\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^9} \, dx\) [1862]

Optimal result

Integrand size = 35, antiderivative size = 73 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=\frac {(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {c d (a e+c d x)^4}{20 \left (c d^2-a e^2\right )^2 (d+e x)^4} \]

[Out]

1/5*(c*d*x+a*e)^4/(-a*e^2+c*d^2)/(e*x+d)^5+1/20*c*d*(c*d*x+a*e)^4/(-a*e^2+c*d^2)^2/(e*x+d)^4

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {640, 47, 37} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=\frac {c d (a e+c d x)^4}{20 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac {(a e+c d x)^4}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^9,x]

[Out]

(a*e + c*d*x)^4/(5*(c*d^2 - a*e^2)*(d + e*x)^5) + (c*d*(a*e + c*d*x)^4)/(20*(c*d^2 - a*e^2)^2*(d + e*x)^4)

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 47

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] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 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)^3}{(d+e x)^6} \, dx \\ & = \frac {(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {(c d) \int \frac {(a e+c d x)^3}{(d+e x)^5} \, dx}{5 \left (c d^2-a e^2\right )} \\ & = \frac {(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {c d (a e+c d x)^4}{20 \left (c d^2-a e^2\right )^2 (d+e x)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=-\frac {4 a^3 e^6+3 a^2 c d e^4 (d+5 e x)+2 a c^2 d^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+c^3 d^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )}{20 e^4 (d+e x)^5} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^9,x]

[Out]

-1/20*(4*a^3*e^6 + 3*a^2*c*d*e^4*(d + 5*e*x) + 2*a*c^2*d^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + c^3*d^3*(d^3 + 5
*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3))/(e^4*(d + e*x)^5)

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.77

method	result	size
risch	\(\frac {-\frac {c^{3} d^{3} x^{3}}{2 e}-\frac {d^{2} c^{2} \left (2 e^{2} a +c \,d^{2}\right ) x^{2}}{2 e^{2}}-\frac {d c \left (3 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{4 e^{3}}-\frac {4 e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +2 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{20 e^{4}}}{\left (e x +d \right )^{5}}\)	\(129\)
gosper	\(-\frac {10 x^{3} c^{3} d^{3} e^{3}+20 x^{2} a \,c^{2} d^{2} e^{4}+10 x^{2} c^{3} d^{4} e^{2}+15 x \,a^{2} c d \,e^{5}+10 x a \,c^{2} d^{3} e^{3}+5 x \,c^{3} d^{5} e +4 e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +2 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{20 e^{4} \left (e x +d \right )^{5}}\)	\(130\)
parallelrisch	\(\frac {-10 c^{3} d^{3} x^{3} e^{4}-20 a \,c^{2} d^{2} e^{5} x^{2}-10 c^{3} d^{4} e^{3} x^{2}-15 a^{2} c d \,e^{6} x -10 a \,c^{2} d^{3} e^{4} x -5 c^{3} d^{5} e^{2} x -4 a^{3} e^{7}-3 d^{2} e^{5} a^{2} c -2 a \,c^{2} d^{4} e^{3}-c^{3} d^{6} e}{20 e^{5} \left (e x +d \right )^{5}}\)	\(134\)
default	\(-\frac {e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}{5 e^{4} \left (e x +d \right )^{5}}-\frac {c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right )}{e^{4} \left (e x +d \right )^{3}}-\frac {3 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{4 e^{4} \left (e x +d \right )^{4}}-\frac {c^{3} d^{3}}{2 e^{4} \left (e x +d \right )^{2}}\)	\(141\)
norman	\(\frac {-\frac {d^{3} \left (4 a^{3} e^{10}+3 a^{2} c \,d^{2} e^{8}+2 d^{4} c^{2} a \,e^{6}+c^{3} d^{6} e^{4}\right )}{20 e^{8}}-\frac {\left (a^{3} e^{10}+12 a^{2} c \,d^{2} e^{8}+23 d^{4} c^{2} a \,e^{6}+14 c^{3} d^{6} e^{4}\right ) x^{3}}{5 e^{5}}-\frac {d \left (3 a^{2} c \,e^{8}+14 a \,c^{2} d^{2} e^{6}+13 c^{3} d^{4} e^{4}\right ) x^{4}}{4 e^{4}}-\frac {d \left (6 a^{3} e^{10}+27 a^{2} c \,d^{2} e^{8}+28 d^{4} c^{2} a \,e^{6}+14 c^{3} d^{6} e^{4}\right ) x^{2}}{10 e^{6}}-\frac {e^{2} c^{3} d^{3} x^{6}}{2}-\frac {d^{2} \left (a \,c^{2} e^{6}+2 c^{3} d^{2} e^{4}\right ) x^{5}}{e^{3}}-\frac {d^{2} \left (3 a^{3} e^{10}+6 a^{2} c \,d^{2} e^{8}+4 d^{4} c^{2} a \,e^{6}+2 c^{3} d^{6} e^{4}\right ) x}{5 e^{7}}}{\left (e x +d \right )^{8}}\)	\(305\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^9,x,method=_RETURNVERBOSE)

[Out]

(-1/2*c^3*d^3/e*x^3-1/2*d^2*c^2/e^2*(2*a*e^2+c*d^2)*x^2-1/4*d*c/e^3*(3*a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)*x-1/20/e
^4*(4*a^3*e^6+3*a^2*c*d^2*e^4+2*a*c^2*d^4*e^2+c^3*d^6))/(e*x+d)^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (69) = 138\).

Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=-\frac {10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \, {\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/20*(10*c^3*d^3*e^3*x^3 + c^3*d^6 + 2*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 4*a^3*e^6 + 10*(c^3*d^4*e^2 + 2*a*c^
2*d^2*e^4)*x^2 + 5*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 +
10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (61) = 122\).

Time = 112.46 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.56 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=\frac {- 4 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 2 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 10 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 20 a c^{2} d^{2} e^{4} - 10 c^{3} d^{4} e^{2}\right ) + x \left (- 15 a^{2} c d e^{5} - 10 a c^{2} d^{3} e^{3} - 5 c^{3} d^{5} e\right )}{20 d^{5} e^{4} + 100 d^{4} e^{5} x + 200 d^{3} e^{6} x^{2} + 200 d^{2} e^{7} x^{3} + 100 d e^{8} x^{4} + 20 e^{9} x^{5}} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**9,x)

[Out]

(-4*a**3*e**6 - 3*a**2*c*d**2*e**4 - 2*a*c**2*d**4*e**2 - c**3*d**6 - 10*c**3*d**3*e**3*x**3 + x**2*(-20*a*c**
2*d**2*e**4 - 10*c**3*d**4*e**2) + x*(-15*a**2*c*d*e**5 - 10*a*c**2*d**3*e**3 - 5*c**3*d**5*e))/(20*d**5*e**4
+ 100*d**4*e**5*x + 200*d**3*e**6*x**2 + 200*d**2*e**7*x**3 + 100*d*e**8*x**4 + 20*e**9*x**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (69) = 138\).

Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=-\frac {10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \, {\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^9,x, algorithm="maxima")

[Out]

-1/20*(10*c^3*d^3*e^3*x^3 + c^3*d^6 + 2*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 4*a^3*e^6 + 10*(c^3*d^4*e^2 + 2*a*c^
2*d^2*e^4)*x^2 + 5*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 +
10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=-\frac {10 \, c^{3} d^{3} e^{3} x^{3} + 10 \, c^{3} d^{4} e^{2} x^{2} + 20 \, a c^{2} d^{2} e^{4} x^{2} + 5 \, c^{3} d^{5} e x + 10 \, a c^{2} d^{3} e^{3} x + 15 \, a^{2} c d e^{5} x + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6}}{20 \, {\left (e x + d\right )}^{5} e^{4}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/20*(10*c^3*d^3*e^3*x^3 + 10*c^3*d^4*e^2*x^2 + 20*a*c^2*d^2*e^4*x^2 + 5*c^3*d^5*e*x + 10*a*c^2*d^3*e^3*x + 1
5*a^2*c*d*e^5*x + c^3*d^6 + 2*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 4*a^3*e^6)/((e*x + d)^5*e^4)

Mupad [B] (verification not implemented)

Time = 9.80 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx=-\frac {d^2\,\left (\frac {3\,a^2\,c}{20}+a\,c^2\,x^2-\frac {c^3\,x^4}{4}\right )-d\,\left (\frac {c^3\,e\,x^5}{20}-\frac {3\,a^2\,c\,e\,x}{4}\right )+\frac {a^3\,e^2}{5}+\frac {a\,c^2\,d^4}{10\,e^2}+\frac {a\,c^2\,d^3\,x}{2\,e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3/(d + e*x)^9,x)

[Out]

-(d^2*((3*a^2*c)/20 - (c^3*x^4)/4 + a*c^2*x^2) - d*((c^3*e*x^5)/20 - (3*a^2*c*e*x)/4) + (a^3*e^2)/5 + (a*c^2*d
^4)/(10*e^2) + (a*c^2*d^3*x)/(2*e))/(d^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d^3*e^2*x^2 + 10*d^2*e^3*x^3 + 5*d^4*e*x
)