\(\int \frac {(a+b x+c x^2)^3}{(d+e x)^6} \, dx\) [2141]

Optimal result

Integrand size = 20, antiderivative size = 256 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^3 x}{e^6}-\frac {\left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^5}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{4 e^7 (d+e x)^4}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{2 e^7 (d+e x)^2}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 256, 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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac {c^3 x}{e^6} \]

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {2 c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+e^3 \left (4 a^3 e^3+3 a^2 b e^2 (d+5 e x)+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+2 c e^2 \left (a^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a b e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+6 b^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+c^2 e \left (12 a e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-b d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 c^2 (2 c d-b e) (d+e x)^5 \log (d+e x)}{20 e^7 (d+e x)^5} \]

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

-1/20*(2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^
6*x^6) + e^3*(4*a^3*e^3 + 3*a^2*b*e^2*(d + 5*e*x) + 2*a*b^2*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*
e*x + 10*d*e^2*x^2 + 10*e^3*x^3)) + 2*c*e^2*(a^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*a*b*e*(d^3 + 5*d^2*e*x +
10*d*e^2*x^2 + 10*e^3*x^3) + 6*b^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + c^2*e*(12
*a*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) - b*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2
*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + 60*c^2*(2*c*d - b*e)*(d + e*x)^5*Log[d + e*x])/(e^7*(d + e*x)^5)

Maple [A] (verified)

Time = 3.24 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.73

method	result	size
norman	\(\frac {\frac {c^{3} x^{6}}{e}-\frac {4 e^{6} a^{3}+3 a^{2} b d \,e^{5}+2 d^{2} e^{4} a^{2} c +2 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+12 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}-137 b \,c^{2} d^{5} e +274 c^{3} d^{6}}{20 e^{7}}-\frac {\left (3 a \,c^{2} e^{2}+3 b^{2} e^{2} c -15 d e b \,c^{2}+30 c^{3} d^{2}\right ) x^{4}}{e^{3}}-\frac {\left (6 a b c \,e^{3}+12 c^{2} a d \,e^{2}+b^{3} e^{3}+12 b^{2} d \,e^{2} c -90 b \,c^{2} d^{2} e +180 c^{3} d^{3}\right ) x^{3}}{2 e^{4}}-\frac {\left (2 e^{4} a^{2} c +2 a \,b^{2} e^{4}+6 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-110 d^{3} e b \,c^{2}+220 d^{4} c^{3}\right ) x^{2}}{2 e^{5}}-\frac {\left (3 a^{2} b \,e^{5}+2 d \,e^{4} a^{2} c +2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+12 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+12 b^{2} c \,d^{3} e^{2}-125 b \,c^{2} d^{4} e +250 d^{5} c^{3}\right ) x}{4 e^{6}}}{\left (e x +d \right )^{5}}+\frac {3 c^{2} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{7}}\)	\(444\)
risch	\(\frac {c^{3} x}{e^{6}}+\frac {\left (-3 a \,c^{2} e^{5}-3 b^{2} c \,e^{5}+15 b \,c^{2} d \,e^{4}-15 c^{3} d^{2} e^{3}\right ) x^{4}-\frac {e^{2} \left (6 a b c \,e^{3}+12 c^{2} a d \,e^{2}+b^{3} e^{3}+12 b^{2} d \,e^{2} c -90 b \,c^{2} d^{2} e +100 c^{3} d^{3}\right ) x^{3}}{2}-\frac {e \left (2 e^{4} a^{2} c +2 a \,b^{2} e^{4}+6 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-110 d^{3} e b \,c^{2}+130 d^{4} c^{3}\right ) x^{2}}{2}+\left (-\frac {3}{4} a^{2} b \,e^{5}-\frac {1}{2} d \,e^{4} a^{2} c -\frac {1}{2} a \,b^{2} d \,e^{4}-\frac {3}{2} a b c \,d^{2} e^{3}-3 d^{3} e^{2} c^{2} a -\frac {1}{4} b^{3} d^{2} e^{3}-3 b^{2} c \,d^{3} e^{2}+\frac {125}{4} b \,c^{2} d^{4} e -\frac {77}{2} d^{5} c^{3}\right ) x -\frac {4 e^{6} a^{3}+3 a^{2} b d \,e^{5}+2 d^{2} e^{4} a^{2} c +2 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+12 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}-137 b \,c^{2} d^{5} e +174 c^{3} d^{6}}{20 e}}{e^{6} \left (e x +d \right )^{5}}+\frac {3 c^{2} \ln \left (e x +d \right ) b}{e^{6}}-\frac {6 c^{3} d \ln \left (e x +d \right )}{e^{7}}\)	\(449\)
default	\(\frac {c^{3} x}{e^{6}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{e^{7} \left (e x +d \right )}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{4 e^{7} \left (e x +d \right )^{4}}-\frac {6 a b c \,e^{3}-12 c^{2} a d \,e^{2}+b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{2 e^{7} \left (e x +d \right )^{2}}+\frac {3 c^{2} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{7}}\)	\(453\)
parallelrisch	\(\frac {-60 a \,c^{2} e^{6} x^{4}-600 c^{3} d^{2} e^{4} x^{4}-6 a b c \,d^{3} e^{3}-b^{3} d^{3} e^{3}-12 d^{4} e^{2} c^{2} a -2 d^{2} e^{4} a^{2} c -1800 x^{3} c^{3} d^{3} e^{3}-2200 x^{2} c^{3} d^{4} e^{2}-274 c^{3} d^{6}-60 x^{2} a b c d \,e^{5}-30 x a b c \,d^{2} e^{4}-10 x \,a^{2} c d \,e^{5}-60 x a \,c^{2} d^{3} e^{3}+20 x^{6} c^{3} e^{6}-12 b^{2} c \,d^{4} e^{2}-3 a^{2} b d \,e^{5}-2 a \,b^{2} d^{2} e^{4}+600 \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{3} e^{3}-4 e^{6} a^{3}-120 \ln \left (e x +d \right ) c^{3} d^{6}-120 x^{3} a \,c^{2} d \,e^{5}-120 x^{2} a \,c^{2} d^{2} e^{4}-20 x^{2} a^{2} c \,e^{6}+137 b \,c^{2} d^{5} e -1250 x \,c^{3} d^{5} e -60 x^{4} b^{2} c \,e^{6}-20 x^{2} a \,b^{2} e^{6}-10 x^{2} b^{3} d \,e^{5}-15 x \,a^{2} b \,e^{6}-5 x \,b^{3} d^{2} e^{4}-10 x^{3} b^{3} e^{6}+60 \ln \left (e x +d \right ) x^{5} b \,c^{2} e^{6}-120 \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{5}-600 \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{4}+60 \ln \left (e x +d \right ) b \,c^{2} d^{5} e -120 x^{2} b^{2} c \,d^{2} e^{4}+1100 x^{2} b \,c^{2} d^{3} e^{3}-10 x a \,b^{2} d \,e^{5}-60 x \,b^{2} c \,d^{3} e^{3}+625 x b \,c^{2} d^{4} e^{2}+300 x^{4} b \,c^{2} d \,e^{5}-60 x^{3} a b c \,e^{6}-120 x^{3} b^{2} c d \,e^{5}+900 x^{3} b \,c^{2} d^{2} e^{4}+600 \ln \left (e x +d \right ) x^{3} b \,c^{2} d^{2} e^{4}-1200 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}-1200 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}-600 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +300 \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{2}+300 \ln \left (e x +d \right ) x^{4} b \,c^{2} d \,e^{5}}{20 e^{7} \left (e x +d \right )^{5}}\)	\(693\)

[In]

int((c*x^2+b*x+a)^3/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

(c^3*x^6/e-1/20*(4*a^3*e^6+3*a^2*b*d*e^5+2*a^2*c*d^2*e^4+2*a*b^2*d^2*e^4+6*a*b*c*d^3*e^3+12*a*c^2*d^4*e^2+b^3*
d^3*e^3+12*b^2*c*d^4*e^2-137*b*c^2*d^5*e+274*c^3*d^6)/e^7-(3*a*c^2*e^2+3*b^2*c*e^2-15*b*c^2*d*e+30*c^3*d^2)/e^
3*x^4-1/2*(6*a*b*c*e^3+12*a*c^2*d*e^2+b^3*e^3+12*b^2*c*d*e^2-90*b*c^2*d^2*e+180*c^3*d^3)/e^4*x^3-1/2*(2*a^2*c*
e^4+2*a*b^2*e^4+6*a*b*c*d*e^3+12*a*c^2*d^2*e^2+b^3*d*e^3+12*b^2*c*d^2*e^2-110*b*c^2*d^3*e+220*c^3*d^4)/e^5*x^2
-1/4*(3*a^2*b*e^5+2*a^2*c*d*e^4+2*a*b^2*d*e^4+6*a*b*c*d^2*e^3+12*a*c^2*d^3*e^2+b^3*d^2*e^3+12*b^2*c*d^3*e^2-12
5*b*c^2*d^4*e+250*c^3*d^5)/e^6*x)/(e*x+d)^5+3*c^2/e^7*(b*e-2*c*d)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (250) = 500\).

Time = 0.42 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.34 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {20 \, c^{3} e^{6} x^{6} + 100 \, c^{3} d e^{5} x^{5} - 174 \, c^{3} d^{6} + 137 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - 4 \, a^{3} e^{6} - 12 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 2 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 20 \, {\left (5 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 10 \, {\left (80 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 10 \, {\left (120 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 5 \, {\left (150 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 60 \, {\left (2 \, c^{3} d^{6} - b c^{2} d^{5} e + {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{3} d^{2} e^{4} - b c^{2} d e^{5}\right )} x^{4} + 10 \, {\left (2 \, c^{3} d^{3} e^{3} - b c^{2} d^{2} e^{4}\right )} x^{3} + 10 \, {\left (2 \, c^{3} d^{4} e^{2} - b c^{2} d^{3} e^{3}\right )} x^{2} + 5 \, {\left (2 \, c^{3} d^{5} e - b c^{2} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^6,x, algorithm="fricas")

[Out]

1/20*(20*c^3*e^6*x^6 + 100*c^3*d*e^5*x^5 - 174*c^3*d^6 + 137*b*c^2*d^5*e - 3*a^2*b*d*e^5 - 4*a^3*e^6 - 12*(b^2
*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - 2*(a*b^2 + a^2*c)*d^2*e^4 - 20*(5*c^3*d^2*e^4 - 15*b*c^2*d*e^5
+ 3*(b^2*c + a*c^2)*e^6)*x^4 - 10*(80*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 + (b^3 + 6*a*
b*c)*e^6)*x^3 - 10*(120*c^3*d^4*e^2 - 110*b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2*e^4 + (b^3 + 6*a*b*c)*d*e^5 +
2*(a*b^2 + a^2*c)*e^6)*x^2 - 5*(150*c^3*d^5*e - 125*b*c^2*d^4*e^2 + 3*a^2*b*e^6 + 12*(b^2*c + a*c^2)*d^3*e^3
+ (b^3 + 6*a*b*c)*d^2*e^4 + 2*(a*b^2 + a^2*c)*d*e^5)*x - 60*(2*c^3*d^6 - b*c^2*d^5*e + (2*c^3*d*e^5 - b*c^2*e^
6)*x^5 + 5*(2*c^3*d^2*e^4 - b*c^2*d*e^5)*x^4 + 10*(2*c^3*d^3*e^3 - b*c^2*d^2*e^4)*x^3 + 10*(2*c^3*d^4*e^2 - b*
c^2*d^3*e^3)*x^2 + 5*(2*c^3*d^5*e - b*c^2*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3
+ 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)**3/(e*x+d)**6,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=-\frac {174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 3 \, a^{2} b d e^{5} + 4 \, a^{3} e^{6} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 60 \, {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 10 \, {\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 10 \, {\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 5 \, {\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{20 \, {\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac {c^{3} x}{e^{6}} - \frac {3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \log \left (e x + d\right )}{e^{7}} \]

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 3*a^2*b*d*e^5 + 4*a^3*e^6 + 12*(b^2*c + a*c^2)*d^4*e^2 + (b^3 + 6*a*b*c
)*d^3*e^3 + 2*(a*b^2 + a^2*c)*d^2*e^4 + 60*(5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 + (b^2*c + a*c^2)*e^6)*x^4 + 10*(100
*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 + (b^3 + 6*a*b*c)*e^6)*x^3 + 10*(130*c^3*d^4*e^2 -
110*b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2*e^4 + (b^3 + 6*a*b*c)*d*e^5 + 2*(a*b^2 + a^2*c)*e^6)*x^2 + 5*(154*c
^3*d^5*e - 125*b*c^2*d^4*e^2 + 3*a^2*b*e^6 + 12*(b^2*c + a*c^2)*d^3*e^3 + (b^3 + 6*a*b*c)*d^2*e^4 + 2*(a*b^2 +
a^2*c)*d*e^5)*x)/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7) + c^3*x
/e^6 - 3*(2*c^3*d - b*c^2*e)*log(e*x + d)/e^7

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^{3} x}{e^{6}} - \frac {3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} - \frac {174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 12 \, a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 2 \, a b^{2} d^{2} e^{4} + 2 \, a^{2} c d^{2} e^{4} + 3 \, a^{2} b d e^{5} + 4 \, a^{3} e^{6} + 60 \, {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6} + a c^{2} e^{6}\right )} x^{4} + 10 \, {\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} + b^{3} e^{6} + 6 \, a b c e^{6}\right )} x^{3} + 10 \, {\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 12 \, a c^{2} d^{2} e^{4} + b^{3} d e^{5} + 6 \, a b c d e^{5} + 2 \, a b^{2} e^{6} + 2 \, a^{2} c e^{6}\right )} x^{2} + 5 \, {\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 12 \, a c^{2} d^{3} e^{3} + b^{3} d^{2} e^{4} + 6 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} + 3 \, a^{2} b e^{6}\right )} x}{20 \, {\left (e x + d\right )}^{5} e^{7}} \]

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^6,x, algorithm="giac")

[Out]

c^3*x/e^6 - 3*(2*c^3*d - b*c^2*e)*log(abs(e*x + d))/e^7 - 1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 12*b^2*c*d^4*e
^2 + 12*a*c^2*d^4*e^2 + b^3*d^3*e^3 + 6*a*b*c*d^3*e^3 + 2*a*b^2*d^2*e^4 + 2*a^2*c*d^2*e^4 + 3*a^2*b*d*e^5 + 4*
a^3*e^6 + 60*(5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 + b^2*c*e^6 + a*c^2*e^6)*x^4 + 10*(100*c^3*d^3*e^3 - 90*b*c^2*d^2*
e^4 + 12*b^2*c*d*e^5 + 12*a*c^2*d*e^5 + b^3*e^6 + 6*a*b*c*e^6)*x^3 + 10*(130*c^3*d^4*e^2 - 110*b*c^2*d^3*e^3 +
12*b^2*c*d^2*e^4 + 12*a*c^2*d^2*e^4 + b^3*d*e^5 + 6*a*b*c*d*e^5 + 2*a*b^2*e^6 + 2*a^2*c*e^6)*x^2 + 5*(154*c^3
*d^5*e - 125*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + 12*a*c^2*d^3*e^3 + b^3*d^2*e^4 + 6*a*b*c*d^2*e^4 + 2*a*b^2*d*e
^5 + 2*a^2*c*d*e^5 + 3*a^2*b*e^6)*x)/((e*x + d)^5*e^7)

Mupad [B] (verification not implemented)

Time = 9.94 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx=\frac {c^3\,x}{e^6}-\frac {x\,\left (\frac {3\,a^2\,b\,e^5}{4}+\frac {a^2\,c\,d\,e^4}{2}+\frac {a\,b^2\,d\,e^4}{2}+\frac {3\,a\,b\,c\,d^2\,e^3}{2}+3\,a\,c^2\,d^3\,e^2+\frac {b^3\,d^2\,e^3}{4}+3\,b^2\,c\,d^3\,e^2-\frac {125\,b\,c^2\,d^4\,e}{4}+\frac {77\,c^3\,d^5}{2}\right )+x^4\,\left (3\,b^2\,c\,e^5-15\,b\,c^2\,d\,e^4+15\,c^3\,d^2\,e^3+3\,a\,c^2\,e^5\right )+\frac {4\,a^3\,e^6+3\,a^2\,b\,d\,e^5+2\,a^2\,c\,d^2\,e^4+2\,a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3+12\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3+12\,b^2\,c\,d^4\,e^2-137\,b\,c^2\,d^5\,e+174\,c^3\,d^6}{20\,e}+x^2\,\left (a^2\,c\,e^5+a\,b^2\,e^5+3\,a\,b\,c\,d\,e^4+6\,a\,c^2\,d^2\,e^3+\frac {b^3\,d\,e^4}{2}+6\,b^2\,c\,d^2\,e^3-55\,b\,c^2\,d^3\,e^2+65\,c^3\,d^4\,e\right )+x^3\,\left (\frac {b^3\,e^5}{2}+6\,b^2\,c\,d\,e^4-45\,b\,c^2\,d^2\,e^3+3\,a\,b\,c\,e^5+50\,c^3\,d^3\,e^2+6\,a\,c^2\,d\,e^4\right )}{d^5\,e^6+5\,d^4\,e^7\,x+10\,d^3\,e^8\,x^2+10\,d^2\,e^9\,x^3+5\,d\,e^{10}\,x^4+e^{11}\,x^5}-\frac {\ln \left (d+e\,x\right )\,\left (6\,c^3\,d-3\,b\,c^2\,e\right )}{e^7} \]

[In]

int((a + b*x + c*x^2)^3/(d + e*x)^6,x)

[Out]

(c^3*x)/e^6 - (x*((77*c^3*d^5)/2 + (3*a^2*b*e^5)/4 + (b^3*d^2*e^3)/4 + 3*a*c^2*d^3*e^2 + 3*b^2*c*d^3*e^2 + (a*
b^2*d*e^4)/2 + (a^2*c*d*e^4)/2 - (125*b*c^2*d^4*e)/4 + (3*a*b*c*d^2*e^3)/2) + x^4*(3*a*c^2*e^5 + 3*b^2*c*e^5 +
15*c^3*d^2*e^3 - 15*b*c^2*d*e^4) + (4*a^3*e^6 + 174*c^3*d^6 + b^3*d^3*e^3 + 2*a*b^2*d^2*e^4 + 12*a*c^2*d^4*e^
2 + 2*a^2*c*d^2*e^4 + 12*b^2*c*d^4*e^2 + 3*a^2*b*d*e^5 - 137*b*c^2*d^5*e + 6*a*b*c*d^3*e^3)/(20*e) + x^2*(a*b^
2*e^5 + a^2*c*e^5 + (b^3*d*e^4)/2 + 65*c^3*d^4*e + 6*a*c^2*d^2*e^3 - 55*b*c^2*d^3*e^2 + 6*b^2*c*d^2*e^3 + 3*a*
b*c*d*e^4) + x^3*((b^3*e^5)/2 + 50*c^3*d^3*e^2 - 45*b*c^2*d^2*e^3 + 3*a*b*c*e^5 + 6*a*c^2*d*e^4 + 6*b^2*c*d*e^
4))/(d^5*e^6 + e^11*x^5 + 5*d^4*e^7*x + 5*d*e^10*x^4 + 10*d^3*e^8*x^2 + 10*d^2*e^9*x^3) - (log(d + e*x)*(6*c^3
*d - 3*b*c^2*e))/e^7