\(\int \frac {(a+b x+c x^2)^3}{(d+e x)^2} \, dx\) [2137]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.22 (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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac {(d+e x)^2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7}+\frac {3 x \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}-\frac {\left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)}-\frac {3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac {3 c^2 (d+e x)^4 (2 c d-b e)}{4 e^7}+\frac {c^3 (d+e x)^5}{5 e^7} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {20 e \left (5 c^3 d^4+3 c^2 d^2 e (-4 b d+3 a e)+b^2 e^3 (-2 b d+3 a e)+3 c e^2 \left (3 b^2 d^2-4 a b d e+a^2 e^2\right )\right ) x+10 e^2 (-c d+b e) \left (4 c^2 d^2+b^2 e^2+c e (-5 b d+6 a e)\right ) x^2+20 c e^3 \left (c^2 d^2+b^2 e^2+c e (-2 b d+a e)\right ) x^3+5 c^2 e^4 (-2 c d+3 b e) x^4+4 c^3 e^5 x^5-\frac {20 \left (c d^2+e (-b d+a e)\right )^3}{d+e x}-60 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^2 \log (d+e x)}{20 e^7} \]

[In]

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

[Out]

(20*e*(5*c^3*d^4 + 3*c^2*d^2*e*(-4*b*d + 3*a*e) + b^2*e^3*(-2*b*d + 3*a*e) + 3*c*e^2*(3*b^2*d^2 - 4*a*b*d*e +
a^2*e^2))*x + 10*e^2*(-(c*d) + b*e)*(4*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + 6*a*e))*x^2 + 20*c*e^3*(c^2*d^2 + b^2
*e^2 + c*e*(-2*b*d + a*e))*x^3 + 5*c^2*e^4*(-2*c*d + 3*b*e)*x^4 + 4*c^3*e^5*x^5 - (20*(c*d^2 + e*(-(b*d) + a*e
))^3)/(d + e*x) - 60*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2*Log[d + e*x])/(20*e^7)

Maple [A] (verified)

Time = 2.95 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.75

method	result	size
norman	\(\frac {\frac {\left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+6 d^{2} e^{4} a^{2} c +6 a \,b^{2} d^{2} e^{4}-18 a b c \,d^{3} e^{3}+12 d^{4} e^{2} c^{2} a -3 b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}-15 b \,c^{2} d^{5} e +6 c^{3} d^{6}\right ) x}{d \,e^{6}}+\frac {c^{3} x^{6}}{5 e}+\frac {\left (6 a b c \,e^{3}-4 c^{2} a d \,e^{2}+b^{3} e^{3}-4 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right ) x^{3}}{2 e^{4}}+\frac {3 \left (2 e^{4} a^{2} c +2 a \,b^{2} e^{4}-6 a b c d \,e^{3}+4 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+4 b^{2} c \,d^{2} e^{2}-5 d^{3} e b \,c^{2}+2 d^{4} c^{3}\right ) x^{2}}{2 e^{5}}+\frac {c \left (4 a c \,e^{2}+4 b^{2} e^{2}-5 b c d e +2 c^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {3 c^{2} \left (5 b e -2 c d \right ) x^{5}}{20 e^{2}}}{e x +d}+\frac {3 \left (a^{2} b \,e^{5}-2 d \,e^{4} a^{2} c -2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 d^{5} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(447\)
default	\(\frac {\frac {1}{5} c^{3} x^{5} e^{4}+\frac {3}{4} b \,c^{2} e^{4} x^{4}-\frac {1}{2} c^{3} d \,e^{3} x^{4}+a \,c^{2} e^{4} x^{3}+b^{2} c \,e^{4} x^{3}-2 b \,c^{2} d \,e^{3} x^{3}+c^{3} d^{2} e^{2} x^{3}+3 a b c \,e^{4} x^{2}-3 a \,c^{2} d \,e^{3} x^{2}+\frac {1}{2} b^{3} e^{4} x^{2}-3 b^{2} c d \,e^{3} x^{2}+\frac {9}{2} b \,c^{2} d^{2} e^{2} x^{2}-2 c^{3} d^{3} e \,x^{2}+3 e^{4} a^{2} c x +3 a \,b^{2} e^{4} x -12 a b c d \,e^{3} x +9 d^{2} e^{2} c^{2} a x -2 b^{3} d \,e^{3} x +9 d^{2} e^{2} b^{2} c x -12 d^{3} e b \,c^{2} x +5 d^{4} c^{3} x}{e^{6}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}}{e^{7} \left (e x +d \right )}+\frac {\left (3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(472\)
risch	\(\frac {c^{3} x^{5}}{5 e^{2}}-\frac {12 a b c d x}{e^{3}}-\frac {6 \ln \left (e x +d \right ) d \,a^{2} c}{e^{3}}-\frac {6 \ln \left (e x +d \right ) a \,b^{2} d}{e^{3}}-\frac {12 \ln \left (e x +d \right ) d^{3} c^{2} a}{e^{5}}+\frac {5 d^{4} c^{3} x}{e^{6}}+\frac {b^{3} d^{3}}{e^{4} \left (e x +d \right )}-\frac {c^{3} d^{6}}{e^{7} \left (e x +d \right )}+\frac {3 \ln \left (e x +d \right ) a^{2} b}{e^{2}}+\frac {3 \ln \left (e x +d \right ) b^{3} d^{2}}{e^{4}}-\frac {6 \ln \left (e x +d \right ) d^{5} c^{3}}{e^{7}}-\frac {12 \ln \left (e x +d \right ) b^{2} c \,d^{3}}{e^{5}}+\frac {15 \ln \left (e x +d \right ) b \,c^{2} d^{4}}{e^{6}}+\frac {18 \ln \left (e x +d \right ) a b c \,d^{2}}{e^{4}}+\frac {6 a b c \,d^{3}}{e^{4} \left (e x +d \right )}-\frac {2 b \,c^{2} d \,x^{3}}{e^{3}}+\frac {3 a b c \,x^{2}}{e^{2}}+\frac {b^{3} x^{2}}{2 e^{2}}-\frac {a^{3}}{e \left (e x +d \right )}-\frac {12 d^{3} b \,c^{2} x}{e^{5}}+\frac {3 a^{2} b d}{e^{2} \left (e x +d \right )}-\frac {3 d^{2} a^{2} c}{e^{3} \left (e x +d \right )}-\frac {3 a \,b^{2} d^{2}}{e^{3} \left (e x +d \right )}-\frac {3 d^{4} c^{2} a}{e^{5} \left (e x +d \right )}-\frac {3 b^{2} c \,d^{4}}{e^{5} \left (e x +d \right )}+\frac {3 b \,c^{2} d^{5}}{e^{6} \left (e x +d \right )}-\frac {3 a \,c^{2} d \,x^{2}}{e^{3}}-\frac {3 b^{2} c d \,x^{2}}{e^{3}}+\frac {9 b \,c^{2} d^{2} x^{2}}{2 e^{4}}+\frac {9 d^{2} c^{2} a x}{e^{4}}+\frac {9 d^{2} b^{2} c x}{e^{4}}+\frac {3 b \,c^{2} x^{4}}{4 e^{2}}+\frac {a \,c^{2} x^{3}}{e^{2}}+\frac {b^{2} c \,x^{3}}{e^{2}}+\frac {c^{3} d^{2} x^{3}}{e^{4}}-\frac {2 c^{3} d^{3} x^{2}}{e^{5}}+\frac {3 a^{2} c x}{e^{2}}+\frac {3 a \,b^{2} x}{e^{2}}-\frac {2 b^{3} d x}{e^{3}}-\frac {c^{3} d \,x^{4}}{2 e^{3}}\)	\(585\)
parallelrisch	\(\frac {20 a \,c^{2} e^{6} x^{4}+10 c^{3} d^{2} e^{4} x^{4}-240 \ln \left (e x +d \right ) x \,b^{2} c \,d^{3} e^{3}+360 a b c \,d^{3} e^{3}+60 b^{3} d^{3} e^{3}-240 d^{4} e^{2} c^{2} a -120 d^{2} e^{4} a^{2} c +360 \ln \left (e x +d \right ) x a b c \,d^{2} e^{4}-20 x^{3} c^{3} d^{3} e^{3}+60 x^{2} c^{3} d^{4} e^{2}+360 \ln \left (e x +d \right ) a b c \,d^{3} e^{3}-120 c^{3} d^{6}-180 x^{2} a b c d \,e^{5}+4 x^{6} c^{3} e^{6}-240 b^{2} c \,d^{4} e^{2}+60 a^{2} b d \,e^{5}-120 a \,b^{2} d^{2} e^{4}-20 e^{6} a^{3}-120 \ln \left (e x +d \right ) c^{3} d^{6}-40 x^{3} a \,c^{2} d \,e^{5}+120 x^{2} a \,c^{2} d^{2} e^{4}-6 x^{5} c^{3} d \,e^{5}+60 x^{2} a^{2} c \,e^{6}+300 b \,c^{2} d^{5} e +15 x^{5} b \,c^{2} e^{6}+20 x^{4} b^{2} c \,e^{6}+60 x^{2} a \,b^{2} e^{6}-30 x^{2} b^{3} d \,e^{5}+60 \ln \left (e x +d \right ) b^{3} d^{3} e^{3}+10 x^{3} b^{3} e^{6}+60 \ln \left (e x +d \right ) x \,a^{2} b \,e^{6}+60 \ln \left (e x +d \right ) x \,b^{3} d^{2} e^{4}-240 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}-240 \ln \left (e x +d \right ) b^{2} c \,d^{4} e^{2}+300 \ln \left (e x +d \right ) b \,c^{2} d^{5} e +120 x^{2} b^{2} c \,d^{2} e^{4}-150 x^{2} b \,c^{2} d^{3} e^{3}-25 x^{4} b \,c^{2} d \,e^{5}+60 x^{3} a b c \,e^{6}-40 x^{3} b^{2} c d \,e^{5}+50 x^{3} b \,c^{2} d^{2} e^{4}+60 \ln \left (e x +d \right ) a^{2} b d \,e^{5}-120 \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{4}-120 \ln \left (e x +d \right ) a \,b^{2} d^{2} e^{4}-120 \ln \left (e x +d \right ) x \,a^{2} c d \,e^{5}-120 \ln \left (e x +d \right ) x a \,b^{2} d \,e^{5}-120 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +300 \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{2}-240 \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{3}}{20 e^{7} \left (e x +d \right )}\)	\(701\)

[In]

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

[Out]

((a^3*e^6-3*a^2*b*d*e^5+6*a^2*c*d^2*e^4+6*a*b^2*d^2*e^4-18*a*b*c*d^3*e^3+12*a*c^2*d^4*e^2-3*b^3*d^3*e^3+12*b^2
*c*d^4*e^2-15*b*c^2*d^5*e+6*c^3*d^6)/d/e^6*x+1/5*c^3*x^6/e+1/2*(6*a*b*c*e^3-4*a*c^2*d*e^2+b^3*e^3-4*b^2*c*d*e^
2+5*b*c^2*d^2*e-2*c^3*d^3)/e^4*x^3+3/2*(2*a^2*c*e^4+2*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^2-5*b*c^2*d^3*e+2*c^3*d^4)/e^5*x^2+1/4*c*(4*a*c*e^2+4*b^2*e^2-5*b*c*d*e+2*c^2*d^2)/e^3*x^4+3/20*c^2*(5
*b*e-2*c*d)/e^2*x^5)/(e*x+d)+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^3-4*a*c^2*d^3*e^2+b^3*d^2*
e^3-4*b^2*c*d^3*e^2+5*b*c^2*d^4*e-2*c^3*d^5)/e^7*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (250) = 500\).

Time = 0.39 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.27 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {4 \, c^{3} e^{6} x^{6} - 20 \, c^{3} d^{6} + 60 \, b c^{2} d^{5} e + 60 \, a^{2} b d e^{5} - 20 \, a^{3} e^{6} - 60 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 20 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 60 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \, {\left (2 \, c^{3} d e^{5} - 5 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 10 \, {\left (2 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 30 \, {\left (2 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 20 \, {\left (5 \, c^{3} d^{5} e - 12 \, b c^{2} d^{4} e^{2} + 9 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 2 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 60 \, {\left (2 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e - a^{2} b d e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{8} x + d e^{7}\right )}} \]

[In]

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

[Out]

1/20*(4*c^3*e^6*x^6 - 20*c^3*d^6 + 60*b*c^2*d^5*e + 60*a^2*b*d*e^5 - 20*a^3*e^6 - 60*(b^2*c + a*c^2)*d^4*e^2 +
20*(b^3 + 6*a*b*c)*d^3*e^3 - 60*(a*b^2 + a^2*c)*d^2*e^4 - 3*(2*c^3*d*e^5 - 5*b*c^2*e^6)*x^5 + 5*(2*c^3*d^2*e^
4 - 5*b*c^2*d*e^5 + 4*(b^2*c + a*c^2)*e^6)*x^4 - 10*(2*c^3*d^3*e^3 - 5*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 + 30*(2*c^3*d^4*e^2 - 5*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 + 2*(a*b^2 + a^2*c)*e^6)*x^2 + 20*(5*c^3*d^5*e - 12*b*c^2*d^4*e^2 + 9*(b^2*c + a*c^2)*d^3*e^3 - 2*(b^3
+ 6*a*b*c)*d^2*e^4 + 3*(a*b^2 + a^2*c)*d*e^5)*x - 60*(2*c^3*d^6 - 5*b*c^2*d^5*e - a^2*b*d*e^5 + 4*(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 + (2*c^3*d^5*e - 5*b*c^2*d^4*e^2 - 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 + 2*(a*b^2 + a^2*c)*d*e^5)*x)*log(e*x + d))/(e^8*x + d
*e^7)

Sympy [A] (verification not implemented)

Time = 1.05 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {c^{3} x^{5}}{5 e^{2}} + x^{4} \cdot \left (\frac {3 b c^{2}}{4 e^{2}} - \frac {c^{3} d}{2 e^{3}}\right ) + x^{3} \left (\frac {a c^{2}}{e^{2}} + \frac {b^{2} c}{e^{2}} - \frac {2 b c^{2} d}{e^{3}} + \frac {c^{3} d^{2}}{e^{4}}\right ) + x^{2} \cdot \left (\frac {3 a b c}{e^{2}} - \frac {3 a c^{2} d}{e^{3}} + \frac {b^{3}}{2 e^{2}} - \frac {3 b^{2} c d}{e^{3}} + \frac {9 b c^{2} d^{2}}{2 e^{4}} - \frac {2 c^{3} d^{3}}{e^{5}}\right ) + x \left (\frac {3 a^{2} c}{e^{2}} + \frac {3 a b^{2}}{e^{2}} - \frac {12 a b c d}{e^{3}} + \frac {9 a c^{2} d^{2}}{e^{4}} - \frac {2 b^{3} d}{e^{3}} + \frac {9 b^{2} c d^{2}}{e^{4}} - \frac {12 b c^{2} d^{3}}{e^{5}} + \frac {5 c^{3} d^{4}}{e^{6}}\right ) + \frac {- a^{3} e^{6} + 3 a^{2} b d e^{5} - 3 a^{2} c d^{2} e^{4} - 3 a b^{2} d^{2} e^{4} + 6 a b c d^{3} e^{3} - 3 a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} - 3 b^{2} c d^{4} e^{2} + 3 b c^{2} d^{5} e - c^{3} d^{6}}{d e^{7} + e^{8} x} + \frac {3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \log {\left (d + e x \right )}}{e^{7}} \]

[In]

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

[Out]

c**3*x**5/(5*e**2) + x**4*(3*b*c**2/(4*e**2) - c**3*d/(2*e**3)) + x**3*(a*c**2/e**2 + b**2*c/e**2 - 2*b*c**2*d
/e**3 + c**3*d**2/e**4) + x**2*(3*a*b*c/e**2 - 3*a*c**2*d/e**3 + b**3/(2*e**2) - 3*b**2*c*d/e**3 + 9*b*c**2*d*
*2/(2*e**4) - 2*c**3*d**3/e**5) + x*(3*a**2*c/e**2 + 3*a*b**2/e**2 - 12*a*b*c*d/e**3 + 9*a*c**2*d**2/e**4 - 2*
b**3*d/e**3 + 9*b**2*c*d**2/e**4 - 12*b*c**2*d**3/e**5 + 5*c**3*d**4/e**6) + (-a**3*e**6 + 3*a**2*b*d*e**5 - 3
*a**2*c*d**2*e**4 - 3*a*b**2*d**2*e**4 + 6*a*b*c*d**3*e**3 - 3*a*c**2*d**4*e**2 + b**3*d**3*e**3 - 3*b**2*c*d*
*4*e**2 + 3*b*c**2*d**5*e - c**3*d**6)/(d*e**7 + e**8*x) + 3*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2*log(d
+ e*x)/e**7

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=-\frac {c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}}{e^{8} x + d e^{7}} + \frac {4 \, c^{3} e^{4} x^{5} - 5 \, {\left (2 \, c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} x^{4} + 20 \, {\left (c^{3} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + {\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{3} - 10 \, {\left (4 \, c^{3} d^{3} e - 9 \, b c^{2} d^{2} e^{2} + 6 \, {\left (b^{2} c + a c^{2}\right )} d e^{3} - {\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{2} + 20 \, {\left (5 \, c^{3} d^{4} - 12 \, b c^{2} d^{3} e + 9 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - 2 \, {\left (b^{3} + 6 \, a b c\right )} d e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x}{20 \, e^{6}} - \frac {3 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]

[In]

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

[Out]

-(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*
(a*b^2 + a^2*c)*d^2*e^4)/(e^8*x + d*e^7) + 1/20*(4*c^3*e^4*x^5 - 5*(2*c^3*d*e^3 - 3*b*c^2*e^4)*x^4 + 20*(c^3*d
^2*e^2 - 2*b*c^2*d*e^3 + (b^2*c + a*c^2)*e^4)*x^3 - 10*(4*c^3*d^3*e - 9*b*c^2*d^2*e^2 + 6*(b^2*c + a*c^2)*d*e^
3 - (b^3 + 6*a*b*c)*e^4)*x^2 + 20*(5*c^3*d^4 - 12*b*c^2*d^3*e + 9*(b^2*c + a*c^2)*d^2*e^2 - 2*(b^3 + 6*a*b*c)*
d*e^3 + 3*(a*b^2 + a^2*c)*e^4)*x)/e^6 - 3*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 -
(b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*log(e*x + d)/e^7

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (250) = 500\).

Time = 0.28 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {{\left (4 \, c^{3} - \frac {15 \, {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {20 \, {\left (5 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + b^{2} c e^{4} + a c^{2} e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {10 \, {\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} - b^{3} e^{6} - 6 \, a b c e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {60 \, {\left (5 \, c^{3} d^{4} e^{4} - 10 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} + 6 \, a c^{2} d^{2} e^{6} - b^{3} d e^{7} - 6 \, a b c d e^{7} + a b^{2} e^{8} + a^{2} c e^{8}\right )}}{{\left (e x + d\right )}^{4} e^{4}}\right )} {\left (e x + d\right )}^{5}}{20 \, e^{7}} + \frac {3 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{7}} - \frac {\frac {c^{3} d^{6} e^{5}}{e x + d} - \frac {3 \, b c^{2} d^{5} e^{6}}{e x + d} + \frac {3 \, b^{2} c d^{4} e^{7}}{e x + d} + \frac {3 \, a c^{2} d^{4} e^{7}}{e x + d} - \frac {b^{3} d^{3} e^{8}}{e x + d} - \frac {6 \, a b c d^{3} e^{8}}{e x + d} + \frac {3 \, a b^{2} d^{2} e^{9}}{e x + d} + \frac {3 \, a^{2} c d^{2} e^{9}}{e x + d} - \frac {3 \, a^{2} b d e^{10}}{e x + d} + \frac {a^{3} e^{11}}{e x + d}}{e^{12}} \]

[In]

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

[Out]

1/20*(4*c^3 - 15*(2*c^3*d*e - b*c^2*e^2)/((e*x + d)*e) + 20*(5*c^3*d^2*e^2 - 5*b*c^2*d*e^3 + b^2*c*e^4 + a*c^2
*e^4)/((e*x + d)^2*e^2) - 10*(20*c^3*d^3*e^3 - 30*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)/((e*x + d)^3*e^3) + 60*(5*c^3*d^4*e^4 - 10*b*c^2*d^3*e^5 + 6*b^2*c*d^2*e^6 + 6*a*c^2*d^2*e^6 - b^
3*d*e^7 - 6*a*b*c*d*e^7 + a*b^2*e^8 + a^2*c*e^8)/((e*x + d)^4*e^4))*(e*x + d)^5/e^7 + 3*(2*c^3*d^5 - 5*b*c^2*d
^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 + 2*a*b^2*d*e^4 + 2*a^2*c*d*e^4 - a^2
*b*e^5)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^7 - (c^3*d^6*e^5/(e*x + d) - 3*b*c^2*d^5*e^6/(e*x + d) + 3*b^
2*c*d^4*e^7/(e*x + d) + 3*a*c^2*d^4*e^7/(e*x + d) - b^3*d^3*e^8/(e*x + d) - 6*a*b*c*d^3*e^8/(e*x + d) + 3*a*b^
2*d^2*e^9/(e*x + d) + 3*a^2*c*d^2*e^9/(e*x + d) - 3*a^2*b*d*e^10/(e*x + d) + a^3*e^11/(e*x + d))/e^12

Mupad [B] (verification not implemented)

Time = 10.12 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.31 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=x^2\,\left (\frac {b^3+6\,a\,c\,b}{2\,e^2}+\frac {d\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}+\frac {c^3\,d^2}{e^4}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{2\,e^2}\right )+x^4\,\left (\frac {3\,b\,c^2}{4\,e^2}-\frac {c^3\,d}{2\,e^3}\right )-x^3\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{3\,e}+\frac {c^3\,d^2}{3\,e^4}-\frac {c\,\left (b^2+a\,c\right )}{e^2}\right )+x\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}+\frac {c^3\,d^2}{e^4}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^2}\right )}{e^2}-\frac {2\,d\,\left (\frac {b^3+6\,a\,c\,b}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}+\frac {c^3\,d^2}{e^4}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e^2}\right )}{e}+\frac {3\,a\,\left (b^2+a\,c\right )}{e^2}\right )-\frac {a^3\,e^6-3\,a^2\,b\,d\,e^5+3\,a^2\,c\,d^2\,e^4+3\,a\,b^2\,d^2\,e^4-6\,a\,b\,c\,d^3\,e^3+3\,a\,c^2\,d^4\,e^2-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}{e\,\left (x\,e^7+d\,e^6\right )}+\frac {c^3\,x^5}{5\,e^2}-\frac {\ln \left (d+e\,x\right )\,\left (-3\,a^2\,b\,e^5+6\,a^2\,c\,d\,e^4+6\,a\,b^2\,d\,e^4-18\,a\,b\,c\,d^2\,e^3+12\,a\,c^2\,d^3\,e^2-3\,b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2-15\,b\,c^2\,d^4\,e+6\,c^3\,d^5\right )}{e^7} \]

[In]

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

[Out]

x^2*((b^3 + 6*a*b*c)/(2*e^2) + (d*((2*d*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/e + (c^3*d^2)/e^4 - (3*c*(a*c + b^2))
/e^2))/e - (d^2*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/(2*e^2)) + x^4*((3*b*c^2)/(4*e^2) - (c^3*d)/(2*e^3)) - x^3*((
2*d*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/(3*e) + (c^3*d^2)/(3*e^4) - (c*(a*c + b^2))/e^2) + x*((d^2*((2*d*((3*b*c^
2)/e^2 - (2*c^3*d)/e^3))/e + (c^3*d^2)/e^4 - (3*c*(a*c + b^2))/e^2))/e^2 - (2*d*((b^3 + 6*a*b*c)/e^2 + (2*d*((
2*d*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/e + (c^3*d^2)/e^4 - (3*c*(a*c + b^2))/e^2))/e - (d^2*((3*b*c^2)/e^2 - (2*
c^3*d)/e^3))/e^2))/e + (3*a*(a*c + b^2))/e^2) - (a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d
^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3)/(e*(d*e^6 + e^7*
x)) + (c^3*x^5)/(5*e^2) - (log(d + e*x)*(6*c^3*d^5 - 3*a^2*b*e^5 - 3*b^3*d^2*e^3 + 12*a*c^2*d^3*e^2 + 12*b^2*c
*d^3*e^2 + 6*a*b^2*d*e^4 + 6*a^2*c*d*e^4 - 15*b*c^2*d^4*e - 18*a*b*c*d^2*e^3))/e^7