\(\int \frac {(a+b x+c x^2)^3}{(d+e x)^8} \, dx\) [2143]

Optimal result

Integrand size = 20, antiderivative size = 268 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^3}{7 e^7 (d+e x)^7}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{2 e^7 (d+e x)^6}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{4 e^7 (d+e x)^4}-\frac {c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {c^3}{e^7 (d+e x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 268, 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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{4 e^7 (d+e x)^4}-\frac {3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right )^3}{7 e^7 (d+e x)^7}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {c^3}{e^7 (d+e x)} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {20 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )+e^3 \left (20 a^3 e^3+10 a^2 b e^2 (d+7 e x)+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c e^2 \left (2 a^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 b^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+2 c^2 e \left (2 a e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 b \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{140 e^7 (d+e x)^7} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.25 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.63

method	result	size
risch	\(\frac {-\frac {c^{3} x^{6}}{e}-\frac {3 c^{2} \left (b e +2 c d \right ) x^{5}}{2 e^{2}}-\frac {c \left (2 a c \,e^{2}+2 b^{2} e^{2}+5 b c d e +10 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (6 a b c \,e^{3}+4 c^{2} a d \,e^{2}+b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x^{3}}{4 e^{4}}-\frac {3 \left (4 e^{4} a^{2} c +4 a \,b^{2} e^{4}+6 a b c d \,e^{3}+4 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+4 b^{2} c \,d^{2} e^{2}+10 d^{3} e b \,c^{2}+20 d^{4} c^{3}\right ) x^{2}}{20 e^{5}}-\frac {\left (10 a^{2} b \,e^{5}+4 d \,e^{4} a^{2} c +4 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+4 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+4 b^{2} c \,d^{3} e^{2}+10 b \,c^{2} d^{4} e +20 d^{5} c^{3}\right ) x}{20 e^{6}}-\frac {20 e^{6} a^{3}+10 a^{2} b d \,e^{5}+4 d^{2} e^{4} a^{2} c +4 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+4 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+4 b^{2} c \,d^{4} e^{2}+10 b \,c^{2} d^{5} e +20 c^{3} d^{6}}{140 e^{7}}}{\left (e x +d \right )^{7}}\)	\(437\)
norman	\(\frac {-\frac {c^{3} x^{6}}{e}-\frac {3 \left (b \,c^{2} e +2 c^{3} d \right ) x^{5}}{2 e^{2}}-\frac {\left (2 a \,c^{2} e^{2}+2 b^{2} e^{2} c +5 d e b \,c^{2}+10 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (6 a b c \,e^{3}+4 c^{2} a d \,e^{2}+b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x^{3}}{4 e^{4}}-\frac {3 \left (4 e^{4} a^{2} c +4 a \,b^{2} e^{4}+6 a b c d \,e^{3}+4 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+4 b^{2} c \,d^{2} e^{2}+10 d^{3} e b \,c^{2}+20 d^{4} c^{3}\right ) x^{2}}{20 e^{5}}-\frac {\left (10 a^{2} b \,e^{5}+4 d \,e^{4} a^{2} c +4 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+4 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+4 b^{2} c \,d^{3} e^{2}+10 b \,c^{2} d^{4} e +20 d^{5} c^{3}\right ) x}{20 e^{6}}-\frac {20 e^{6} a^{3}+10 a^{2} b d \,e^{5}+4 d^{2} e^{4} a^{2} c +4 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+4 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+4 b^{2} c \,d^{4} e^{2}+10 b \,c^{2} d^{5} e +20 c^{3} d^{6}}{140 e^{7}}}{\left (e x +d \right )^{7}}\)	\(443\)
default	\(-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {c^{3}}{e^{7} \left (e x +d \right )}-\frac {c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{e^{7} \left (e x +d \right )^{3}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}}{7 e^{7} \left (e x +d \right )^{7}}-\frac {6 a b c \,e^{3}-12 c^{2} a d \,e^{2}+b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{4 e^{7} \left (e x +d \right )^{4}}-\frac {3 c^{2} \left (b e -2 c d \right )}{2 e^{7} \left (e x +d \right )^{2}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{6 e^{7} \left (e x +d \right )^{6}}\)	\(461\)
gosper	\(-\frac {140 x^{6} c^{3} e^{6}+210 x^{5} b \,c^{2} e^{6}+420 x^{5} c^{3} d \,e^{5}+140 a \,c^{2} e^{6} x^{4}+140 x^{4} b^{2} c \,e^{6}+350 x^{4} b \,c^{2} d \,e^{5}+700 c^{3} d^{2} e^{4} x^{4}+210 x^{3} a b c \,e^{6}+140 x^{3} a \,c^{2} d \,e^{5}+35 x^{3} b^{3} e^{6}+140 x^{3} b^{2} c d \,e^{5}+350 x^{3} b \,c^{2} d^{2} e^{4}+700 x^{3} c^{3} d^{3} e^{3}+84 x^{2} a^{2} c \,e^{6}+84 x^{2} a \,b^{2} e^{6}+126 x^{2} a b c d \,e^{5}+84 x^{2} a \,c^{2} d^{2} e^{4}+21 x^{2} b^{3} d \,e^{5}+84 x^{2} b^{2} c \,d^{2} e^{4}+210 x^{2} b \,c^{2} d^{3} e^{3}+420 x^{2} c^{3} d^{4} e^{2}+70 x \,a^{2} b \,e^{6}+28 x \,a^{2} c d \,e^{5}+28 x a \,b^{2} d \,e^{5}+42 x a b c \,d^{2} e^{4}+28 x a \,c^{2} d^{3} e^{3}+7 x \,b^{3} d^{2} e^{4}+28 x \,b^{2} c \,d^{3} e^{3}+70 x b \,c^{2} d^{4} e^{2}+140 x \,c^{3} d^{5} e +20 e^{6} a^{3}+10 a^{2} b d \,e^{5}+4 d^{2} e^{4} a^{2} c +4 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+4 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+4 b^{2} c \,d^{4} e^{2}+10 b \,c^{2} d^{5} e +20 c^{3} d^{6}}{140 \left (e x +d \right )^{7} e^{7}}\)	\(494\)
parallelrisch	\(\frac {-140 x^{6} c^{3} e^{6}-210 x^{5} b \,c^{2} e^{6}-420 x^{5} c^{3} d \,e^{5}-140 a \,c^{2} e^{6} x^{4}-140 x^{4} b^{2} c \,e^{6}-350 x^{4} b \,c^{2} d \,e^{5}-700 c^{3} d^{2} e^{4} x^{4}-210 x^{3} a b c \,e^{6}-140 x^{3} a \,c^{2} d \,e^{5}-35 x^{3} b^{3} e^{6}-140 x^{3} b^{2} c d \,e^{5}-350 x^{3} b \,c^{2} d^{2} e^{4}-700 x^{3} c^{3} d^{3} e^{3}-84 x^{2} a^{2} c \,e^{6}-84 x^{2} a \,b^{2} e^{6}-126 x^{2} a b c d \,e^{5}-84 x^{2} a \,c^{2} d^{2} e^{4}-21 x^{2} b^{3} d \,e^{5}-84 x^{2} b^{2} c \,d^{2} e^{4}-210 x^{2} b \,c^{2} d^{3} e^{3}-420 x^{2} c^{3} d^{4} e^{2}-70 x \,a^{2} b \,e^{6}-28 x \,a^{2} c d \,e^{5}-28 x a \,b^{2} d \,e^{5}-42 x a b c \,d^{2} e^{4}-28 x a \,c^{2} d^{3} e^{3}-7 x \,b^{3} d^{2} e^{4}-28 x \,b^{2} c \,d^{3} e^{3}-70 x b \,c^{2} d^{4} e^{2}-140 x \,c^{3} d^{5} e -20 e^{6} a^{3}-10 a^{2} b d \,e^{5}-4 d^{2} e^{4} a^{2} c -4 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}-4 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}-4 b^{2} c \,d^{4} e^{2}-10 b \,c^{2} d^{5} e -20 c^{3} d^{6}}{140 e^{7} \left (e x +d \right )^{7}}\)	\(495\)

[In]

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

[Out]

(-c^3*x^6/e-3/2*c^2*(b*e+2*c*d)/e^2*x^5-1/2*c*(2*a*c*e^2+2*b^2*e^2+5*b*c*d*e+10*c^2*d^2)/e^3*x^4-1/4*(6*a*b*c*
e^3+4*a*c^2*d*e^2+b^3*e^3+4*b^2*c*d*e^2+10*b*c^2*d^2*e+20*c^3*d^3)/e^4*x^3-3/20*(4*a^2*c*e^4+4*a*b^2*e^4+6*a*b
*c*d*e^3+4*a*c^2*d^2*e^2+b^3*d*e^3+4*b^2*c*d^2*e^2+10*b*c^2*d^3*e+20*c^3*d^4)/e^5*x^2-1/20*(10*a^2*b*e^5+4*a^2
*c*d*e^4+4*a*b^2*d*e^4+6*a*b*c*d^2*e^3+4*a*c^2*d^3*e^2+b^3*d^2*e^3+4*b^2*c*d^3*e^2+10*b*c^2*d^4*e+20*c^3*d^5)/
e^6*x-1/140*(20*a^3*e^6+10*a^2*b*d*e^5+4*a^2*c*d^2*e^4+4*a*b^2*d^2*e^4+6*a*b*c*d^3*e^3+4*a*c^2*d^4*e^2+b^3*d^3
*e^3+4*b^2*c*d^4*e^2+10*b*c^2*d^5*e+20*c^3*d^6)/e^7)/(e*x+d)^7

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 10 \, a^{2} b d e^{5} + 20 \, a^{3} e^{6} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 4 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 210 \, {\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 35 \, {\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 21 \, {\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 4 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 7 \, {\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 10 \, a^{2} b e^{6} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 4 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{140 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]

[In]

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

[Out]

-1/140*(140*c^3*e^6*x^6 + 20*c^3*d^6 + 10*b*c^2*d^5*e + 10*a^2*b*d*e^5 + 20*a^3*e^6 + 4*(b^2*c + a*c^2)*d^4*e^
2 + (b^3 + 6*a*b*c)*d^3*e^3 + 4*(a*b^2 + a^2*c)*d^2*e^4 + 210*(2*c^3*d*e^5 + b*c^2*e^6)*x^5 + 70*(10*c^3*d^2*e
^4 + 5*b*c^2*d*e^5 + 2*(b^2*c + a*c^2)*e^6)*x^4 + 35*(20*c^3*d^3*e^3 + 10*b*c^2*d^2*e^4 + 4*(b^2*c + a*c^2)*d*
e^5 + (b^3 + 6*a*b*c)*e^6)*x^3 + 21*(20*c^3*d^4*e^2 + 10*b*c^2*d^3*e^3 + 4*(b^2*c + a*c^2)*d^2*e^4 + (b^3 + 6*
a*b*c)*d*e^5 + 4*(a*b^2 + a^2*c)*e^6)*x^2 + 7*(20*c^3*d^5*e + 10*b*c^2*d^4*e^2 + 10*a^2*b*e^6 + 4*(b^2*c + a*c
^2)*d^3*e^3 + (b^3 + 6*a*b*c)*d^2*e^4 + 4*(a*b^2 + a^2*c)*d*e^5)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5
+ 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 10 \, a^{2} b d e^{5} + 20 \, a^{3} e^{6} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 4 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 210 \, {\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 35 \, {\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 21 \, {\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 4 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 7 \, {\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 10 \, a^{2} b e^{6} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 4 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{140 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]

[In]

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

[Out]

-1/140*(140*c^3*e^6*x^6 + 20*c^3*d^6 + 10*b*c^2*d^5*e + 10*a^2*b*d*e^5 + 20*a^3*e^6 + 4*(b^2*c + a*c^2)*d^4*e^
2 + (b^3 + 6*a*b*c)*d^3*e^3 + 4*(a*b^2 + a^2*c)*d^2*e^4 + 210*(2*c^3*d*e^5 + b*c^2*e^6)*x^5 + 70*(10*c^3*d^2*e
^4 + 5*b*c^2*d*e^5 + 2*(b^2*c + a*c^2)*e^6)*x^4 + 35*(20*c^3*d^3*e^3 + 10*b*c^2*d^2*e^4 + 4*(b^2*c + a*c^2)*d*
e^5 + (b^3 + 6*a*b*c)*e^6)*x^3 + 21*(20*c^3*d^4*e^2 + 10*b*c^2*d^3*e^3 + 4*(b^2*c + a*c^2)*d^2*e^4 + (b^3 + 6*
a*b*c)*d*e^5 + 4*(a*b^2 + a^2*c)*e^6)*x^2 + 7*(20*c^3*d^5*e + 10*b*c^2*d^4*e^2 + 10*a^2*b*e^6 + 4*(b^2*c + a*c
^2)*d^3*e^3 + (b^3 + 6*a*b*c)*d^2*e^4 + 4*(a*b^2 + a^2*c)*d*e^5)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5
+ 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 420 \, c^{3} d e^{5} x^{5} + 210 \, b c^{2} e^{6} x^{5} + 700 \, c^{3} d^{2} e^{4} x^{4} + 350 \, b c^{2} d e^{5} x^{4} + 140 \, b^{2} c e^{6} x^{4} + 140 \, a c^{2} e^{6} x^{4} + 700 \, c^{3} d^{3} e^{3} x^{3} + 350 \, b c^{2} d^{2} e^{4} x^{3} + 140 \, b^{2} c d e^{5} x^{3} + 140 \, a c^{2} d e^{5} x^{3} + 35 \, b^{3} e^{6} x^{3} + 210 \, a b c e^{6} x^{3} + 420 \, c^{3} d^{4} e^{2} x^{2} + 210 \, b c^{2} d^{3} e^{3} x^{2} + 84 \, b^{2} c d^{2} e^{4} x^{2} + 84 \, a c^{2} d^{2} e^{4} x^{2} + 21 \, b^{3} d e^{5} x^{2} + 126 \, a b c d e^{5} x^{2} + 84 \, a b^{2} e^{6} x^{2} + 84 \, a^{2} c e^{6} x^{2} + 140 \, c^{3} d^{5} e x + 70 \, b c^{2} d^{4} e^{2} x + 28 \, b^{2} c d^{3} e^{3} x + 28 \, a c^{2} d^{3} e^{3} x + 7 \, b^{3} d^{2} e^{4} x + 42 \, a b c d^{2} e^{4} x + 28 \, a b^{2} d e^{5} x + 28 \, a^{2} c d e^{5} x + 70 \, a^{2} b e^{6} x + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} + 4 \, a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 4 \, a b^{2} d^{2} e^{4} + 4 \, a^{2} c d^{2} e^{4} + 10 \, a^{2} b d e^{5} + 20 \, a^{3} e^{6}}{140 \, {\left (e x + d\right )}^{7} e^{7}} \]

[In]

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

[Out]

-1/140*(140*c^3*e^6*x^6 + 420*c^3*d*e^5*x^5 + 210*b*c^2*e^6*x^5 + 700*c^3*d^2*e^4*x^4 + 350*b*c^2*d*e^5*x^4 +
140*b^2*c*e^6*x^4 + 140*a*c^2*e^6*x^4 + 700*c^3*d^3*e^3*x^3 + 350*b*c^2*d^2*e^4*x^3 + 140*b^2*c*d*e^5*x^3 + 14
0*a*c^2*d*e^5*x^3 + 35*b^3*e^6*x^3 + 210*a*b*c*e^6*x^3 + 420*c^3*d^4*e^2*x^2 + 210*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 + 21*b^3*d*e^5*x^2 + 126*a*b*c*d*e^5*x^2 + 84*a*b^2*e^6*x^2 + 84*a^2*c*e^
6*x^2 + 140*c^3*d^5*e*x + 70*b*c^2*d^4*e^2*x + 28*b^2*c*d^3*e^3*x + 28*a*c^2*d^3*e^3*x + 7*b^3*d^2*e^4*x + 42*
a*b*c*d^2*e^4*x + 28*a*b^2*d*e^5*x + 28*a^2*c*d*e^5*x + 70*a^2*b*e^6*x + 20*c^3*d^6 + 10*b*c^2*d^5*e + 4*b^2*c
*d^4*e^2 + 4*a*c^2*d^4*e^2 + b^3*d^3*e^3 + 6*a*b*c*d^3*e^3 + 4*a*b^2*d^2*e^4 + 4*a^2*c*d^2*e^4 + 10*a^2*b*d*e^
5 + 20*a^3*e^6)/((e*x + d)^7*e^7)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {\frac {20\,a^3\,e^6+10\,a^2\,b\,d\,e^5+4\,a^2\,c\,d^2\,e^4+4\,a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3+4\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3+4\,b^2\,c\,d^4\,e^2+10\,b\,c^2\,d^5\,e+20\,c^3\,d^6}{140\,e^7}+\frac {x^3\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3+20\,c^3\,d^3+4\,a\,c^2\,d\,e^2\right )}{4\,e^4}+\frac {3\,x^2\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4+6\,a\,b\,c\,d\,e^3+4\,a\,c^2\,d^2\,e^2+b^3\,d\,e^3+4\,b^2\,c\,d^2\,e^2+10\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{20\,e^5}+\frac {c^3\,x^6}{e}+\frac {x\,\left (10\,a^2\,b\,e^5+4\,a^2\,c\,d\,e^4+4\,a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3+4\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3+4\,b^2\,c\,d^3\,e^2+10\,b\,c^2\,d^4\,e+20\,c^3\,d^5\right )}{20\,e^6}+\frac {c\,x^4\,\left (2\,b^2\,e^2+5\,b\,c\,d\,e+10\,c^2\,d^2+2\,a\,c\,e^2\right )}{2\,e^3}+\frac {3\,c^2\,x^5\,\left (b\,e+2\,c\,d\right )}{2\,e^2}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]

[In]

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

[Out]

-((20*a^3*e^6 + 20*c^3*d^6 + b^3*d^3*e^3 + 4*a*b^2*d^2*e^4 + 4*a*c^2*d^4*e^2 + 4*a^2*c*d^2*e^4 + 4*b^2*c*d^4*e
^2 + 10*a^2*b*d*e^5 + 10*b*c^2*d^5*e + 6*a*b*c*d^3*e^3)/(140*e^7) + (x^3*(b^3*e^3 + 20*c^3*d^3 + 6*a*b*c*e^3 +
4*a*c^2*d*e^2 + 10*b*c^2*d^2*e + 4*b^2*c*d*e^2))/(4*e^4) + (3*x^2*(20*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 + b
^3*d*e^3 + 4*a*c^2*d^2*e^2 + 4*b^2*c*d^2*e^2 + 10*b*c^2*d^3*e + 6*a*b*c*d*e^3))/(20*e^5) + (c^3*x^6)/e + (x*(2
0*c^3*d^5 + 10*a^2*b*e^5 + b^3*d^2*e^3 + 4*a*c^2*d^3*e^2 + 4*b^2*c*d^3*e^2 + 4*a*b^2*d*e^4 + 4*a^2*c*d*e^4 + 1
0*b*c^2*d^4*e + 6*a*b*c*d^2*e^3))/(20*e^6) + (c*x^4*(2*b^2*e^2 + 10*c^2*d^2 + 2*a*c*e^2 + 5*b*c*d*e))/(2*e^3)
+ (3*c^2*x^5*(b*e + 2*c*d))/(2*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)