\(\int \frac {(a+b x+c x^2)^4}{(d+e x)^4} \, dx\) [2154]

Optimal result

Integrand size = 20, antiderivative size = 417 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx=\frac {\left (35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x}{e^8}-\frac {2 c \left (5 c^3 d^3-b^3 e^3-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)\right ) x^2}{e^7}+\frac {2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^3}{3 e^6}-\frac {c^3 (c d-b e) x^4}{e^5}+\frac {c^4 x^5}{5 e^4}-\frac {\left (c d^2-b d e+a e^2\right )^4}{3 e^9 (d+e x)^3}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^2}-\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^9 (d+e x)}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \log (d+e x)}{e^9} \]

[Out]

(35*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+4*b*d)-40*c^3*d^2*e*(-a*e+2*b*d)+6*c^2*e^2*(a^2*e^2-8*a*b*d*e+10*b^2*d
^2))*x/e^8-2*c*(5*c^3*d^3-b^3*e^3-2*c^2*d*e*(-2*a*e+5*b*d)+3*b*c*e^2*(-a*e+2*b*d))*x^2/e^7+2/3*c^2*(5*c^2*d^2+
3*b^2*e^2-2*c*e*(-a*e+4*b*d))*x^3/e^6-c^3*(-b*e+c*d)*x^4/e^5+1/5*c^4*x^5/e^4-1/3*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*
x+d)^3+2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^2-2*(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)-4*(-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))*ln(e*x+d)/
e^9

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 417, 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} \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx=\frac {x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4\right )}{e^8}-\frac {2 c x^2 \left (-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)-b^3 e^3+5 c^3 d^3\right )}{e^7}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}-\frac {4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac {2 c^2 x^3 \left (-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right )}{3 e^6}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right )^4}{3 e^9 (d+e x)^3}-\frac {c^3 x^4 (c d-b e)}{e^5}+\frac {c^4 x^5}{5 e^4} \]

[In]

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

[Out]

((35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) - 40*c^3*d^2*e*(2*b*d - a*e) + 6*c^2*e^2*(10*b^2*d^2 - 8*
a*b*d*e + a^2*e^2))*x)/e^8 - (2*c*(5*c^3*d^3 - b^3*e^3 - 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(2*b*d - a*e))*
x^2)/e^7 + (2*c^2*(5*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(4*b*d - a*e))*x^3)/(3*e^6) - (c^3*(c*d - b*e)*x^4)/e^5 + (c^
4*x^5)/(5*e^4) - (c*d^2 - b*d*e + a*e^2)^4/(3*e^9*(d + e*x)^3) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(
e^9*(d + e*x)^2) - (2*(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)))/(e^9*(d + e*x)
) - (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))*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 {35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )}{e^8}+\frac {4 c \left (-5 c^3 d^3+b^3 e^3+2 c^2 d e (5 b d-2 a e)-3 b c e^2 (2 b d-a e)\right ) x}{e^7}+\frac {2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^2}{e^6}-\frac {4 c^3 (c d-b e) x^3}{e^5}+\frac {c^4 x^4}{e^4}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^4}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^3}+\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)^2}+\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)}\right ) \, dx \\ & = \frac {\left (35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x}{e^8}-\frac {2 c \left (5 c^3 d^3-b^3 e^3-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)\right ) x^2}{e^7}+\frac {2 c^2 \left (5 c^2 d^2+3 b^2 e^2-2 c e (4 b d-a e)\right ) x^3}{3 e^6}-\frac {c^3 (c d-b e) x^4}{e^5}+\frac {c^4 x^5}{5 e^4}-\frac {\left (c d^2-b d e+a e^2\right )^4}{3 e^9 (d+e x)^3}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^9 (d+e x)^2}-\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^9 (d+e x)}-\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 ) \log (d+e x)}{e^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx=\frac {15 e \left (35 c^4 d^4+b^4 e^4-4 b^2 c e^3 (4 b d-3 a e)+40 c^3 d^2 e (-2 b d+a e)+6 c^2 e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right ) x+30 c e^2 \left (-5 c^3 d^3+b^3 e^3+2 c^2 d e (5 b d-2 a e)+3 b c e^2 (-2 b d+a e)\right ) x^2+10 c^2 e^3 \left (5 c^2 d^2+3 b^2 e^2+2 c e (-4 b d+a e)\right ) x^3+15 c^3 e^4 (-c d+b e) x^4+3 c^4 e^5 x^5-\frac {5 \left (c d^2+e (-b d+a e)\right )^4}{(d+e x)^3}+\frac {30 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^2}-\frac {30 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2}{d+e x}-60 (2 c d-b e) \left (7 c^3 d^4-2 c^2 d^2 e (7 b d-5 a e)+b^2 e^3 (-b d+a e)+c e^2 \left (8 b^2 d^2-10 a b d e+3 a^2 e^2\right )\right ) \log (d+e x)}{15 e^9} \]

[In]

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

[Out]

(15*e*(35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) + 40*c^3*d^2*e*(-2*b*d + a*e) + 6*c^2*e^2*(10*b^2*d^
2 - 8*a*b*d*e + a^2*e^2))*x + 30*c*e^2*(-5*c^3*d^3 + b^3*e^3 + 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(-2*b*d +
a*e))*x^2 + 10*c^2*e^3*(5*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-4*b*d + a*e))*x^3 + 15*c^3*e^4*(-(c*d) + b*e)*x^4 + 3
*c^4*e^5*x^5 - (5*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^3 + (30*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/
(d + e*x)^2 - (30*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x) - 60
*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d
*e + 3*a^2*e^2))*Log[d + e*x])/(15*e^9)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(886\) vs. \(2(411)=822\).

Time = 3.21 (sec) , antiderivative size = 887, normalized size of antiderivative = 2.13

method	result	size
norman	\(\frac {\frac {\left (6 c^{2} a^{2} e^{4}+12 a \,b^{2} c \,e^{4}-30 a b \,c^{2} d \,e^{3}+20 c^{3} a \,d^{2} e^{2}+b^{4} e^{4}-10 b^{3} c d \,e^{3}+30 b^{2} c^{2} d^{2} e^{2}-35 b \,c^{3} d^{3} e +14 c^{4} d^{4}\right ) x^{4}}{e^{5}}-\frac {a^{4} e^{8}+2 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-66 a^{2} b c \,d^{3} e^{5}+132 a^{2} c^{2} d^{4} e^{4}-22 a \,b^{3} d^{3} e^{5}+264 a \,b^{2} c \,d^{4} e^{4}-660 a b \,c^{2} d^{5} e^{3}+440 a \,c^{3} d^{6} e^{2}+22 b^{4} d^{4} e^{4}-220 b^{3} c \,d^{5} e^{3}+660 b^{2} c^{2} d^{6} e^{2}-770 b \,c^{3} d^{7} e +308 c^{4} d^{8}}{3 e^{9}}+\frac {c^{4} x^{8}}{5 e}-\frac {\left (4 e^{6} c \,a^{3}+6 a^{2} b^{2} e^{6}-36 a^{2} b c d \,e^{5}+72 d^{2} e^{4} a^{2} c^{2}-12 a \,b^{3} d \,e^{5}+144 a \,b^{2} c \,d^{2} e^{4}-360 a b \,c^{2} d^{3} e^{3}+240 d^{4} e^{2} c^{3} a +12 b^{4} d^{2} e^{4}-120 b^{3} c \,d^{3} e^{3}+360 b^{2} c^{2} d^{4} e^{2}-420 b \,c^{3} d^{5} e +168 d^{6} c^{4}\right ) x^{2}}{e^{7}}-\frac {\left (2 a^{3} b \,e^{7}+4 d \,e^{6} c \,a^{3}+6 a^{2} b^{2} d \,e^{6}-54 a^{2} b c \,d^{2} e^{5}+108 d^{3} e^{4} a^{2} c^{2}-18 a \,b^{3} d^{2} e^{5}+216 a \,b^{2} c \,d^{3} e^{4}-540 a b \,c^{2} d^{4} e^{3}+360 d^{5} e^{2} c^{3} a +18 b^{4} d^{3} e^{4}-180 b^{3} c \,d^{4} e^{3}+540 b^{2} c^{2} d^{5} e^{2}-630 b \,c^{3} d^{6} e +252 d^{7} c^{4}\right ) x}{e^{8}}+\frac {c \left (30 a b c \,e^{3}-20 c^{2} a d \,e^{2}+10 b^{3} e^{3}-30 b^{2} d \,e^{2} c +35 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) x^{5}}{5 e^{4}}+\frac {c^{2} \left (20 a c \,e^{2}+30 b^{2} e^{2}-35 b c d e +14 c^{2} d^{2}\right ) x^{6}}{15 e^{3}}+\frac {c^{3} \left (5 b e -2 c d \right ) x^{7}}{5 e^{2}}}{\left (e x +d \right )^{3}}+\frac {4 \left (3 a^{2} b c \,e^{5}-6 d \,e^{4} a^{2} c^{2}+a \,b^{3} e^{5}-12 a \,b^{2} c d \,e^{4}+30 a b \,c^{2} d^{2} e^{3}-20 d^{3} e^{2} c^{3} a -b^{4} d \,e^{4}+10 b^{3} c \,d^{2} e^{3}-30 b^{2} c^{2} d^{3} e^{2}+35 b \,c^{3} d^{4} e -14 c^{4} d^{5}\right ) \ln \left (e x +d \right )}{e^{9}}\)	\(887\)
default	\(\frac {\frac {1}{5} c^{4} e^{4} x^{5}+e^{4} x^{4} b \,c^{3}-c^{4} d \,e^{3} x^{4}+\frac {4}{3} a \,c^{3} e^{4} x^{3}+2 x^{3} b^{2} c^{2} e^{4}-\frac {16}{3} x^{3} b \,c^{3} d \,e^{3}+\frac {10}{3} c^{4} d^{2} e^{2} x^{3}+6 a b \,c^{2} e^{4} x^{2}-8 a \,c^{3} d \,e^{3} x^{2}+2 x^{2} b^{3} c \,e^{4}-12 x^{2} b^{2} c^{2} d \,e^{3}+20 x^{2} b \,c^{3} d^{2} e^{2}-10 c^{4} d^{3} e \,x^{2}+6 c^{2} a^{2} e^{4} x +12 a \,b^{2} c \,e^{4} x -48 a b \,c^{2} d \,e^{3} x +40 c^{3} a \,d^{2} e^{2} x +b^{4} e^{4} x -16 b^{3} c d \,e^{3} x +60 b^{2} c^{2} d^{2} e^{2} x -80 b \,c^{3} d^{3} e x +35 c^{4} d^{4} x}{e^{8}}-\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}}{e^{9} \left (e x +d \right )}-\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}}{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}}{2 e^{9} \left (e x +d \right )^{2}}+\frac {\left (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}\right ) \ln \left (e x +d \right )}{e^{9}}\)	\(930\)
risch	\(\frac {6 a b \,c^{2} x^{2}}{e^{4}}-\frac {24 \ln \left (e x +d \right ) d \,a^{2} c^{2}}{e^{5}}-\frac {80 \ln \left (e x +d \right ) d^{3} c^{3} a}{e^{7}}+\frac {40 \ln \left (e x +d \right ) b^{3} c \,d^{2}}{e^{6}}-\frac {120 \ln \left (e x +d \right ) b^{2} c^{2} d^{3}}{e^{7}}+\frac {140 \ln \left (e x +d \right ) b \,c^{3} d^{4}}{e^{8}}+\frac {c^{4} x^{5}}{5 e^{4}}+\frac {x^{4} b \,c^{3}}{e^{4}}-\frac {c^{4} d \,x^{4}}{e^{5}}+\frac {4 a \,c^{3} x^{3}}{3 e^{4}}+\frac {2 x^{3} b^{2} c^{2}}{e^{4}}+\frac {10 c^{4} d^{2} x^{3}}{3 e^{6}}+\frac {2 x^{2} b^{3} c}{e^{4}}-\frac {10 c^{4} d^{3} x^{2}}{e^{7}}+\frac {6 c^{2} a^{2} x}{e^{4}}+\frac {35 c^{4} d^{4} x}{e^{8}}-\frac {16 x^{3} b \,c^{3} d}{3 e^{5}}+\frac {4 \ln \left (e x +d \right ) a \,b^{3}}{e^{4}}-\frac {4 \ln \left (e x +d \right ) b^{4} d}{e^{5}}-\frac {56 \ln \left (e x +d \right ) c^{4} d^{5}}{e^{9}}+\frac {\left (-4 e^{7} c \,a^{3}-6 a^{2} b^{2} e^{7}+36 a^{2} b c d \,e^{6}-36 d^{2} e^{5} a^{2} c^{2}+12 a \,b^{3} d \,e^{6}-72 a \,b^{2} c \,d^{2} e^{5}+120 a b \,c^{2} d^{3} e^{4}-60 d^{4} e^{3} c^{3} a -6 b^{4} d^{2} e^{5}+40 b^{3} c \,d^{3} e^{4}-90 b^{2} c^{2} d^{4} e^{3}+84 c^{3} b \,d^{5} e^{2}-28 d^{6} e \,c^{4}\right ) x^{2}+\left (-2 a^{3} b \,e^{7}-4 d \,e^{6} c \,a^{3}-6 a^{2} b^{2} d \,e^{6}+54 a^{2} b c \,d^{2} e^{5}-60 d^{3} e^{4} a^{2} c^{2}+18 a \,b^{3} d^{2} e^{5}-120 a \,b^{2} c \,d^{3} e^{4}+210 a b \,c^{2} d^{4} e^{3}-108 d^{5} e^{2} c^{3} a -10 b^{4} d^{3} e^{4}+70 b^{3} c \,d^{4} e^{3}-162 b^{2} c^{2} d^{5} e^{2}+154 b \,c^{3} d^{6} e -52 d^{7} c^{4}\right ) x -\frac {a^{4} e^{8}+2 a^{3} b d \,e^{7}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} b^{2} d^{2} e^{6}-66 a^{2} b c \,d^{3} e^{5}+78 a^{2} c^{2} d^{4} e^{4}-22 a \,b^{3} d^{3} e^{5}+156 a \,b^{2} c \,d^{4} e^{4}-282 a b \,c^{2} d^{5} e^{3}+148 a \,c^{3} d^{6} e^{2}+13 b^{4} d^{4} e^{4}-94 b^{3} c \,d^{5} e^{3}+222 b^{2} c^{2} d^{6} e^{2}-214 b \,c^{3} d^{7} e +73 c^{4} d^{8}}{3 e}}{e^{8} \left (e x +d \right )^{3}}+\frac {b^{4} x}{e^{4}}-\frac {8 a \,c^{3} d \,x^{2}}{e^{5}}-\frac {12 x^{2} b^{2} c^{2} d}{e^{5}}+\frac {20 x^{2} b \,c^{3} d^{2}}{e^{6}}+\frac {12 a \,b^{2} c x}{e^{4}}+\frac {40 c^{3} a \,d^{2} x}{e^{6}}-\frac {16 b^{3} c d x}{e^{5}}+\frac {60 b^{2} c^{2} d^{2} x}{e^{6}}-\frac {80 b \,c^{3} d^{3} x}{e^{7}}-\frac {48 a b \,c^{2} d x}{e^{5}}-\frac {48 \ln \left (e x +d \right ) a \,b^{2} c d}{e^{5}}+\frac {120 \ln \left (e x +d \right ) a b \,c^{2} d^{2}}{e^{6}}+\frac {12 \ln \left (e x +d \right ) a^{2} b c}{e^{4}}\)	\(984\)
parallelrisch	\(\text {Expression too large to display}\)	\(1724\)

[In]

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

[Out]

((6*a^2*c^2*e^4+12*a*b^2*c*e^4-30*a*b*c^2*d*e^3+20*a*c^3*d^2*e^2+b^4*e^4-10*b^3*c*d*e^3+30*b^2*c^2*d^2*e^2-35*
b*c^3*d^3*e+14*c^4*d^4)/e^5*x^4-1/3*(a^4*e^8+2*a^3*b*d*e^7+4*a^3*c*d^2*e^6+6*a^2*b^2*d^2*e^6-66*a^2*b*c*d^3*e^
5+132*a^2*c^2*d^4*e^4-22*a*b^3*d^3*e^5+264*a*b^2*c*d^4*e^4-660*a*b*c^2*d^5*e^3+440*a*c^3*d^6*e^2+22*b^4*d^4*e^
4-220*b^3*c*d^5*e^3+660*b^2*c^2*d^6*e^2-770*b*c^3*d^7*e+308*c^4*d^8)/e^9+1/5/e*c^4*x^8-(4*a^3*c*e^6+6*a^2*b^2*
e^6-36*a^2*b*c*d*e^5+72*a^2*c^2*d^2*e^4-12*a*b^3*d*e^5+144*a*b^2*c*d^2*e^4-360*a*b*c^2*d^3*e^3+240*a*c^3*d^4*e
^2+12*b^4*d^2*e^4-120*b^3*c*d^3*e^3+360*b^2*c^2*d^4*e^2-420*b*c^3*d^5*e+168*c^4*d^6)/e^7*x^2-(2*a^3*b*e^7+4*a^
3*c*d*e^6+6*a^2*b^2*d*e^6-54*a^2*b*c*d^2*e^5+108*a^2*c^2*d^3*e^4-18*a*b^3*d^2*e^5+216*a*b^2*c*d^3*e^4-540*a*b*
c^2*d^4*e^3+360*a*c^3*d^5*e^2+18*b^4*d^3*e^4-180*b^3*c*d^4*e^3+540*b^2*c^2*d^5*e^2-630*b*c^3*d^6*e+252*c^4*d^7
)/e^8*x+1/5*c*(30*a*b*c*e^3-20*a*c^2*d*e^2+10*b^3*e^3-30*b^2*c*d*e^2+35*b*c^2*d^2*e-14*c^3*d^3)/e^4*x^5+1/15*c
^2*(20*a*c*e^2+30*b^2*e^2-35*b*c*d*e+14*c^2*d^2)/e^3*x^6+1/5*c^3*(5*b*e-2*c*d)/e^2*x^7)/(e*x+d)^3+4*(3*a^2*b*c
*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b^2*c*d*e^4+30*a*b*c^2*d^2*e^3-20*a*c^3*d^3*e^2-b^4*d*e^4+10*b^3*c*d^2*e^3
-30*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e-14*c^4*d^5)/e^9*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1282 vs. \(2 (411) = 822\).

Time = 0.29 (sec) , antiderivative size = 1282, normalized size of antiderivative = 3.07 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/15*(3*c^4*e^8*x^8 - 365*c^4*d^8 + 1070*b*c^3*d^7*e - 10*a^3*b*d*e^7 - 5*a^4*e^8 - 370*(3*b^2*c^2 + 2*a*c^3)*
d^6*e^2 + 470*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 65*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 110*(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 - 3*(2*c^4*d*e^7 - 5*b*c^3*e^8)*x^7 + (14*c^4*d^2*e^6 - 35*b*c^3*
d*e^7 + 10*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 3*(14*c^4*d^3*e^5 - 35*b*c^3*d^2*e^6 + 10*(3*b^2*c^2 + 2*a*c^3)*d*
e^7 - 10*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 15*(14*c^4*d^4*e^4 - 35*b*c^3*d^3*e^5 + 10*(3*b^2*c^2 + 2*a*c^3)*d^2*e
^6 - 10*(b^3*c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 5*(235*c^4*d^5*e^3 - 556*b*c^3*d
^4*e^4 + 146*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 126*(b^3*c + 3*a*b*c^2)*d^2*e^6 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2
)*d*e^7)*x^3 + 15*(67*c^4*d^6*e^2 - 136*b*c^3*d^5*e^3 + 26*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 6*(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 - 2*(3*a^2*b^2 + 2*a^3*c)
*e^8)*x^2 - 15*(17*c^4*d^7*e - 74*b*c^3*d^6*e^2 + 2*a^3*b*e^8 + 34*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 54*(b^3*c +
3*a*b*c^2)*d^4*e^4 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 18*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 2*(3*a^2*b^2
+ 2*a^3*c)*d*e^7)*x - 60*(14*c^4*d^8 - 35*b*c^3*d^7*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 10*(b^3*c + 3*a*b*
c^2)*d^5*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - (a*b^3 + 3*a^2*b*c)*d^3*e^5 + (14*c^4*d^5*e^3 - 35*b*c
^3*d^4*e^4 + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2
)*d*e^7 - (a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 3*(14*c^4*d^6*e^2 - 35*b*c^3*d^5*e^3 + 10*(3*b^2*c^2 + 2*a*c^3)*d^4*e
^4 - 10*(b^3*c + 3*a*b*c^2)*d^3*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - (a*b^3 + 3*a^2*b*c)*d*e^7)*x^2
+ 3*(14*c^4*d^7*e - 35*b*c^3*d^6*e^2 + 10*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 10*(b^3*c + 3*a*b*c^2)*d^4*e^4 + (b^
4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - (a*b^3 + 3*a^2*b*c)*d^2*e^6)*x)*log(e*x + d))/(e^12*x^3 + 3*d*e^11*x^2 +
3*d^2*e^10*x + d^3*e^9)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 827 vs. \(2 (411) = 822\).

Time = 0.22 (sec) , antiderivative size = 827, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx=-\frac {73 \, c^{4} d^{8} - 214 \, b c^{3} d^{7} e + 2 \, a^{3} b d e^{7} + a^{4} e^{8} + 74 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 94 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 13 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 22 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 6 \, {\left (14 \, c^{4} d^{6} e^{2} - 42 \, b c^{3} d^{5} e^{3} + 15 \, {\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} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 6 \, {\left (26 \, c^{4} d^{7} e - 77 \, b c^{3} d^{6} e^{2} + a^{3} b e^{8} + 27 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 35 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + 5 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} - 9 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{3 \, {\left (e^{12} x^{3} + 3 \, d e^{11} x^{2} + 3 \, d^{2} e^{10} x + d^{3} e^{9}\right )}} + \frac {3 \, c^{4} e^{4} x^{5} - 15 \, {\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{4} + 10 \, {\left (5 \, c^{4} d^{2} e^{2} - 8 \, b c^{3} d e^{3} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{4}\right )} x^{3} - 30 \, {\left (5 \, c^{4} d^{3} e - 10 \, b c^{3} d^{2} e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{3} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{4}\right )} x^{2} + 15 \, {\left (35 \, c^{4} d^{4} - 80 \, b c^{3} d^{3} e + 20 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 16 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} x}{15 \, e^{8}} - \frac {4 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} \log \left (e x + d\right )}{e^{9}} \]

[In]

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

[Out]

-1/3*(73*c^4*d^8 - 214*b*c^3*d^7*e + 2*a^3*b*d*e^7 + a^4*e^8 + 74*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 94*(b^3*c +
3*a*b*c^2)*d^5*e^3 + 13*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 22*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^2*b^2
+ 2*a^3*c)*d^2*e^6 + 6*(14*c^4*d^6*e^2 - 42*b*c^3*d^5*e^3 + 15*(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 - 6*(a*b^3 + 3*a^2*b*c)*d*e^7 + (3*a^2*b^2 + 2*a^3
*c)*e^8)*x^2 + 6*(26*c^4*d^7*e - 77*b*c^3*d^6*e^2 + a^3*b*e^8 + 27*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 35*(b^3*c +
3*a*b*c^2)*d^4*e^4 + 5*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 9*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + (3*a^2*b^2 +
2*a^3*c)*d*e^7)*x)/(e^12*x^3 + 3*d*e^11*x^2 + 3*d^2*e^10*x + d^3*e^9) + 1/15*(3*c^4*e^4*x^5 - 15*(c^4*d*e^3 -
b*c^3*e^4)*x^4 + 10*(5*c^4*d^2*e^2 - 8*b*c^3*d*e^3 + (3*b^2*c^2 + 2*a*c^3)*e^4)*x^3 - 30*(5*c^4*d^3*e - 10*b*c
^3*d^2*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^3 - (b^3*c + 3*a*b*c^2)*e^4)*x^2 + 15*(35*c^4*d^4 - 80*b*c^3*d^3*e +
20*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 16*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*x)/e^8 -
4*(14*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3 + (b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b*c)*e^5)*log(e*x + d)/e^9

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 933 vs. \(2 (411) = 822\).

Time = 0.26 (sec) , antiderivative size = 933, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx=-\frac {4 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 30 \, b^{2} c^{2} d^{3} e^{2} + 20 \, a c^{3} d^{3} e^{2} - 10 \, b^{3} c d^{2} e^{3} - 30 \, a b c^{2} d^{2} e^{3} + b^{4} d e^{4} + 12 \, a b^{2} c d e^{4} + 6 \, a^{2} c^{2} d e^{4} - a b^{3} e^{5} - 3 \, a^{2} b c e^{5}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{9}} - \frac {73 \, c^{4} d^{8} - 214 \, b c^{3} d^{7} e + 222 \, b^{2} c^{2} d^{6} e^{2} + 148 \, a c^{3} d^{6} e^{2} - 94 \, b^{3} c d^{5} e^{3} - 282 \, a b c^{2} d^{5} e^{3} + 13 \, b^{4} d^{4} e^{4} + 156 \, a b^{2} c d^{4} e^{4} + 78 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a b^{3} d^{3} e^{5} - 66 \, a^{2} b c d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} + 4 \, a^{3} c d^{2} e^{6} + 2 \, a^{3} b d e^{7} + a^{4} e^{8} + 6 \, {\left (14 \, c^{4} d^{6} e^{2} - 42 \, b c^{3} d^{5} e^{3} + 45 \, b^{2} c^{2} d^{4} e^{4} + 30 \, a c^{3} d^{4} e^{4} - 20 \, b^{3} c d^{3} e^{5} - 60 \, a b c^{2} d^{3} e^{5} + 3 \, b^{4} d^{2} e^{6} + 36 \, a b^{2} c d^{2} e^{6} + 18 \, a^{2} c^{2} d^{2} e^{6} - 6 \, a b^{3} d e^{7} - 18 \, a^{2} b c d e^{7} + 3 \, a^{2} b^{2} e^{8} + 2 \, a^{3} c e^{8}\right )} x^{2} + 6 \, {\left (26 \, c^{4} d^{7} e - 77 \, b c^{3} d^{6} e^{2} + 81 \, b^{2} c^{2} d^{5} e^{3} + 54 \, a c^{3} d^{5} e^{3} - 35 \, b^{3} c d^{4} e^{4} - 105 \, a b c^{2} d^{4} e^{4} + 5 \, b^{4} d^{3} e^{5} + 60 \, a b^{2} c d^{3} e^{5} + 30 \, a^{2} c^{2} d^{3} e^{5} - 9 \, a b^{3} d^{2} e^{6} - 27 \, a^{2} b c d^{2} e^{6} + 3 \, a^{2} b^{2} d e^{7} + 2 \, a^{3} c d e^{7} + a^{3} b e^{8}\right )} x}{3 \, {\left (e x + d\right )}^{3} e^{9}} + \frac {3 \, c^{4} e^{16} x^{5} - 15 \, c^{4} d e^{15} x^{4} + 15 \, b c^{3} e^{16} x^{4} + 50 \, c^{4} d^{2} e^{14} x^{3} - 80 \, b c^{3} d e^{15} x^{3} + 30 \, b^{2} c^{2} e^{16} x^{3} + 20 \, a c^{3} e^{16} x^{3} - 150 \, c^{4} d^{3} e^{13} x^{2} + 300 \, b c^{3} d^{2} e^{14} x^{2} - 180 \, b^{2} c^{2} d e^{15} x^{2} - 120 \, a c^{3} d e^{15} x^{2} + 30 \, b^{3} c e^{16} x^{2} + 90 \, a b c^{2} e^{16} x^{2} + 525 \, c^{4} d^{4} e^{12} x - 1200 \, b c^{3} d^{3} e^{13} x + 900 \, b^{2} c^{2} d^{2} e^{14} x + 600 \, a c^{3} d^{2} e^{14} x - 240 \, b^{3} c d e^{15} x - 720 \, a b c^{2} d e^{15} x + 15 \, b^{4} e^{16} x + 180 \, a b^{2} c e^{16} x + 90 \, a^{2} c^{2} e^{16} x}{15 \, e^{20}} \]

[In]

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

[Out]

-4*(14*c^4*d^5 - 35*b*c^3*d^4*e + 30*b^2*c^2*d^3*e^2 + 20*a*c^3*d^3*e^2 - 10*b^3*c*d^2*e^3 - 30*a*b*c^2*d^2*e^
3 + b^4*d*e^4 + 12*a*b^2*c*d*e^4 + 6*a^2*c^2*d*e^4 - a*b^3*e^5 - 3*a^2*b*c*e^5)*log(abs(e*x + d))/e^9 - 1/3*(7
3*c^4*d^8 - 214*b*c^3*d^7*e + 222*b^2*c^2*d^6*e^2 + 148*a*c^3*d^6*e^2 - 94*b^3*c*d^5*e^3 - 282*a*b*c^2*d^5*e^3
+ 13*b^4*d^4*e^4 + 156*a*b^2*c*d^4*e^4 + 78*a^2*c^2*d^4*e^4 - 22*a*b^3*d^3*e^5 - 66*a^2*b*c*d^3*e^5 + 6*a^2*b
^2*d^2*e^6 + 4*a^3*c*d^2*e^6 + 2*a^3*b*d*e^7 + a^4*e^8 + 6*(14*c^4*d^6*e^2 - 42*b*c^3*d^5*e^3 + 45*b^2*c^2*d^4
*e^4 + 30*a*c^3*d^4*e^4 - 20*b^3*c*d^3*e^5 - 60*a*b*c^2*d^3*e^5 + 3*b^4*d^2*e^6 + 36*a*b^2*c*d^2*e^6 + 18*a^2*
c^2*d^2*e^6 - 6*a*b^3*d*e^7 - 18*a^2*b*c*d*e^7 + 3*a^2*b^2*e^8 + 2*a^3*c*e^8)*x^2 + 6*(26*c^4*d^7*e - 77*b*c^3
*d^6*e^2 + 81*b^2*c^2*d^5*e^3 + 54*a*c^3*d^5*e^3 - 35*b^3*c*d^4*e^4 - 105*a*b*c^2*d^4*e^4 + 5*b^4*d^3*e^5 + 60
*a*b^2*c*d^3*e^5 + 30*a^2*c^2*d^3*e^5 - 9*a*b^3*d^2*e^6 - 27*a^2*b*c*d^2*e^6 + 3*a^2*b^2*d*e^7 + 2*a^3*c*d*e^7
+ a^3*b*e^8)*x)/((e*x + d)^3*e^9) + 1/15*(3*c^4*e^16*x^5 - 15*c^4*d*e^15*x^4 + 15*b*c^3*e^16*x^4 + 50*c^4*d^2
*e^14*x^3 - 80*b*c^3*d*e^15*x^3 + 30*b^2*c^2*e^16*x^3 + 20*a*c^3*e^16*x^3 - 150*c^4*d^3*e^13*x^2 + 300*b*c^3*d
^2*e^14*x^2 - 180*b^2*c^2*d*e^15*x^2 - 120*a*c^3*d*e^15*x^2 + 30*b^3*c*e^16*x^2 + 90*a*b*c^2*e^16*x^2 + 525*c^
4*d^4*e^12*x - 1200*b*c^3*d^3*e^13*x + 900*b^2*c^2*d^2*e^14*x + 600*a*c^3*d^2*e^14*x - 240*b^3*c*d*e^15*x - 72
0*a*b*c^2*d*e^15*x + 15*b^4*e^16*x + 180*a*b^2*c*e^16*x + 90*a^2*c^2*e^16*x)/e^20

Mupad [B] (verification not implemented)

Time = 9.97 (sec) , antiderivative size = 1143, normalized size of antiderivative = 2.74 \[ \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx=x^4\,\left (\frac {b\,c^3}{e^4}-\frac {c^4\,d}{e^5}\right )-x^2\,\left (\frac {2\,c^4\,d^3}{e^7}+\frac {3\,d^2\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e^2}-\frac {2\,d\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^4}+\frac {6\,c^4\,d^2}{e^6}\right )}{e}-\frac {2\,b\,c\,\left (b^2+3\,a\,c\right )}{e^4}\right )-x^3\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{3\,e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{3\,e^4}+\frac {2\,c^4\,d^2}{e^6}\right )+x\,\left (\frac {6\,a^2\,c^2+12\,a\,b^2\,c+b^4}{e^4}+\frac {6\,d^2\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^4}+\frac {6\,c^4\,d^2}{e^6}\right )}{e^2}+\frac {4\,d\,\left (\frac {4\,c^4\,d^3}{e^7}+\frac {6\,d^2\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e^2}-\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^4}+\frac {6\,c^4\,d^2}{e^6}\right )}{e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^4}\right )}{e}-\frac {c^4\,d^4}{e^8}-\frac {4\,d^3\,\left (\frac {4\,b\,c^3}{e^4}-\frac {4\,c^4\,d}{e^5}\right )}{e^3}\right )-\frac {x\,\left (2\,a^3\,b\,e^7+4\,a^3\,c\,d\,e^6+6\,a^2\,b^2\,d\,e^6-54\,a^2\,b\,c\,d^2\,e^5+60\,a^2\,c^2\,d^3\,e^4-18\,a\,b^3\,d^2\,e^5+120\,a\,b^2\,c\,d^3\,e^4-210\,a\,b\,c^2\,d^4\,e^3+108\,a\,c^3\,d^5\,e^2+10\,b^4\,d^3\,e^4-70\,b^3\,c\,d^4\,e^3+162\,b^2\,c^2\,d^5\,e^2-154\,b\,c^3\,d^6\,e+52\,c^4\,d^7\right )+\frac {a^4\,e^8+2\,a^3\,b\,d\,e^7+4\,a^3\,c\,d^2\,e^6+6\,a^2\,b^2\,d^2\,e^6-66\,a^2\,b\,c\,d^3\,e^5+78\,a^2\,c^2\,d^4\,e^4-22\,a\,b^3\,d^3\,e^5+156\,a\,b^2\,c\,d^4\,e^4-282\,a\,b\,c^2\,d^5\,e^3+148\,a\,c^3\,d^6\,e^2+13\,b^4\,d^4\,e^4-94\,b^3\,c\,d^5\,e^3+222\,b^2\,c^2\,d^6\,e^2-214\,b\,c^3\,d^7\,e+73\,c^4\,d^8}{3\,e}+x^2\,\left (4\,a^3\,c\,e^7+6\,a^2\,b^2\,e^7-36\,a^2\,b\,c\,d\,e^6+36\,a^2\,c^2\,d^2\,e^5-12\,a\,b^3\,d\,e^6+72\,a\,b^2\,c\,d^2\,e^5-120\,a\,b\,c^2\,d^3\,e^4+60\,a\,c^3\,d^4\,e^3+6\,b^4\,d^2\,e^5-40\,b^3\,c\,d^3\,e^4+90\,b^2\,c^2\,d^4\,e^3-84\,b\,c^3\,d^5\,e^2+28\,c^4\,d^6\,e\right )}{d^3\,e^8+3\,d^2\,e^9\,x+3\,d\,e^{10}\,x^2+e^{11}\,x^3}+\frac {c^4\,x^5}{5\,e^4}-\frac {\ln \left (d+e\,x\right )\,\left (-12\,a^2\,b\,c\,e^5+24\,a^2\,c^2\,d\,e^4-4\,a\,b^3\,e^5+48\,a\,b^2\,c\,d\,e^4-120\,a\,b\,c^2\,d^2\,e^3+80\,a\,c^3\,d^3\,e^2+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\right )}{e^9} \]

[In]

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

[Out]

x^4*((b*c^3)/e^4 - (c^4*d)/e^5) - x^2*((2*c^4*d^3)/e^7 + (3*d^2*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e^2 - (2*d*((
4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - (4*a*c^3 + 6*b^2*c^2)/e^4 + (6*c^4*d^2)/e^6))/e - (2*b*c*(3*a*c + b^2
))/e^4) - x^3*((4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/(3*e) - (4*a*c^3 + 6*b^2*c^2)/(3*e^4) + (2*c^4*d^2)/e^6)
+ x*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^4 + (6*d^2*((4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - (4*a*c^3 + 6*b^2*c
^2)/e^4 + (6*c^4*d^2)/e^6))/e^2 + (4*d*((4*c^4*d^3)/e^7 + (6*d^2*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e^2 - (4*d*(
(4*d*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e - (4*a*c^3 + 6*b^2*c^2)/e^4 + (6*c^4*d^2)/e^6))/e - (4*b*c*(3*a*c + b^
2))/e^4))/e - (c^4*d^4)/e^8 - (4*d^3*((4*b*c^3)/e^4 - (4*c^4*d)/e^5))/e^3) - (x*(52*c^4*d^7 + 2*a^3*b*e^7 + 10
*b^4*d^3*e^4 - 18*a*b^3*d^2*e^5 + 6*a^2*b^2*d*e^6 + 108*a*c^3*d^5*e^2 - 70*b^3*c*d^4*e^3 + 60*a^2*c^2*d^3*e^4
+ 162*b^2*c^2*d^5*e^2 + 4*a^3*c*d*e^6 - 154*b*c^3*d^6*e - 210*a*b*c^2*d^4*e^3 + 120*a*b^2*c*d^3*e^4 - 54*a^2*b
*c*d^2*e^5) + (a^4*e^8 + 73*c^4*d^8 + 13*b^4*d^4*e^4 - 22*a*b^3*d^3*e^5 + 148*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6
- 94*b^3*c*d^5*e^3 + 6*a^2*b^2*d^2*e^6 + 78*a^2*c^2*d^4*e^4 + 222*b^2*c^2*d^6*e^2 + 2*a^3*b*d*e^7 - 214*b*c^3*
d^7*e - 282*a*b*c^2*d^5*e^3 + 156*a*b^2*c*d^4*e^4 - 66*a^2*b*c*d^3*e^5)/(3*e) + x^2*(4*a^3*c*e^7 + 28*c^4*d^6*
e + 6*a^2*b^2*e^7 + 6*b^4*d^2*e^5 + 60*a*c^3*d^4*e^3 - 84*b*c^3*d^5*e^2 - 40*b^3*c*d^3*e^4 + 36*a^2*c^2*d^2*e^
5 + 90*b^2*c^2*d^4*e^3 - 12*a*b^3*d*e^6 - 36*a^2*b*c*d*e^6 - 120*a*b*c^2*d^3*e^4 + 72*a*b^2*c*d^2*e^5))/(d^3*e
^8 + e^11*x^3 + 3*d^2*e^9*x + 3*d*e^10*x^2) + (c^4*x^5)/(5*e^4) - (log(d + e*x)*(56*c^4*d^5 - 4*a*b^3*e^5 + 4*
b^4*d*e^4 + 80*a*c^3*d^3*e^2 + 24*a^2*c^2*d*e^4 - 40*b^3*c*d^2*e^3 + 120*b^2*c^2*d^3*e^2 - 12*a^2*b*c*e^5 - 14
0*b*c^3*d^4*e + 48*a*b^2*c*d*e^4 - 120*a*b*c^2*d^2*e^3))/e^9