\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^8} \, dx\) [1861]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^8} \, dx=\frac {(a e+c d x)^4}{4 \left (c d^2-a e^2\right ) (d+e x)^4} \]

[Out]

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

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} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^8} \, dx=\frac {(a e+c d x)^4}{4 (d+e x)^4 \left (c d^2-a e^2\right )} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^8,x]

[Out]

(a*e + c*d*x)^4/(4*(c*d^2 - a*e^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 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)^5} \, dx \\ & = \frac {(a e+c d x)^4}{4 \left (c d^2-a e^2\right ) (d+e x)^4} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(100\) vs. \(2(35)=70\).

Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.86 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^8} \, dx=-\frac {a^3 e^6+a^2 c d e^4 (d+4 e x)+a c^2 d^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^3 d^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{4 e^4 (d+e x)^4} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^8,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(33)=66\).

Time = 2.60 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.51

method	result	size
risch	\(\frac {-\frac {c^{3} d^{3} x^{3}}{e}-\frac {3 c^{2} d^{2} \left (e^{2} a +c \,d^{2}\right ) x^{2}}{2 e^{2}}-\frac {d c \left (a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{e^{3}}-\frac {e^{6} a^{3}+d^{2} e^{4} a^{2} c +d^{4} e^{2} c^{2} a +c^{3} d^{6}}{4 e^{4}}}{\left (e x +d \right )^{4}}\)	\(123\)
gosper	\(-\frac {4 x^{3} c^{3} d^{3} e^{3}+6 x^{2} a \,c^{2} d^{2} e^{4}+6 x^{2} c^{3} d^{4} e^{2}+4 x \,a^{2} c d \,e^{5}+4 x a \,c^{2} d^{3} e^{3}+4 x \,c^{3} d^{5} e +e^{6} a^{3}+d^{2} e^{4} a^{2} c +d^{4} e^{2} c^{2} a +c^{3} d^{6}}{4 e^{4} \left (e x +d \right )^{4}}\)	\(127\)
parallelrisch	\(\frac {-4 x^{3} c^{3} d^{3} e^{3}-6 x^{2} a \,c^{2} d^{2} e^{4}-6 x^{2} c^{3} d^{4} e^{2}-4 x \,a^{2} c d \,e^{5}-4 x a \,c^{2} d^{3} e^{3}-4 x \,c^{3} d^{5} e -e^{6} a^{3}-d^{2} e^{4} a^{2} c -d^{4} e^{2} c^{2} a -c^{3} d^{6}}{4 e^{4} \left (e x +d \right )^{4}}\)	\(131\)
default	\(-\frac {c^{3} d^{3}}{e^{4} \left (e x +d \right )}-\frac {c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{e^{4} \left (e x +d \right )^{3}}-\frac {e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}{4 e^{4} \left (e x +d \right )^{4}}-\frac {3 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right )}{2 e^{4} \left (e x +d \right )^{2}}\)	\(141\)
norman	\(\frac {-\frac {d^{3} \left (a^{3} e^{9}+a^{2} c \,d^{2} e^{7}+d^{4} c^{2} a \,e^{5}+c^{3} d^{6} e^{3}\right )}{4 e^{7}}-\frac {\left (a^{3} e^{9}+13 a^{2} c \,d^{2} e^{7}+31 d^{4} c^{2} a \,e^{5}+35 c^{3} d^{6} e^{3}\right ) x^{3}}{4 e^{4}}-e^{2} c^{3} d^{3} x^{6}-\frac {d \left (2 a^{2} c \,e^{7}+11 a \,c^{2} d^{2} e^{5}+17 c^{3} d^{4} e^{3}\right ) x^{4}}{2 e^{3}}-\frac {3 d \left (a^{3} e^{9}+5 a^{2} c \,d^{2} e^{7}+7 d^{4} c^{2} a \,e^{5}+7 c^{3} d^{6} e^{3}\right ) x^{2}}{4 e^{5}}-\frac {3 d^{2} \left (a \,c^{2} e^{5}+3 c^{3} d^{2} e^{3}\right ) x^{5}}{2 e^{2}}-\frac {d^{2} \left (3 a^{3} e^{9}+7 a^{2} c \,d^{2} e^{7}+7 d^{4} c^{2} a \,e^{5}+7 c^{3} d^{6} e^{3}\right ) x}{4 e^{6}}}{\left (e x +d \right )^{7}}\)	\(301\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^8,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (33) = 66\).

Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 4.51 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^8} \, dx=-\frac {4 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{4 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^8,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (27) = 54\).

Time = 47.48 (sec) , antiderivative size = 170, normalized size of antiderivative = 4.86 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^8} \, dx=\frac {- a^{3} e^{6} - a^{2} c d^{2} e^{4} - a c^{2} d^{4} e^{2} - c^{3} d^{6} - 4 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 6 a c^{2} d^{2} e^{4} - 6 c^{3} d^{4} e^{2}\right ) + x \left (- 4 a^{2} c d e^{5} - 4 a c^{2} d^{3} e^{3} - 4 c^{3} d^{5} e\right )}{4 d^{4} e^{4} + 16 d^{3} e^{5} x + 24 d^{2} e^{6} x^{2} + 16 d e^{7} x^{3} + 4 e^{8} x^{4}} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**8,x)

[Out]

(-a**3*e**6 - a**2*c*d**2*e**4 - a*c**2*d**4*e**2 - c**3*d**6 - 4*c**3*d**3*e**3*x**3 + x**2*(-6*a*c**2*d**2*e
**4 - 6*c**3*d**4*e**2) + x*(-4*a**2*c*d*e**5 - 4*a*c**2*d**3*e**3 - 4*c**3*d**5*e))/(4*d**4*e**4 + 16*d**3*e*
*5*x + 24*d**2*e**6*x**2 + 16*d*e**7*x**3 + 4*e**8*x**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (33) = 66\).

Time = 0.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 4.51 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^8} \, dx=-\frac {4 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{4 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^8,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (33) = 66\).

Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.60 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^8} \, dx=-\frac {4 \, c^{3} d^{3} e^{3} x^{3} + 6 \, c^{3} d^{4} e^{2} x^{2} + 6 \, a c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d^{5} e x + 4 \, a c^{2} d^{3} e^{3} x + 4 \, a^{2} c d e^{5} x + c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + a^{3} e^{6}}{4 \, {\left (e x + d\right )}^{4} e^{4}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^8,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.89 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.91 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^8} \, dx=-\frac {d\,\left (a^2\,c\,e\,x-a\,c^2\,e\,x^3\right )+\frac {a^3\,e^2}{4}+d^2\,\left (\frac {a^2\,c}{4}-\frac {c^3\,x^4}{4}\right )-\frac {a\,c^2\,e^2\,x^4}{4}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3/(d + e*x)^8,x)

[Out]

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