\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{11}} \, dx\) [1864]

Optimal result

Integrand size = 35, antiderivative size = 111 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=\frac {\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac {c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac {c^3 d^3}{4 e^4 (d+e x)^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac {c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac {\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac {c^3 d^3}{4 e^4 (d+e x)^4} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^11,x]

[Out]

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

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])

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)^8} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^8}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^7}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^6}+\frac {c^3 d^3}{e^3 (d+e x)^5}\right ) \, dx \\ & = \frac {\left (c d^2-a e^2\right )^3}{7 e^4 (d+e x)^7}-\frac {c d \left (c d^2-a e^2\right )^2}{2 e^4 (d+e x)^6}+\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{5 e^4 (d+e x)^5}-\frac {c^3 d^3}{4 e^4 (d+e x)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {20 a^3 e^6+10 a^2 c d e^4 (d+7 e x)+4 a c^2 d^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+c^3 d^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )}{140 e^4 (d+e x)^7} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^11,x]

[Out]

-1/140*(20*a^3*e^6 + 10*a^2*c*d*e^4*(d + 7*e*x) + 4*a*c^2*d^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + c^3*d^3*(d^3
+ 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3))/(e^4*(d + e*x)^7)

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16

method	result	size
risch	\(\frac {-\frac {c^{3} d^{3} x^{3}}{4 e}-\frac {3 d^{2} c^{2} \left (4 e^{2} a +c \,d^{2}\right ) x^{2}}{20 e^{2}}-\frac {d c \left (10 a^{2} e^{4}+4 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{20 e^{3}}-\frac {20 e^{6} a^{3}+10 d^{2} e^{4} a^{2} c +4 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{140 e^{4}}}{\left (e x +d \right )^{7}}\)	\(129\)
gosper	\(-\frac {35 x^{3} c^{3} d^{3} e^{3}+84 x^{2} a \,c^{2} d^{2} e^{4}+21 x^{2} c^{3} d^{4} e^{2}+70 x \,a^{2} c d \,e^{5}+28 x a \,c^{2} d^{3} e^{3}+7 x \,c^{3} d^{5} e +20 e^{6} a^{3}+10 d^{2} e^{4} a^{2} c +4 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{140 e^{4} \left (e x +d \right )^{7}}\)	\(130\)
parallelrisch	\(\frac {-35 c^{3} d^{3} x^{3} e^{6}-84 a \,c^{2} d^{2} e^{7} x^{2}-21 c^{3} d^{4} e^{5} x^{2}-70 a^{2} c d \,e^{8} x -28 a \,c^{2} d^{3} e^{6} x -7 c^{3} d^{5} e^{4} x -20 a^{3} e^{9}-10 a^{2} c \,d^{2} e^{7}-4 d^{4} c^{2} a \,e^{5}-c^{3} d^{6} e^{3}}{140 e^{7} \left (e x +d \right )^{7}}\)	\(136\)
default	\(-\frac {3 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right )}{5 e^{4} \left (e x +d \right )^{5}}-\frac {e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}}{7 e^{4} \left (e x +d \right )^{7}}-\frac {c^{3} d^{3}}{4 e^{4} \left (e x +d \right )^{4}}-\frac {c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{2 e^{4} \left (e x +d \right )^{6}}\)	\(141\)
norman	\(\frac {-\frac {d^{3} \left (20 a^{3} e^{12}+10 a^{2} c \,d^{2} e^{10}+4 d^{4} c^{2} a \,e^{8}+c^{3} d^{6} e^{6}\right )}{140 e^{10}}-\frac {\left (a^{3} e^{12}+11 a^{2} c \,d^{2} e^{10}+17 d^{4} c^{2} a \,e^{8}+6 c^{3} d^{6} e^{6}\right ) x^{3}}{7 e^{7}}-\frac {d \left (2 a^{2} c \,e^{10}+8 a \,c^{2} d^{2} e^{8}+5 d^{4} c^{3} e^{6}\right ) x^{4}}{4 e^{6}}-\frac {3 d \left (4 a^{3} e^{12}+16 a^{2} c \,d^{2} e^{10}+12 d^{4} c^{2} a \,e^{8}+3 c^{3} d^{6} e^{6}\right ) x^{2}}{28 e^{8}}-\frac {3 d^{2} \left (2 a \,c^{2} e^{8}+3 c^{3} d^{2} e^{6}\right ) x^{5}}{10 e^{5}}-\frac {d^{2} \left (6 a^{3} e^{12}+10 a^{2} c \,d^{2} e^{10}+4 d^{4} c^{2} a \,e^{8}+c^{3} d^{6} e^{6}\right ) x}{14 e^{9}}-\frac {e^{2} c^{3} d^{3} x^{6}}{4}}{\left (e x +d \right )^{10}}\)	\(305\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^11,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {35 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6} + 21 \, {\left (c^{3} d^{4} e^{2} + 4 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \, {\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + 10 \, a^{2} c d e^{5}\right )} x}{140 \, {\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^11,x, algorithm="fricas")

[Out]

-1/140*(35*c^3*d^3*e^3*x^3 + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4 + 20*a^3*e^6 + 21*(c^3*d^4*e^2 + 4*a
*c^2*d^2*e^4)*x^2 + 7*(c^3*d^5*e + 4*a*c^2*d^3*e^3 + 10*a^2*c*d*e^5)*x)/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*
x^5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3 + 21*d^5*e^6*x^2 + 7*d^6*e^5*x + d^7*e^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3/(e*x+d)**11,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {35 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6} + 21 \, {\left (c^{3} d^{4} e^{2} + 4 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \, {\left (c^{3} d^{5} e + 4 \, a c^{2} d^{3} e^{3} + 10 \, a^{2} c d e^{5}\right )} x}{140 \, {\left (e^{11} x^{7} + 7 \, d e^{10} x^{6} + 21 \, d^{2} e^{9} x^{5} + 35 \, d^{3} e^{8} x^{4} + 35 \, d^{4} e^{7} x^{3} + 21 \, d^{5} e^{6} x^{2} + 7 \, d^{6} e^{5} x + d^{7} e^{4}\right )}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^11,x, algorithm="maxima")

[Out]

-1/140*(35*c^3*d^3*e^3*x^3 + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4 + 20*a^3*e^6 + 21*(c^3*d^4*e^2 + 4*a
*c^2*d^2*e^4)*x^2 + 7*(c^3*d^5*e + 4*a*c^2*d^3*e^3 + 10*a^2*c*d*e^5)*x)/(e^11*x^7 + 7*d*e^10*x^6 + 21*d^2*e^9*
x^5 + 35*d^3*e^8*x^4 + 35*d^4*e^7*x^3 + 21*d^5*e^6*x^2 + 7*d^6*e^5*x + d^7*e^4)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {35 \, c^{3} d^{3} e^{3} x^{3} + 21 \, c^{3} d^{4} e^{2} x^{2} + 84 \, a c^{2} d^{2} e^{4} x^{2} + 7 \, c^{3} d^{5} e x + 28 \, a c^{2} d^{3} e^{3} x + 70 \, a^{2} c d e^{5} x + c^{3} d^{6} + 4 \, a c^{2} d^{4} e^{2} + 10 \, a^{2} c d^{2} e^{4} + 20 \, a^{3} e^{6}}{140 \, {\left (e x + d\right )}^{7} e^{4}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3/(e*x+d)^11,x, algorithm="giac")

[Out]

-1/140*(35*c^3*d^3*e^3*x^3 + 21*c^3*d^4*e^2*x^2 + 84*a*c^2*d^2*e^4*x^2 + 7*c^3*d^5*e*x + 28*a*c^2*d^3*e^3*x +
70*a^2*c*d*e^5*x + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4 + 20*a^3*e^6)/((e*x + d)^7*e^4)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{11}} \, dx=-\frac {\frac {20\,a^3\,e^6+10\,a^2\,c\,d^2\,e^4+4\,a\,c^2\,d^4\,e^2+c^3\,d^6}{140\,e^4}+\frac {c^3\,d^3\,x^3}{4\,e}+\frac {c\,d\,x\,\left (10\,a^2\,e^4+4\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{20\,e^3}+\frac {3\,c^2\,d^2\,x^2\,\left (c\,d^2+4\,a\,e^2\right )}{20\,e^2}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3/(d + e*x)^11,x)

[Out]

-((20*a^3*e^6 + c^3*d^6 + 4*a*c^2*d^4*e^2 + 10*a^2*c*d^2*e^4)/(140*e^4) + (c^3*d^3*x^3)/(4*e) + (c*d*x*(10*a^2
*e^4 + c^2*d^4 + 4*a*c*d^2*e^2))/(20*e^3) + (3*c^2*d^2*x^2*(4*a*e^2 + c*d^2))/(20*e^2))/(d^7 + e^7*x^7 + 7*d*e
^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)