=\frac {(a e+c d x)^3}{3 \left (c d^2-a e^2\right ) (d+e x)^3} \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 37} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^6} \, dx=\frac {(a e+c d x)^3}{3 (d+e x)^3 \left (c d^2-a e^2\right )} \]

[In]

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

[Out]

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

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]

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^6} \, dx=-\frac {a^2 e^4+a c d e^2 (d+3 e x)+c^2 d^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(33)=66\).

Time = 2.37 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.00

method	result	size
gosper	\(-\frac {3 x^{2} c^{2} d^{2} e^{2}+3 x a c d \,e^{3}+3 x \,c^{2} d^{3} e +a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{3 e^{3} \left (e x +d \right )^{3}}\)	\(70\)
risch	\(\frac {-\frac {d^{2} c^{2} x^{2}}{e}-\frac {c d \left (e^{2} a +c \,d^{2}\right ) x}{e^{2}}-\frac {a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{3 e^{3}}}{\left (e x +d \right )^{3}}\)	\(72\)
parallelrisch	\(\frac {-3 x^{2} c^{2} d^{2} e^{2}-3 x a c d \,e^{3}-3 x \,c^{2} d^{3} e -a^{2} e^{4}-a c \,d^{2} e^{2}-c^{2} d^{4}}{3 e^{3} \left (e x +d \right )^{3}}\)	\(73\)
default	\(-\frac {d^{2} c^{2}}{e^{3} \left (e x +d \right )}-\frac {a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {c d \left (e^{2} a -c \,d^{2}\right )}{e^{3} \left (e x +d \right )^{2}}\)	\(83\)
norman	\(\frac {-\frac {d^{2} \left (a^{2} e^{6}+a c \,d^{2} e^{4}+c^{2} d^{4} e^{2}\right )}{3 e^{5}}-\frac {\left (a^{2} e^{6}+7 a c \,d^{2} e^{4}+10 c^{2} d^{4} e^{2}\right ) x^{2}}{3 e^{3}}-e \,c^{2} d^{2} x^{4}-\frac {d \left (a c \,e^{4}+3 c^{2} d^{2} e^{2}\right ) x^{3}}{e^{2}}-\frac {d \left (2 a^{2} e^{6}+5 a c \,d^{2} e^{4}+5 c^{2} d^{4} e^{2}\right ) x}{3 e^{4}}}{\left (e x +d \right )^{5}}\)	\(158\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (33) = 66\).

Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.69 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^6} \, dx=-\frac {3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (27) = 54\).

Time = 0.44 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^6} \, dx=\frac {- a^{2} e^{4} - a c d^{2} e^{2} - c^{2} d^{4} - 3 c^{2} d^{2} e^{2} x^{2} + x \left (- 3 a c d e^{3} - 3 c^{2} d^{3} e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \]

[In]

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

[Out]

(-a**2*e**4 - a*c*d**2*e**2 - c**2*d**4 - 3*c**2*d**2*e**2*x**2 + x*(-3*a*c*d*e**3 - 3*c**2*d**3*e))/(3*d**3*e
**3 + 9*d**2*e**4*x + 9*d*e**5*x**2 + 3*e**6*x**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (33) = 66\).

Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.69 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^6} \, dx=-\frac {3 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4} + 3 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (33) = 66\).

Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^6} \, dx=-\frac {3 \, c^{2} d^{2} e^{2} x^{2} + 3 \, c^{2} d^{3} e x + 3 \, a c d e^{3} x + c^{2} d^{4} + a c d^{2} e^{2} + a^{2} e^{4}}{3 \, {\left (e x + d\right )}^{3} e^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^6} \, dx=-\frac {\frac {a^2\,e}{3}-d\,\left (\frac {c^2\,x^3}{3}-a\,c\,x\right )+\frac {a\,c\,d^2}{3\,e}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]

[In]

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

[Out]

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