\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^9} \, dx\) [2159]

Optimal result

Integrand size = 20, antiderivative size = 435 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^4}{8 e^9 (d+e x)^8}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{7 e^9 (d+e x)^7}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{3 e^9 (d+e x)^6}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{5 e^9 (d+e x)^5}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{4 e^9 (d+e x)^4}+\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (d+e x)^3}-\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)^2}+\frac {4 c^3 (2 c d-b e)}{e^9 (d+e x)}+\frac {c^4 \log (d+e x)}{e^9} \]

[Out]

-1/8*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x+d)^8+4/7*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^7-1/3*(a*e^2-b*d*e
+c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))/e^9/(e*x+d)^6+4/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*
d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))/e^9/(e*x+d)^5+1/4*(-70*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(-3*a*e+5*b*d)+20*c^3*d^2*e
*(-3*a*e+7*b*d)-6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))/e^9/(e*x+d)^4+4/3*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2
-c*e*(-3*a*e+7*b*d))/e^9/(e*x+d)^3-c^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d))/e^9/(e*x+d)^2+4*c^3*(-b*e+2*c
*d)/e^9/(e*x+d)+c^4*ln(e*x+d)/e^9

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 435, 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} \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=-\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{4 e^9 (d+e x)^4}-\frac {c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)^2}+\frac {4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^3}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^6}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{7 e^9 (d+e x)^7}-\frac {\left (a e^2-b d e+c d^2\right )^4}{8 e^9 (d+e x)^8}+\frac {4 c^3 (2 c d-b e)}{e^9 (d+e x)}+\frac {c^4 \log (d+e x)}{e^9} \]

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^9,x]

[Out]

-1/8*(c*d^2 - b*d*e + a*e^2)^4/(e^9*(d + e*x)^8) + (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(7*e^9*(d + e*x
)^7) - ((c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(3*e^9*(d + e*x)^6) + (4*(2*
c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(5*e^9*(d + e*x)^5) - (70*c^4*
d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*
e + a^2*e^2))/(4*e^9*(d + e*x)^4) + (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(3*e^9*(d
+ e*x)^3) - (c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(e^9*(d + e*x)^2) + (4*c^3*(2*c*d - b*e))/(e^
9*(d + e*x)) + (c^4*Log[d + e*x])/e^9

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 )^4}{e^8 (d+e x)^9}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^8 (d+e x)^7}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right )}{e^8 (d+e x)^6}+\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{e^8 (d+e x)^5}+\frac {4 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right )}{e^8 (d+e x)^4}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^8 (d+e x)^3}-\frac {4 c^3 (2 c d-b e)}{e^8 (d+e x)^2}+\frac {c^4}{e^8 (d+e x)}\right ) \, dx \\ & = -\frac {\left (c d^2-b d e+a e^2\right )^4}{8 e^9 (d+e x)^8}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{7 e^9 (d+e x)^7}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{3 e^9 (d+e x)^6}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{5 e^9 (d+e x)^5}-\frac {70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )}{4 e^9 (d+e x)^4}+\frac {4 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )}{3 e^9 (d+e x)^3}-\frac {c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)^2}+\frac {4 c^3 (2 c d-b e)}{e^9 (d+e x)}+\frac {c^4 \log (d+e x)}{e^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 740, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=\frac {c^4 d \left (2283 d^7+17424 d^6 e x+57624 d^5 e^2 x^2+107408 d^4 e^3 x^3+122500 d^3 e^4 x^4+86240 d^2 e^5 x^5+35280 d e^6 x^6+6720 e^7 x^7\right )-3 e^4 \left (35 a^4 e^4+20 a^3 b e^3 (d+8 e x)+10 a^2 b^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a b^3 e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b^4 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )-4 c e^3 \left (5 a^3 e^3 \left (d^2+8 d e x+28 e^2 x^2\right )+9 a^2 b e^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+9 a b^2 e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b^3 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )-6 c^2 e^2 \left (3 a^2 e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+10 a b e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+15 b^2 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )-60 c^3 e \left (a e \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+7 b \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )\right )+840 c^4 (d+e x)^8 \log (d+e x)}{840 e^9 (d+e x)^8} \]

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^9,x]

[Out]

(c^4*d*(2283*d^7 + 17424*d^6*e*x + 57624*d^5*e^2*x^2 + 107408*d^4*e^3*x^3 + 122500*d^3*e^4*x^4 + 86240*d^2*e^5
*x^5 + 35280*d*e^6*x^6 + 6720*e^7*x^7) - 3*e^4*(35*a^4*e^4 + 20*a^3*b*e^3*(d + 8*e*x) + 10*a^2*b^2*e^2*(d^2 +
8*d*e*x + 28*e^2*x^2) + 4*a*b^3*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + b^4*(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)) - 4*c*e^3*(5*a^3*e^3*(d^2 + 8*d*e*x + 28*e^2*x^2) + 9*a^2*b*e^2*(d^3 +
8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 9*a*b^2*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^3*(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)) - 6*c^2*e^2*(3*a
^2*e^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) + 10*a*b*e*(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) + 15*b^2*(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)) - 60*c^3*e*(a*e*(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) + 7*b*(d^7 + 8*d^6*e*x + 28*d^5*e^2*x^2 + 56*d^4*e^3*x^3 +
70*d^3*e^4*x^4 + 56*d^2*e^5*x^5 + 28*d*e^6*x^6 + 8*e^7*x^7)) + 840*c^4*(d + e*x)^8*Log[d + e*x])/(840*e^9*(d +
e*x)^8)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(888\) vs. \(2(424)=848\).

Time = 26.77 (sec) , antiderivative size = 889, normalized size of antiderivative = 2.04

method	result	size
risch	\(\frac {-\frac {4 c^{3} \left (b e -2 c d \right ) x^{7}}{e^{2}}-\frac {c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}+14 b c d e -42 c^{2} d^{2}\right ) x^{6}}{e^{3}}-\frac {2 c \left (6 a b c \,e^{3}+6 c^{2} a d \,e^{2}+2 b^{3} e^{3}+9 b^{2} d \,e^{2} c +42 b \,c^{2} d^{2} e -154 c^{3} d^{3}\right ) x^{5}}{3 e^{4}}-\frac {\left (18 c^{2} a^{2} e^{4}+36 a \,b^{2} c \,e^{4}+60 a b \,c^{2} d \,e^{3}+60 c^{3} a \,d^{2} e^{2}+3 b^{4} e^{4}+20 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}+420 b \,c^{3} d^{3} e -1750 c^{4} d^{4}\right ) x^{4}}{12 e^{5}}-\frac {\left (36 a^{2} b c \,e^{5}+18 d \,e^{4} a^{2} c^{2}+12 a \,b^{3} e^{5}+36 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}+60 d^{3} e^{2} c^{3} a +3 b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}+90 b^{2} c^{2} d^{3} e^{2}+420 b \,c^{3} d^{4} e -1918 c^{4} d^{5}\right ) x^{3}}{15 e^{6}}-\frac {\left (20 e^{6} c \,a^{3}+30 a^{2} b^{2} e^{6}+36 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}+12 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}+60 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} c^{3} a +3 b^{4} d^{2} e^{4}+20 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}+420 b \,c^{3} d^{5} e -2058 d^{6} c^{4}\right ) x^{2}}{30 e^{7}}-\frac {\left (60 a^{3} b \,e^{7}+20 d \,e^{6} c \,a^{3}+30 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}+18 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}+36 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}+60 d^{5} e^{2} c^{3} a +3 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}+90 b^{2} c^{2} d^{5} e^{2}+420 b \,c^{3} d^{6} e -2178 d^{7} c^{4}\right ) x}{105 e^{8}}-\frac {105 a^{4} e^{8}+60 a^{3} b d \,e^{7}+20 a^{3} c \,d^{2} e^{6}+30 a^{2} b^{2} d^{2} e^{6}+36 a^{2} b c \,d^{3} e^{5}+18 a^{2} c^{2} d^{4} e^{4}+12 a \,b^{3} d^{3} e^{5}+36 a \,b^{2} c \,d^{4} e^{4}+60 a b \,c^{2} d^{5} e^{3}+60 a \,c^{3} d^{6} e^{2}+3 b^{4} d^{4} e^{4}+20 b^{3} c \,d^{5} e^{3}+90 b^{2} c^{2} d^{6} e^{2}+420 b \,c^{3} d^{7} e -2283 c^{4} d^{8}}{840 e^{9}}}{\left (e x +d \right )^{8}}+\frac {c^{4} \ln \left (e x +d \right )}{e^{9}}\)	\(889\)
norman	\(\frac {-\frac {105 a^{4} e^{8}+60 a^{3} b d \,e^{7}+20 a^{3} c \,d^{2} e^{6}+30 a^{2} b^{2} d^{2} e^{6}+36 a^{2} b c \,d^{3} e^{5}+18 a^{2} c^{2} d^{4} e^{4}+12 a \,b^{3} d^{3} e^{5}+36 a \,b^{2} c \,d^{4} e^{4}+60 a b \,c^{2} d^{5} e^{3}+60 a \,c^{3} d^{6} e^{2}+3 b^{4} d^{4} e^{4}+20 b^{3} c \,d^{5} e^{3}+90 b^{2} c^{2} d^{6} e^{2}+420 b \,c^{3} d^{7} e -2283 c^{4} d^{8}}{840 e^{9}}-\frac {4 \left (b \,c^{3} e -2 c^{4} d \right ) x^{7}}{e^{2}}-\frac {\left (2 a \,c^{3} e^{2}+3 b^{2} c^{2} e^{2}+14 d e \,c^{3} b -42 c^{4} d^{2}\right ) x^{6}}{e^{3}}-\frac {2 \left (6 a b \,c^{2} e^{3}+6 a \,c^{3} d \,e^{2}+2 b^{3} c \,e^{3}+9 b^{2} c^{2} d \,e^{2}+42 b \,c^{3} d^{2} e -154 d^{3} c^{4}\right ) x^{5}}{3 e^{4}}-\frac {\left (18 c^{2} a^{2} e^{4}+36 a \,b^{2} c \,e^{4}+60 a b \,c^{2} d \,e^{3}+60 c^{3} a \,d^{2} e^{2}+3 b^{4} e^{4}+20 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}+420 b \,c^{3} d^{3} e -1750 c^{4} d^{4}\right ) x^{4}}{12 e^{5}}-\frac {\left (36 a^{2} b c \,e^{5}+18 d \,e^{4} a^{2} c^{2}+12 a \,b^{3} e^{5}+36 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}+60 d^{3} e^{2} c^{3} a +3 b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}+90 b^{2} c^{2} d^{3} e^{2}+420 b \,c^{3} d^{4} e -1918 c^{4} d^{5}\right ) x^{3}}{15 e^{6}}-\frac {\left (20 e^{6} c \,a^{3}+30 a^{2} b^{2} e^{6}+36 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}+12 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}+60 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} c^{3} a +3 b^{4} d^{2} e^{4}+20 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}+420 b \,c^{3} d^{5} e -2058 d^{6} c^{4}\right ) x^{2}}{30 e^{7}}-\frac {\left (60 a^{3} b \,e^{7}+20 d \,e^{6} c \,a^{3}+30 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}+18 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}+36 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}+60 d^{5} e^{2} c^{3} a +3 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}+90 b^{2} c^{2} d^{5} e^{2}+420 b \,c^{3} d^{6} e -2178 d^{7} c^{4}\right ) x}{105 e^{8}}}{\left (e x +d \right )^{8}}+\frac {c^{4} \ln \left (e x +d \right )}{e^{9}}\)	\(899\)
default	\(-\frac {12 a^{2} b c \,e^{5}-24 d \,e^{4} a^{2} c^{2}+4 a \,b^{3} e^{5}-48 a \,b^{2} c d \,e^{4}+120 a b \,c^{2} d^{2} e^{3}-80 d^{3} e^{2} c^{3} a -4 b^{4} d \,e^{4}+40 b^{3} c \,d^{2} e^{3}-120 b^{2} c^{2} d^{3} e^{2}+140 b \,c^{3} d^{4} e -56 c^{4} d^{5}}{5 e^{9} \left (e x +d \right )^{5}}-\frac {a^{4} e^{8}-4 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-12 a^{2} b c \,d^{3} e^{5}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,b^{3} d^{3} e^{5}+12 a \,b^{2} c \,d^{4} e^{4}-12 a b \,c^{2} d^{5} e^{3}+4 a \,c^{3} d^{6} e^{2}+b^{4} d^{4} e^{4}-4 b^{3} c \,d^{5} e^{3}+6 b^{2} c^{2} d^{6} e^{2}-4 b \,c^{3} d^{7} e +c^{4} d^{8}}{8 e^{9} \left (e x +d \right )^{8}}-\frac {4 c^{3} \left (b e -2 c d \right )}{e^{9} \left (e x +d \right )}-\frac {4 c \left (3 a b c \,e^{3}-6 c^{2} a d \,e^{2}+b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right )}{3 e^{9} \left (e x +d \right )^{3}}-\frac {4 a^{3} b \,e^{7}-8 d \,e^{6} c \,a^{3}-12 a^{2} b^{2} d \,e^{6}+36 a^{2} b c \,d^{2} e^{5}-24 d^{3} e^{4} a^{2} c^{2}+12 a \,b^{3} d^{2} e^{5}-48 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}-24 d^{5} e^{2} c^{3} a -4 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}-36 b^{2} c^{2} d^{5} e^{2}+28 b \,c^{3} d^{6} e -8 d^{7} c^{4}}{7 e^{9} \left (e x +d \right )^{7}}-\frac {6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+60 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}-20 b^{3} c d \,e^{3}+90 b^{2} c^{2} d^{2} e^{2}-140 b \,c^{3} d^{3} e +70 c^{4} d^{4}}{4 e^{9} \left (e x +d \right )^{4}}-\frac {c^{2} \left (2 a c \,e^{2}+3 b^{2} e^{2}-14 b c d e +14 c^{2} d^{2}\right )}{e^{9} \left (e x +d \right )^{2}}+\frac {c^{4} \ln \left (e x +d \right )}{e^{9}}-\frac {4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}-36 a^{2} b c d \,e^{5}+36 d^{2} e^{4} a^{2} c^{2}-12 a \,b^{3} d \,e^{5}+72 a \,b^{2} c \,d^{2} e^{4}-120 a b \,c^{2} d^{3} e^{3}+60 d^{4} e^{2} c^{3} a +6 b^{4} d^{2} e^{4}-40 b^{3} c \,d^{3} e^{3}+90 b^{2} c^{2} d^{4} e^{2}-84 b \,c^{3} d^{5} e +28 d^{6} c^{4}}{6 e^{9} \left (e x +d \right )^{6}}\)	\(912\)
parallelrisch	\(\text {Expression too large to display}\)	\(1170\)

[In]

int((c*x^2+b*x+a)^4/(e*x+d)^9,x,method=_RETURNVERBOSE)

[Out]

(-4*c^3*(b*e-2*c*d)/e^2*x^7-c^2*(2*a*c*e^2+3*b^2*e^2+14*b*c*d*e-42*c^2*d^2)/e^3*x^6-2/3*c*(6*a*b*c*e^3+6*a*c^2
*d*e^2+2*b^3*e^3+9*b^2*c*d*e^2+42*b*c^2*d^2*e-154*c^3*d^3)/e^4*x^5-1/12*(18*a^2*c^2*e^4+36*a*b^2*c*e^4+60*a*b*
c^2*d*e^3+60*a*c^3*d^2*e^2+3*b^4*e^4+20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2+420*b*c^3*d^3*e-1750*c^4*d^4)/e^5*x^4-1
/15*(36*a^2*b*c*e^5+18*a^2*c^2*d*e^4+12*a*b^3*e^5+36*a*b^2*c*d*e^4+60*a*b*c^2*d^2*e^3+60*a*c^3*d^3*e^2+3*b^4*d
*e^4+20*b^3*c*d^2*e^3+90*b^2*c^2*d^3*e^2+420*b*c^3*d^4*e-1918*c^4*d^5)/e^6*x^3-1/30*(20*a^3*c*e^6+30*a^2*b^2*e
^6+36*a^2*b*c*d*e^5+18*a^2*c^2*d^2*e^4+12*a*b^3*d*e^5+36*a*b^2*c*d^2*e^4+60*a*b*c^2*d^3*e^3+60*a*c^3*d^4*e^2+3
*b^4*d^2*e^4+20*b^3*c*d^3*e^3+90*b^2*c^2*d^4*e^2+420*b*c^3*d^5*e-2058*c^4*d^6)/e^7*x^2-1/105*(60*a^3*b*e^7+20*
a^3*c*d*e^6+30*a^2*b^2*d*e^6+36*a^2*b*c*d^2*e^5+18*a^2*c^2*d^3*e^4+12*a*b^3*d^2*e^5+36*a*b^2*c*d^3*e^4+60*a*b*
c^2*d^4*e^3+60*a*c^3*d^5*e^2+3*b^4*d^3*e^4+20*b^3*c*d^4*e^3+90*b^2*c^2*d^5*e^2+420*b*c^3*d^6*e-2178*c^4*d^7)/e
^8*x-1/840*(105*a^4*e^8+60*a^3*b*d*e^7+20*a^3*c*d^2*e^6+30*a^2*b^2*d^2*e^6+36*a^2*b*c*d^3*e^5+18*a^2*c^2*d^4*e
^4+12*a*b^3*d^3*e^5+36*a*b^2*c*d^4*e^4+60*a*b*c^2*d^5*e^3+60*a*c^3*d^6*e^2+3*b^4*d^4*e^4+20*b^3*c*d^5*e^3+90*b
^2*c^2*d^6*e^2+420*b*c^3*d^7*e-2283*c^4*d^8)/e^9)/(e*x+d)^8+c^4*ln(e*x+d)/e^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (423) = 846\).

Time = 0.29 (sec) , antiderivative size = 998, normalized size of antiderivative = 2.29 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=\frac {2283 \, c^{4} d^{8} - 420 \, b c^{3} d^{7} e - 60 \, a^{3} b d e^{7} - 105 \, a^{4} e^{8} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} - 10 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 3360 \, {\left (2 \, c^{4} d e^{7} - b c^{3} e^{8}\right )} x^{7} + 840 \, {\left (42 \, c^{4} d^{2} e^{6} - 14 \, b c^{3} d e^{7} - {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 560 \, {\left (154 \, c^{4} d^{3} e^{5} - 42 \, b c^{3} d^{2} e^{6} - 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} - 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 70 \, {\left (1750 \, c^{4} d^{4} e^{4} - 420 \, b c^{3} d^{3} e^{5} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 56 \, {\left (1918 \, c^{4} d^{5} e^{3} - 420 \, b c^{3} d^{4} e^{4} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} - 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 28 \, {\left (2058 \, c^{4} d^{6} e^{2} - 420 \, b c^{3} d^{5} e^{3} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} - 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} - 10 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 8 \, {\left (2178 \, c^{4} d^{7} e - 420 \, b c^{3} d^{6} e^{2} - 60 \, a^{3} b e^{8} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} - 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} - 10 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x + 840 \, {\left (c^{4} e^{8} x^{8} + 8 \, c^{4} d e^{7} x^{7} + 28 \, c^{4} d^{2} e^{6} x^{6} + 56 \, c^{4} d^{3} e^{5} x^{5} + 70 \, c^{4} d^{4} e^{4} x^{4} + 56 \, c^{4} d^{5} e^{3} x^{3} + 28 \, c^{4} d^{6} e^{2} x^{2} + 8 \, c^{4} d^{7} e x + c^{4} d^{8}\right )} \log \left (e x + d\right )}{840 \, {\left (e^{17} x^{8} + 8 \, d e^{16} x^{7} + 28 \, d^{2} e^{15} x^{6} + 56 \, d^{3} e^{14} x^{5} + 70 \, d^{4} e^{13} x^{4} + 56 \, d^{5} e^{12} x^{3} + 28 \, d^{6} e^{11} x^{2} + 8 \, d^{7} e^{10} x + d^{8} e^{9}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^9,x, algorithm="fricas")

[Out]

1/840*(2283*c^4*d^8 - 420*b*c^3*d^7*e - 60*a^3*b*d*e^7 - 105*a^4*e^8 - 30*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 20*(
b^3*c + 3*a*b*c^2)*d^5*e^3 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 12*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 10*(3
*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 3360*(2*c^4*d*e^7 - b*c^3*e^8)*x^7 + 840*(42*c^4*d^2*e^6 - 14*b*c^3*d*e^7 - (3*b
^2*c^2 + 2*a*c^3)*e^8)*x^6 + 560*(154*c^4*d^3*e^5 - 42*b*c^3*d^2*e^6 - 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - 2*(b^3*
c + 3*a*b*c^2)*e^8)*x^5 + 70*(1750*c^4*d^4*e^4 - 420*b*c^3*d^3*e^5 - 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 20*(b^
3*c + 3*a*b*c^2)*d*e^7 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 56*(1918*c^4*d^5*e^3 - 420*b*c^3*d^4*e^4
- 30*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 20*(b^3*c + 3*a*b*c^2)*d^2*e^6 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 -
12*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 28*(2058*c^4*d^6*e^2 - 420*b*c^3*d^5*e^3 - 30*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4
- 20*(b^3*c + 3*a*b*c^2)*d^3*e^5 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 12*(a*b^3 + 3*a^2*b*c)*d*e^7 -
10*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 8*(2178*c^4*d^7*e - 420*b*c^3*d^6*e^2 - 60*a^3*b*e^8 - 30*(3*b^2*c^2 + 2*a
*c^3)*d^5*e^3 - 20*(b^3*c + 3*a*b*c^2)*d^4*e^4 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 12*(a*b^3 + 3*a^2*
b*c)*d^2*e^6 - 10*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x + 840*(c^4*e^8*x^8 + 8*c^4*d*e^7*x^7 + 28*c^4*d^2*e^6*x^6 + 5
6*c^4*d^3*e^5*x^5 + 70*c^4*d^4*e^4*x^4 + 56*c^4*d^5*e^3*x^3 + 28*c^4*d^6*e^2*x^2 + 8*c^4*d^7*e*x + c^4*d^8)*lo
g(e*x + d))/(e^17*x^8 + 8*d*e^16*x^7 + 28*d^2*e^15*x^6 + 56*d^3*e^14*x^5 + 70*d^4*e^13*x^4 + 56*d^5*e^12*x^3 +
28*d^6*e^11*x^2 + 8*d^7*e^10*x + d^8*e^9)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)**4/(e*x+d)**9,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (423) = 846\).

Time = 0.24 (sec) , antiderivative size = 894, normalized size of antiderivative = 2.06 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=\frac {2283 \, c^{4} d^{8} - 420 \, b c^{3} d^{7} e - 60 \, a^{3} b d e^{7} - 105 \, a^{4} e^{8} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} - 10 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 3360 \, {\left (2 \, c^{4} d e^{7} - b c^{3} e^{8}\right )} x^{7} + 840 \, {\left (42 \, c^{4} d^{2} e^{6} - 14 \, b c^{3} d e^{7} - {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 560 \, {\left (154 \, c^{4} d^{3} e^{5} - 42 \, b c^{3} d^{2} e^{6} - 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} - 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 70 \, {\left (1750 \, c^{4} d^{4} e^{4} - 420 \, b c^{3} d^{3} e^{5} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 56 \, {\left (1918 \, c^{4} d^{5} e^{3} - 420 \, b c^{3} d^{4} e^{4} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} - 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 28 \, {\left (2058 \, c^{4} d^{6} e^{2} - 420 \, b c^{3} d^{5} e^{3} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} - 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} - 10 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 8 \, {\left (2178 \, c^{4} d^{7} e - 420 \, b c^{3} d^{6} e^{2} - 60 \, a^{3} b e^{8} - 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} - 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} - 10 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{840 \, {\left (e^{17} x^{8} + 8 \, d e^{16} x^{7} + 28 \, d^{2} e^{15} x^{6} + 56 \, d^{3} e^{14} x^{5} + 70 \, d^{4} e^{13} x^{4} + 56 \, d^{5} e^{12} x^{3} + 28 \, d^{6} e^{11} x^{2} + 8 \, d^{7} e^{10} x + d^{8} e^{9}\right )}} + \frac {c^{4} \log \left (e x + d\right )}{e^{9}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^9,x, algorithm="maxima")

[Out]

1/840*(2283*c^4*d^8 - 420*b*c^3*d^7*e - 60*a^3*b*d*e^7 - 105*a^4*e^8 - 30*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 20*(
b^3*c + 3*a*b*c^2)*d^5*e^3 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 12*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 10*(3
*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 3360*(2*c^4*d*e^7 - b*c^3*e^8)*x^7 + 840*(42*c^4*d^2*e^6 - 14*b*c^3*d*e^7 - (3*b
^2*c^2 + 2*a*c^3)*e^8)*x^6 + 560*(154*c^4*d^3*e^5 - 42*b*c^3*d^2*e^6 - 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - 2*(b^3*
c + 3*a*b*c^2)*e^8)*x^5 + 70*(1750*c^4*d^4*e^4 - 420*b*c^3*d^3*e^5 - 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 20*(b^
3*c + 3*a*b*c^2)*d*e^7 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 56*(1918*c^4*d^5*e^3 - 420*b*c^3*d^4*e^4
- 30*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 20*(b^3*c + 3*a*b*c^2)*d^2*e^6 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 -
12*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 28*(2058*c^4*d^6*e^2 - 420*b*c^3*d^5*e^3 - 30*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4
- 20*(b^3*c + 3*a*b*c^2)*d^3*e^5 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 12*(a*b^3 + 3*a^2*b*c)*d*e^7 -
10*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 8*(2178*c^4*d^7*e - 420*b*c^3*d^6*e^2 - 60*a^3*b*e^8 - 30*(3*b^2*c^2 + 2*a
*c^3)*d^5*e^3 - 20*(b^3*c + 3*a*b*c^2)*d^4*e^4 - 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 12*(a*b^3 + 3*a^2*
b*c)*d^2*e^6 - 10*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^17*x^8 + 8*d*e^16*x^7 + 28*d^2*e^15*x^6 + 56*d^3*e^14*x^5
+ 70*d^4*e^13*x^4 + 56*d^5*e^12*x^3 + 28*d^6*e^11*x^2 + 8*d^7*e^10*x + d^8*e^9) + c^4*log(e*x + d)/e^9

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (423) = 846\).

Time = 0.26 (sec) , antiderivative size = 910, normalized size of antiderivative = 2.09 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=\frac {c^{4} \log \left ({\left | e x + d \right |}\right )}{e^{9}} + \frac {3360 \, {\left (2 \, c^{4} d e^{6} - b c^{3} e^{7}\right )} x^{7} + 840 \, {\left (42 \, c^{4} d^{2} e^{5} - 14 \, b c^{3} d e^{6} - 3 \, b^{2} c^{2} e^{7} - 2 \, a c^{3} e^{7}\right )} x^{6} + 560 \, {\left (154 \, c^{4} d^{3} e^{4} - 42 \, b c^{3} d^{2} e^{5} - 9 \, b^{2} c^{2} d e^{6} - 6 \, a c^{3} d e^{6} - 2 \, b^{3} c e^{7} - 6 \, a b c^{2} e^{7}\right )} x^{5} + 70 \, {\left (1750 \, c^{4} d^{4} e^{3} - 420 \, b c^{3} d^{3} e^{4} - 90 \, b^{2} c^{2} d^{2} e^{5} - 60 \, a c^{3} d^{2} e^{5} - 20 \, b^{3} c d e^{6} - 60 \, a b c^{2} d e^{6} - 3 \, b^{4} e^{7} - 36 \, a b^{2} c e^{7} - 18 \, a^{2} c^{2} e^{7}\right )} x^{4} + 56 \, {\left (1918 \, c^{4} d^{5} e^{2} - 420 \, b c^{3} d^{4} e^{3} - 90 \, b^{2} c^{2} d^{3} e^{4} - 60 \, a c^{3} d^{3} e^{4} - 20 \, b^{3} c d^{2} e^{5} - 60 \, a b c^{2} d^{2} e^{5} - 3 \, b^{4} d e^{6} - 36 \, a b^{2} c d e^{6} - 18 \, a^{2} c^{2} d e^{6} - 12 \, a b^{3} e^{7} - 36 \, a^{2} b c e^{7}\right )} x^{3} + 28 \, {\left (2058 \, c^{4} d^{6} e - 420 \, b c^{3} d^{5} e^{2} - 90 \, b^{2} c^{2} d^{4} e^{3} - 60 \, a c^{3} d^{4} e^{3} - 20 \, b^{3} c d^{3} e^{4} - 60 \, a b c^{2} d^{3} e^{4} - 3 \, b^{4} d^{2} e^{5} - 36 \, a b^{2} c d^{2} e^{5} - 18 \, a^{2} c^{2} d^{2} e^{5} - 12 \, a b^{3} d e^{6} - 36 \, a^{2} b c d e^{6} - 30 \, a^{2} b^{2} e^{7} - 20 \, a^{3} c e^{7}\right )} x^{2} + 8 \, {\left (2178 \, c^{4} d^{7} - 420 \, b c^{3} d^{6} e - 90 \, b^{2} c^{2} d^{5} e^{2} - 60 \, a c^{3} d^{5} e^{2} - 20 \, b^{3} c d^{4} e^{3} - 60 \, a b c^{2} d^{4} e^{3} - 3 \, b^{4} d^{3} e^{4} - 36 \, a b^{2} c d^{3} e^{4} - 18 \, a^{2} c^{2} d^{3} e^{4} - 12 \, a b^{3} d^{2} e^{5} - 36 \, a^{2} b c d^{2} e^{5} - 30 \, a^{2} b^{2} d e^{6} - 20 \, a^{3} c d e^{6} - 60 \, a^{3} b e^{7}\right )} x + \frac {2283 \, c^{4} d^{8} - 420 \, b c^{3} d^{7} e - 90 \, b^{2} c^{2} d^{6} e^{2} - 60 \, a c^{3} d^{6} e^{2} - 20 \, b^{3} c d^{5} e^{3} - 60 \, a b c^{2} d^{5} e^{3} - 3 \, b^{4} d^{4} e^{4} - 36 \, a b^{2} c d^{4} e^{4} - 18 \, a^{2} c^{2} d^{4} e^{4} - 12 \, a b^{3} d^{3} e^{5} - 36 \, a^{2} b c d^{3} e^{5} - 30 \, a^{2} b^{2} d^{2} e^{6} - 20 \, a^{3} c d^{2} e^{6} - 60 \, a^{3} b d e^{7} - 105 \, a^{4} e^{8}}{e}}{840 \, {\left (e x + d\right )}^{8} e^{8}} \]

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^9,x, algorithm="giac")

[Out]

c^4*log(abs(e*x + d))/e^9 + 1/840*(3360*(2*c^4*d*e^6 - b*c^3*e^7)*x^7 + 840*(42*c^4*d^2*e^5 - 14*b*c^3*d*e^6 -
3*b^2*c^2*e^7 - 2*a*c^3*e^7)*x^6 + 560*(154*c^4*d^3*e^4 - 42*b*c^3*d^2*e^5 - 9*b^2*c^2*d*e^6 - 6*a*c^3*d*e^6
- 2*b^3*c*e^7 - 6*a*b*c^2*e^7)*x^5 + 70*(1750*c^4*d^4*e^3 - 420*b*c^3*d^3*e^4 - 90*b^2*c^2*d^2*e^5 - 60*a*c^3*
d^2*e^5 - 20*b^3*c*d*e^6 - 60*a*b*c^2*d*e^6 - 3*b^4*e^7 - 36*a*b^2*c*e^7 - 18*a^2*c^2*e^7)*x^4 + 56*(1918*c^4*
d^5*e^2 - 420*b*c^3*d^4*e^3 - 90*b^2*c^2*d^3*e^4 - 60*a*c^3*d^3*e^4 - 20*b^3*c*d^2*e^5 - 60*a*b*c^2*d^2*e^5 -
3*b^4*d*e^6 - 36*a*b^2*c*d*e^6 - 18*a^2*c^2*d*e^6 - 12*a*b^3*e^7 - 36*a^2*b*c*e^7)*x^3 + 28*(2058*c^4*d^6*e -
420*b*c^3*d^5*e^2 - 90*b^2*c^2*d^4*e^3 - 60*a*c^3*d^4*e^3 - 20*b^3*c*d^3*e^4 - 60*a*b*c^2*d^3*e^4 - 3*b^4*d^2*
e^5 - 36*a*b^2*c*d^2*e^5 - 18*a^2*c^2*d^2*e^5 - 12*a*b^3*d*e^6 - 36*a^2*b*c*d*e^6 - 30*a^2*b^2*e^7 - 20*a^3*c*
e^7)*x^2 + 8*(2178*c^4*d^7 - 420*b*c^3*d^6*e - 90*b^2*c^2*d^5*e^2 - 60*a*c^3*d^5*e^2 - 20*b^3*c*d^4*e^3 - 60*a
*b*c^2*d^4*e^3 - 3*b^4*d^3*e^4 - 36*a*b^2*c*d^3*e^4 - 18*a^2*c^2*d^3*e^4 - 12*a*b^3*d^2*e^5 - 36*a^2*b*c*d^2*e
^5 - 30*a^2*b^2*d*e^6 - 20*a^3*c*d*e^6 - 60*a^3*b*e^7)*x + (2283*c^4*d^8 - 420*b*c^3*d^7*e - 90*b^2*c^2*d^6*e^
2 - 60*a*c^3*d^6*e^2 - 20*b^3*c*d^5*e^3 - 60*a*b*c^2*d^5*e^3 - 3*b^4*d^4*e^4 - 36*a*b^2*c*d^4*e^4 - 18*a^2*c^2
*d^4*e^4 - 12*a*b^3*d^3*e^5 - 36*a^2*b*c*d^3*e^5 - 30*a^2*b^2*d^2*e^6 - 20*a^3*c*d^2*e^6 - 60*a^3*b*d*e^7 - 10
5*a^4*e^8)/e)/((e*x + d)^8*e^8)

Mupad [B] (verification not implemented)

Time = 10.07 (sec) , antiderivative size = 1168, normalized size of antiderivative = 2.69 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^9} \, dx=-\frac {\frac {a^4\,e^8}{8}-\frac {761\,c^4\,d^8}{280}-c^4\,d^8\,\ln \left (d+e\,x\right )+\frac {b^4\,d^4\,e^4}{280}+\frac {b^4\,e^8\,x^4}{4}+\frac {a\,b^3\,d^3\,e^5}{70}+\frac {a\,c^3\,d^6\,e^2}{14}+\frac {a^3\,c\,d^2\,e^6}{42}+\frac {b^3\,c\,d^5\,e^3}{42}+\frac {4\,a\,b^3\,e^8\,x^3}{5}+\frac {2\,a^3\,c\,e^8\,x^2}{3}+2\,a\,c^3\,e^8\,x^6+\frac {4\,b^3\,c\,e^8\,x^5}{3}+4\,b\,c^3\,e^8\,x^7+\frac {b^4\,d^3\,e^5\,x}{35}+\frac {b^4\,d\,e^7\,x^3}{5}-8\,c^4\,d\,e^7\,x^7-c^4\,e^8\,x^8\,\ln \left (d+e\,x\right )+\frac {a^2\,b^2\,d^2\,e^6}{28}+\frac {3\,a^2\,c^2\,d^4\,e^4}{140}+\frac {3\,b^2\,c^2\,d^6\,e^2}{28}+a^2\,b^2\,e^8\,x^2+\frac {3\,a^2\,c^2\,e^8\,x^4}{2}+3\,b^2\,c^2\,e^8\,x^6+\frac {b^4\,d^2\,e^6\,x^2}{10}-\frac {343\,c^4\,d^6\,e^2\,x^2}{5}-\frac {1918\,c^4\,d^5\,e^3\,x^3}{15}-\frac {875\,c^4\,d^4\,e^4\,x^4}{6}-\frac {308\,c^4\,d^3\,e^5\,x^5}{3}-42\,c^4\,d^2\,e^6\,x^6+\frac {a^3\,b\,d\,e^7}{14}+\frac {b\,c^3\,d^7\,e}{2}+\frac {4\,a^3\,b\,e^8\,x}{7}-\frac {726\,c^4\,d^7\,e\,x}{35}+\frac {4\,a^3\,c\,d\,e^7\,x}{21}-8\,c^4\,d^7\,e\,x\,\ln \left (d+e\,x\right )+\frac {3\,a^2\,c^2\,d^2\,e^6\,x^2}{5}+3\,b^2\,c^2\,d^4\,e^4\,x^2+6\,b^2\,c^2\,d^3\,e^5\,x^3+\frac {15\,b^2\,c^2\,d^2\,e^6\,x^4}{2}+\frac {a\,b\,c^2\,d^5\,e^3}{14}+\frac {3\,a\,b^2\,c\,d^4\,e^4}{70}+\frac {3\,a^2\,b\,c\,d^3\,e^5}{70}+\frac {12\,a^2\,b\,c\,e^8\,x^3}{5}+3\,a\,b^2\,c\,e^8\,x^4+4\,a\,b\,c^2\,e^8\,x^5+\frac {4\,a\,b^3\,d^2\,e^6\,x}{35}+\frac {2\,a^2\,b^2\,d\,e^7\,x}{7}+\frac {2\,a\,b^3\,d\,e^7\,x^2}{5}+\frac {4\,a\,c^3\,d^5\,e^3\,x}{7}+4\,a\,c^3\,d\,e^7\,x^5+4\,b\,c^3\,d^6\,e^2\,x+\frac {4\,b^3\,c\,d^4\,e^4\,x}{21}+\frac {5\,b^3\,c\,d\,e^7\,x^4}{3}+14\,b\,c^3\,d\,e^7\,x^6-8\,c^4\,d\,e^7\,x^7\,\ln \left (d+e\,x\right )+\frac {6\,a^2\,c^2\,d^3\,e^5\,x}{35}+2\,a\,c^3\,d^4\,e^4\,x^2+4\,a\,c^3\,d^3\,e^5\,x^3+\frac {6\,a^2\,c^2\,d\,e^7\,x^3}{5}+5\,a\,c^3\,d^2\,e^6\,x^4+\frac {6\,b^2\,c^2\,d^5\,e^3\,x}{7}+14\,b\,c^3\,d^5\,e^3\,x^2+\frac {2\,b^3\,c\,d^3\,e^5\,x^2}{3}+28\,b\,c^3\,d^4\,e^4\,x^3+\frac {4\,b^3\,c\,d^2\,e^6\,x^3}{3}+35\,b\,c^3\,d^3\,e^5\,x^4+28\,b\,c^3\,d^2\,e^6\,x^5+6\,b^2\,c^2\,d\,e^7\,x^5-28\,c^4\,d^6\,e^2\,x^2\,\ln \left (d+e\,x\right )-56\,c^4\,d^5\,e^3\,x^3\,\ln \left (d+e\,x\right )-70\,c^4\,d^4\,e^4\,x^4\,\ln \left (d+e\,x\right )-56\,c^4\,d^3\,e^5\,x^5\,\ln \left (d+e\,x\right )-28\,c^4\,d^2\,e^6\,x^6\,\ln \left (d+e\,x\right )+2\,a\,b\,c^2\,d^3\,e^5\,x^2+\frac {6\,a\,b^2\,c\,d^2\,e^6\,x^2}{5}+4\,a\,b\,c^2\,d^2\,e^6\,x^3+\frac {4\,a\,b\,c^2\,d^4\,e^4\,x}{7}+\frac {12\,a\,b^2\,c\,d^3\,e^5\,x}{35}+\frac {12\,a^2\,b\,c\,d^2\,e^6\,x}{35}+\frac {6\,a^2\,b\,c\,d\,e^7\,x^2}{5}+\frac {12\,a\,b^2\,c\,d\,e^7\,x^3}{5}+5\,a\,b\,c^2\,d\,e^7\,x^4}{e^9\,{\left (d+e\,x\right )}^8} \]

[In]

int((a + b*x + c*x^2)^4/(d + e*x)^9,x)

[Out]

-((a^4*e^8)/8 - (761*c^4*d^8)/280 - c^4*d^8*log(d + e*x) + (b^4*d^4*e^4)/280 + (b^4*e^8*x^4)/4 + (a*b^3*d^3*e^
5)/70 + (a*c^3*d^6*e^2)/14 + (a^3*c*d^2*e^6)/42 + (b^3*c*d^5*e^3)/42 + (4*a*b^3*e^8*x^3)/5 + (2*a^3*c*e^8*x^2)
/3 + 2*a*c^3*e^8*x^6 + (4*b^3*c*e^8*x^5)/3 + 4*b*c^3*e^8*x^7 + (b^4*d^3*e^5*x)/35 + (b^4*d*e^7*x^3)/5 - 8*c^4*
d*e^7*x^7 - c^4*e^8*x^8*log(d + e*x) + (a^2*b^2*d^2*e^6)/28 + (3*a^2*c^2*d^4*e^4)/140 + (3*b^2*c^2*d^6*e^2)/28
+ a^2*b^2*e^8*x^2 + (3*a^2*c^2*e^8*x^4)/2 + 3*b^2*c^2*e^8*x^6 + (b^4*d^2*e^6*x^2)/10 - (343*c^4*d^6*e^2*x^2)/
5 - (1918*c^4*d^5*e^3*x^3)/15 - (875*c^4*d^4*e^4*x^4)/6 - (308*c^4*d^3*e^5*x^5)/3 - 42*c^4*d^2*e^6*x^6 + (a^3*
b*d*e^7)/14 + (b*c^3*d^7*e)/2 + (4*a^3*b*e^8*x)/7 - (726*c^4*d^7*e*x)/35 + (4*a^3*c*d*e^7*x)/21 - 8*c^4*d^7*e*
x*log(d + e*x) + (3*a^2*c^2*d^2*e^6*x^2)/5 + 3*b^2*c^2*d^4*e^4*x^2 + 6*b^2*c^2*d^3*e^5*x^3 + (15*b^2*c^2*d^2*e
^6*x^4)/2 + (a*b*c^2*d^5*e^3)/14 + (3*a*b^2*c*d^4*e^4)/70 + (3*a^2*b*c*d^3*e^5)/70 + (12*a^2*b*c*e^8*x^3)/5 +
3*a*b^2*c*e^8*x^4 + 4*a*b*c^2*e^8*x^5 + (4*a*b^3*d^2*e^6*x)/35 + (2*a^2*b^2*d*e^7*x)/7 + (2*a*b^3*d*e^7*x^2)/5
+ (4*a*c^3*d^5*e^3*x)/7 + 4*a*c^3*d*e^7*x^5 + 4*b*c^3*d^6*e^2*x + (4*b^3*c*d^4*e^4*x)/21 + (5*b^3*c*d*e^7*x^4
)/3 + 14*b*c^3*d*e^7*x^6 - 8*c^4*d*e^7*x^7*log(d + e*x) + (6*a^2*c^2*d^3*e^5*x)/35 + 2*a*c^3*d^4*e^4*x^2 + 4*a
*c^3*d^3*e^5*x^3 + (6*a^2*c^2*d*e^7*x^3)/5 + 5*a*c^3*d^2*e^6*x^4 + (6*b^2*c^2*d^5*e^3*x)/7 + 14*b*c^3*d^5*e^3*
x^2 + (2*b^3*c*d^3*e^5*x^2)/3 + 28*b*c^3*d^4*e^4*x^3 + (4*b^3*c*d^2*e^6*x^3)/3 + 35*b*c^3*d^3*e^5*x^4 + 28*b*c
^3*d^2*e^6*x^5 + 6*b^2*c^2*d*e^7*x^5 - 28*c^4*d^6*e^2*x^2*log(d + e*x) - 56*c^4*d^5*e^3*x^3*log(d + e*x) - 70*
c^4*d^4*e^4*x^4*log(d + e*x) - 56*c^4*d^3*e^5*x^5*log(d + e*x) - 28*c^4*d^2*e^6*x^6*log(d + e*x) + 2*a*b*c^2*d
^3*e^5*x^2 + (6*a*b^2*c*d^2*e^6*x^2)/5 + 4*a*b*c^2*d^2*e^6*x^3 + (4*a*b*c^2*d^4*e^4*x)/7 + (12*a*b^2*c*d^3*e^5
*x)/35 + (12*a^2*b*c*d^2*e^6*x)/35 + (6*a^2*b*c*d*e^7*x^2)/5 + (12*a*b^2*c*d*e^7*x^3)/5 + 5*a*b*c^2*d*e^7*x^4)
/(e^9*(d + e*x)^8)