\(\int \frac {(a+b x+c x^2)^3}{(d+e x)^9} \, dx\) [2144]

Optimal result

Integrand size = 20, antiderivative size = 269 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^3}{8 e^7 (d+e x)^8}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{7 e^7 (d+e x)^7}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{2 e^7 (d+e x)^6}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{5 e^7 (d+e x)^5}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^7 (d+e x)^4}+\frac {c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac {c^3}{2 e^7 (d+e x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^7 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^6}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac {\left (a e^2-b d e+c d^2\right )^3}{8 e^7 (d+e x)^8}+\frac {c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac {c^3}{2 e^7 (d+e x)^2} \]

[In]

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

[Out]

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

Rule 712

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^9}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^8}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^7}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^6 (d+e x)^6}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)^5}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)^4}+\frac {c^3}{e^6 (d+e x)^3}\right ) \, dx \\ & = -\frac {\left (c d^2-b d e+a e^2\right )^3}{8 e^7 (d+e x)^8}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{7 e^7 (d+e x)^7}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{2 e^7 (d+e x)^6}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{5 e^7 (d+e x)^5}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^7 (d+e x)^4}+\frac {c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac {c^3}{2 e^7 (d+e x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+e^3 \left (35 a^3 e^3+15 a^2 b e^2 (d+8 e x)+5 a b^2 e \left (d^2+8 d e x+28 e^2 x^2\right )+b^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+c e^2 \left (5 a^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+6 a b e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 b^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+c^2 e \left (3 a e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{280 e^7 (d+e x)^8} \]

[In]

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

[Out]

-1/280*(5*c^3*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6)
+ e^3*(35*a^3*e^3 + 15*a^2*b*e^2*(d + 8*e*x) + 5*a*b^2*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + b^3*(d^3 + 8*d^2*e*x
+ 28*d*e^2*x^2 + 56*e^3*x^3)) + c*e^2*(5*a^2*e^2*(d^2 + 8*d*e*x + 28*e^2*x^2) + 6*a*b*e*(d^3 + 8*d^2*e*x + 28*
d*e^2*x^2 + 56*e^3*x^3) + 3*b^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4)) + c^2*e*(3*a*e
*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*b*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d
^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)))/(e^7*(d + e*x)^8)

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.62

method	result	size
risch	\(\frac {-\frac {c^{3} x^{6}}{2 e}-\frac {c^{2} \left (b e +c d \right ) x^{5}}{e^{2}}-\frac {c \left (3 a c \,e^{2}+3 b^{2} e^{2}+5 b c d e +5 c^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {\left (6 a b c \,e^{3}+3 c^{2} a d \,e^{2}+b^{3} e^{3}+3 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e +5 c^{3} d^{3}\right ) x^{3}}{5 e^{4}}-\frac {\left (5 e^{4} a^{2} c +5 a \,b^{2} e^{4}+6 a b c d \,e^{3}+3 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+3 b^{2} c \,d^{2} e^{2}+5 d^{3} e b \,c^{2}+5 d^{4} c^{3}\right ) x^{2}}{10 e^{5}}-\frac {\left (15 a^{2} b \,e^{5}+5 d \,e^{4} a^{2} c +5 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+3 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+3 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e +5 d^{5} c^{3}\right ) x}{35 e^{6}}-\frac {35 e^{6} a^{3}+15 a^{2} b d \,e^{5}+5 d^{2} e^{4} a^{2} c +5 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}+5 b \,c^{2} d^{5} e +5 c^{3} d^{6}}{280 e^{7}}}{\left (e x +d \right )^{8}}\)	\(436\)
norman	\(\frac {-\frac {c^{3} x^{6}}{2 e}-\frac {\left (e^{2} b \,c^{2}+d e \,c^{3}\right ) x^{5}}{e^{3}}-\frac {\left (3 a \,c^{2} e^{3}+3 b^{2} c \,e^{3}+5 d \,e^{2} b \,c^{2}+5 d^{2} e \,c^{3}\right ) x^{4}}{4 e^{4}}-\frac {\left (6 a b c \,e^{4}+3 d \,e^{3} c^{2} a +b^{3} e^{4}+3 b^{2} d \,e^{3} c +5 d^{2} e^{2} b \,c^{2}+5 d^{3} e \,c^{3}\right ) x^{3}}{5 e^{5}}-\frac {\left (5 a^{2} c \,e^{5}+5 a \,b^{2} e^{5}+6 a b c d \,e^{4}+3 e^{3} a \,c^{2} d^{2}+b^{3} d \,e^{4}+3 b^{2} c \,d^{2} e^{3}+5 d^{3} b \,c^{2} e^{2}+5 c^{3} d^{4} e \right ) x^{2}}{10 e^{6}}-\frac {\left (15 a^{2} b \,e^{6}+5 d \,e^{5} a^{2} c +5 a \,b^{2} d \,e^{5}+6 a b c \,d^{2} e^{4}+3 a \,d^{3} e^{3} c^{2}+b^{3} d^{2} e^{4}+3 b^{2} c \,d^{3} e^{3}+5 b \,c^{2} d^{4} e^{2}+5 c^{3} d^{5} e \right ) x}{35 e^{7}}-\frac {35 a^{3} e^{7}+15 a^{2} b d \,e^{6}+5 a^{2} c \,d^{2} e^{5}+5 a \,b^{2} d^{2} e^{5}+6 a b c \,d^{3} e^{4}+3 a \,c^{2} d^{4} e^{3}+b^{3} d^{3} e^{4}+3 b^{2} c \,d^{4} e^{3}+5 b \,c^{2} d^{5} e^{2}+5 c^{3} d^{6} e}{280 e^{8}}}{\left (e x +d \right )^{8}}\)	\(460\)
default	\(-\frac {6 a b c \,e^{3}-12 c^{2} a d \,e^{2}+b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}}{8 e^{7} \left (e x +d \right )^{8}}-\frac {c^{2} \left (b e -2 c d \right )}{e^{7} \left (e x +d \right )^{3}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{7 e^{7} \left (e x +d \right )^{7}}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {c^{3}}{2 e^{7} \left (e x +d \right )^{2}}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{6 e^{7} \left (e x +d \right )^{6}}\)	\(461\)
gosper	\(-\frac {140 x^{6} c^{3} e^{6}+280 x^{5} b \,c^{2} e^{6}+280 x^{5} c^{3} d \,e^{5}+210 a \,c^{2} e^{6} x^{4}+210 x^{4} b^{2} c \,e^{6}+350 x^{4} b \,c^{2} d \,e^{5}+350 c^{3} d^{2} e^{4} x^{4}+336 x^{3} a b c \,e^{6}+168 x^{3} a \,c^{2} d \,e^{5}+56 x^{3} b^{3} e^{6}+168 x^{3} b^{2} c d \,e^{5}+280 x^{3} b \,c^{2} d^{2} e^{4}+280 x^{3} c^{3} d^{3} e^{3}+140 x^{2} a^{2} c \,e^{6}+140 x^{2} a \,b^{2} e^{6}+168 x^{2} a b c d \,e^{5}+84 x^{2} a \,c^{2} d^{2} e^{4}+28 x^{2} b^{3} d \,e^{5}+84 x^{2} b^{2} c \,d^{2} e^{4}+140 x^{2} b \,c^{2} d^{3} e^{3}+140 x^{2} c^{3} d^{4} e^{2}+120 x \,a^{2} b \,e^{6}+40 x \,a^{2} c d \,e^{5}+40 x a \,b^{2} d \,e^{5}+48 x a b c \,d^{2} e^{4}+24 x a \,c^{2} d^{3} e^{3}+8 x \,b^{3} d^{2} e^{4}+24 x \,b^{2} c \,d^{3} e^{3}+40 x b \,c^{2} d^{4} e^{2}+40 x \,c^{3} d^{5} e +35 e^{6} a^{3}+15 a^{2} b d \,e^{5}+5 d^{2} e^{4} a^{2} c +5 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}+5 b \,c^{2} d^{5} e +5 c^{3} d^{6}}{280 e^{7} \left (e x +d \right )^{8}}\)	\(494\)
parallelrisch	\(\frac {-140 c^{3} x^{6} e^{7}-280 b \,c^{2} e^{7} x^{5}-280 c^{3} d \,e^{6} x^{5}-210 a \,c^{2} e^{7} x^{4}-210 b^{2} c \,e^{7} x^{4}-350 b \,c^{2} d \,e^{6} x^{4}-350 c^{3} d^{2} e^{5} x^{4}-336 a b c \,e^{7} x^{3}-168 a \,c^{2} d \,e^{6} x^{3}-56 b^{3} e^{7} x^{3}-168 b^{2} c d \,e^{6} x^{3}-280 b \,c^{2} d^{2} e^{5} x^{3}-280 c^{3} d^{3} e^{4} x^{3}-140 a^{2} c \,e^{7} x^{2}-140 a \,b^{2} e^{7} x^{2}-168 a b c d \,e^{6} x^{2}-84 a \,c^{2} d^{2} e^{5} x^{2}-28 b^{3} d \,e^{6} x^{2}-84 b^{2} c \,d^{2} e^{5} x^{2}-140 b \,c^{2} d^{3} e^{4} x^{2}-140 c^{3} d^{4} e^{3} x^{2}-120 a^{2} b \,e^{7} x -40 a^{2} c d \,e^{6} x -40 a \,b^{2} d \,e^{6} x -48 a b c \,d^{2} e^{5} x -24 a \,c^{2} d^{3} e^{4} x -8 b^{3} d^{2} e^{5} x -24 b^{2} c \,d^{3} e^{4} x -40 b \,c^{2} d^{4} e^{3} x -40 c^{3} d^{5} e^{2} x -35 a^{3} e^{7}-15 a^{2} b d \,e^{6}-5 a^{2} c \,d^{2} e^{5}-5 a \,b^{2} d^{2} e^{5}-6 a b c \,d^{3} e^{4}-3 a \,c^{2} d^{4} e^{3}-b^{3} d^{3} e^{4}-3 b^{2} c \,d^{4} e^{3}-5 b \,c^{2} d^{5} e^{2}-5 c^{3} d^{6} e}{280 e^{8} \left (e x +d \right )^{8}}\)	\(500\)

[In]

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

[Out]

(-1/2*c^3*x^6/e-c^2/e^2*(b*e+c*d)*x^5-1/4*c/e^3*(3*a*c*e^2+3*b^2*e^2+5*b*c*d*e+5*c^2*d^2)*x^4-1/5/e^4*(6*a*b*c
*e^3+3*a*c^2*d*e^2+b^3*e^3+3*b^2*c*d*e^2+5*b*c^2*d^2*e+5*c^3*d^3)*x^3-1/10/e^5*(5*a^2*c*e^4+5*a*b^2*e^4+6*a*b*
c*d*e^3+3*a*c^2*d^2*e^2+b^3*d*e^3+3*b^2*c*d^2*e^2+5*b*c^2*d^3*e+5*c^3*d^4)*x^2-1/35/e^6*(15*a^2*b*e^5+5*a^2*c*
d*e^4+5*a*b^2*d*e^4+6*a*b*c*d^2*e^3+3*a*c^2*d^3*e^2+b^3*d^2*e^3+3*b^2*c*d^3*e^2+5*b*c^2*d^4*e+5*c^3*d^5)*x-1/2
80/e^7*(35*a^3*e^6+15*a^2*b*d*e^5+5*a^2*c*d^2*e^4+5*a*b^2*d^2*e^4+6*a*b*c*d^3*e^3+3*a*c^2*d^4*e^2+b^3*d^3*e^3+
3*b^2*c*d^4*e^2+5*b*c^2*d^5*e+5*c^3*d^6))/(e*x+d)^8

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 15 \, a^{2} b d e^{5} + 35 \, a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 5 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 280 \, {\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 56 \, {\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + 3 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 28 \, {\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 5 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 8 \, {\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 15 \, a^{2} b e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 5 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{280 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]

[In]

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

[Out]

-1/280*(140*c^3*e^6*x^6 + 5*c^3*d^6 + 5*b*c^2*d^5*e + 15*a^2*b*d*e^5 + 35*a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2
+ (b^3 + 6*a*b*c)*d^3*e^3 + 5*(a*b^2 + a^2*c)*d^2*e^4 + 280*(c^3*d*e^5 + b*c^2*e^6)*x^5 + 70*(5*c^3*d^2*e^4 +
5*b*c^2*d*e^5 + 3*(b^2*c + a*c^2)*e^6)*x^4 + 56*(5*c^3*d^3*e^3 + 5*b*c^2*d^2*e^4 + 3*(b^2*c + a*c^2)*d*e^5 + (
b^3 + 6*a*b*c)*e^6)*x^3 + 28*(5*c^3*d^4*e^2 + 5*b*c^2*d^3*e^3 + 3*(b^2*c + a*c^2)*d^2*e^4 + (b^3 + 6*a*b*c)*d*
e^5 + 5*(a*b^2 + a^2*c)*e^6)*x^2 + 8*(5*c^3*d^5*e + 5*b*c^2*d^4*e^2 + 15*a^2*b*e^6 + 3*(b^2*c + a*c^2)*d^3*e^3
+ (b^3 + 6*a*b*c)*d^2*e^4 + 5*(a*b^2 + a^2*c)*d*e^5)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e
^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 15 \, a^{2} b d e^{5} + 35 \, a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 5 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 280 \, {\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 56 \, {\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + 3 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 28 \, {\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 5 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 8 \, {\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 15 \, a^{2} b e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 5 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{280 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]

[In]

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

[Out]

-1/280*(140*c^3*e^6*x^6 + 5*c^3*d^6 + 5*b*c^2*d^5*e + 15*a^2*b*d*e^5 + 35*a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2
+ (b^3 + 6*a*b*c)*d^3*e^3 + 5*(a*b^2 + a^2*c)*d^2*e^4 + 280*(c^3*d*e^5 + b*c^2*e^6)*x^5 + 70*(5*c^3*d^2*e^4 +
5*b*c^2*d*e^5 + 3*(b^2*c + a*c^2)*e^6)*x^4 + 56*(5*c^3*d^3*e^3 + 5*b*c^2*d^2*e^4 + 3*(b^2*c + a*c^2)*d*e^5 + (
b^3 + 6*a*b*c)*e^6)*x^3 + 28*(5*c^3*d^4*e^2 + 5*b*c^2*d^3*e^3 + 3*(b^2*c + a*c^2)*d^2*e^4 + (b^3 + 6*a*b*c)*d*
e^5 + 5*(a*b^2 + a^2*c)*e^6)*x^2 + 8*(5*c^3*d^5*e + 5*b*c^2*d^4*e^2 + 15*a^2*b*e^6 + 3*(b^2*c + a*c^2)*d^3*e^3
+ (b^3 + 6*a*b*c)*d^2*e^4 + 5*(a*b^2 + a^2*c)*d*e^5)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e
^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 280 \, c^{3} d e^{5} x^{5} + 280 \, b c^{2} e^{6} x^{5} + 350 \, c^{3} d^{2} e^{4} x^{4} + 350 \, b c^{2} d e^{5} x^{4} + 210 \, b^{2} c e^{6} x^{4} + 210 \, a c^{2} e^{6} x^{4} + 280 \, c^{3} d^{3} e^{3} x^{3} + 280 \, b c^{2} d^{2} e^{4} x^{3} + 168 \, b^{2} c d e^{5} x^{3} + 168 \, a c^{2} d e^{5} x^{3} + 56 \, b^{3} e^{6} x^{3} + 336 \, a b c e^{6} x^{3} + 140 \, c^{3} d^{4} e^{2} x^{2} + 140 \, b c^{2} d^{3} e^{3} x^{2} + 84 \, b^{2} c d^{2} e^{4} x^{2} + 84 \, a c^{2} d^{2} e^{4} x^{2} + 28 \, b^{3} d e^{5} x^{2} + 168 \, a b c d e^{5} x^{2} + 140 \, a b^{2} e^{6} x^{2} + 140 \, a^{2} c e^{6} x^{2} + 40 \, c^{3} d^{5} e x + 40 \, b c^{2} d^{4} e^{2} x + 24 \, b^{2} c d^{3} e^{3} x + 24 \, a c^{2} d^{3} e^{3} x + 8 \, b^{3} d^{2} e^{4} x + 48 \, a b c d^{2} e^{4} x + 40 \, a b^{2} d e^{5} x + 40 \, a^{2} c d e^{5} x + 120 \, a^{2} b e^{6} x + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 5 \, a b^{2} d^{2} e^{4} + 5 \, a^{2} c d^{2} e^{4} + 15 \, a^{2} b d e^{5} + 35 \, a^{3} e^{6}}{280 \, {\left (e x + d\right )}^{8} e^{7}} \]

[In]

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

[Out]

-1/280*(140*c^3*e^6*x^6 + 280*c^3*d*e^5*x^5 + 280*b*c^2*e^6*x^5 + 350*c^3*d^2*e^4*x^4 + 350*b*c^2*d*e^5*x^4 +
210*b^2*c*e^6*x^4 + 210*a*c^2*e^6*x^4 + 280*c^3*d^3*e^3*x^3 + 280*b*c^2*d^2*e^4*x^3 + 168*b^2*c*d*e^5*x^3 + 16
8*a*c^2*d*e^5*x^3 + 56*b^3*e^6*x^3 + 336*a*b*c*e^6*x^3 + 140*c^3*d^4*e^2*x^2 + 140*b*c^2*d^3*e^3*x^2 + 84*b^2*
c*d^2*e^4*x^2 + 84*a*c^2*d^2*e^4*x^2 + 28*b^3*d*e^5*x^2 + 168*a*b*c*d*e^5*x^2 + 140*a*b^2*e^6*x^2 + 140*a^2*c*
e^6*x^2 + 40*c^3*d^5*e*x + 40*b*c^2*d^4*e^2*x + 24*b^2*c*d^3*e^3*x + 24*a*c^2*d^3*e^3*x + 8*b^3*d^2*e^4*x + 48
*a*b*c*d^2*e^4*x + 40*a*b^2*d*e^5*x + 40*a^2*c*d*e^5*x + 120*a^2*b*e^6*x + 5*c^3*d^6 + 5*b*c^2*d^5*e + 3*b^2*c
*d^4*e^2 + 3*a*c^2*d^4*e^2 + b^3*d^3*e^3 + 6*a*b*c*d^3*e^3 + 5*a*b^2*d^2*e^4 + 5*a^2*c*d^2*e^4 + 15*a^2*b*d*e^
5 + 35*a^3*e^6)/((e*x + d)^8*e^7)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx=-\frac {\frac {35\,a^3\,e^6+15\,a^2\,b\,d\,e^5+5\,a^2\,c\,d^2\,e^4+5\,a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3+3\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2+5\,b\,c^2\,d^5\,e+5\,c^3\,d^6}{280\,e^7}+\frac {x^3\,\left (b^3\,e^3+3\,b^2\,c\,d\,e^2+5\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3+5\,c^3\,d^3+3\,a\,c^2\,d\,e^2\right )}{5\,e^4}+\frac {x^2\,\left (5\,a^2\,c\,e^4+5\,a\,b^2\,e^4+6\,a\,b\,c\,d\,e^3+3\,a\,c^2\,d^2\,e^2+b^3\,d\,e^3+3\,b^2\,c\,d^2\,e^2+5\,b\,c^2\,d^3\,e+5\,c^3\,d^4\right )}{10\,e^5}+\frac {c^3\,x^6}{2\,e}+\frac {x\,\left (15\,a^2\,b\,e^5+5\,a^2\,c\,d\,e^4+5\,a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3+3\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3+3\,b^2\,c\,d^3\,e^2+5\,b\,c^2\,d^4\,e+5\,c^3\,d^5\right )}{35\,e^6}+\frac {c\,x^4\,\left (3\,b^2\,e^2+5\,b\,c\,d\,e+5\,c^2\,d^2+3\,a\,c\,e^2\right )}{4\,e^3}+\frac {c^2\,x^5\,\left (b\,e+c\,d\right )}{e^2}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]

[In]

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

[Out]

-((35*a^3*e^6 + 5*c^3*d^6 + b^3*d^3*e^3 + 5*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 5*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^
2 + 15*a^2*b*d*e^5 + 5*b*c^2*d^5*e + 6*a*b*c*d^3*e^3)/(280*e^7) + (x^3*(b^3*e^3 + 5*c^3*d^3 + 6*a*b*c*e^3 + 3*
a*c^2*d*e^2 + 5*b*c^2*d^2*e + 3*b^2*c*d*e^2))/(5*e^4) + (x^2*(5*c^3*d^4 + 5*a*b^2*e^4 + 5*a^2*c*e^4 + b^3*d*e^
3 + 3*a*c^2*d^2*e^2 + 3*b^2*c*d^2*e^2 + 5*b*c^2*d^3*e + 6*a*b*c*d*e^3))/(10*e^5) + (c^3*x^6)/(2*e) + (x*(5*c^3
*d^5 + 15*a^2*b*e^5 + b^3*d^2*e^3 + 3*a*c^2*d^3*e^2 + 3*b^2*c*d^3*e^2 + 5*a*b^2*d*e^4 + 5*a^2*c*d*e^4 + 5*b*c^
2*d^4*e + 6*a*b*c*d^2*e^3))/(35*e^6) + (c*x^4*(3*b^2*e^2 + 5*c^2*d^2 + 3*a*c*e^2 + 5*b*c*d*e))/(4*e^3) + (c^2*
x^5*(b*e + c*d))/e^2)/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^5*e^3*x^3 + 70*d^4*e^4*x^4 + 56*d^3
*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*e*x)