\(\int \frac {(a+b x+c x^2)^3}{(d+e x)^3} \, dx\) [2138]

Optimal result

Integrand size = 20, antiderivative size = 255 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {\left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right ) x}{e^6}+\frac {3 c \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right ) x^2}{2 e^5}-\frac {c^2 (c d-b e) x^3}{e^4}+\frac {c^3 x^4}{4 e^3}-\frac {\left (c d^2-b d e+a e^2\right )^3}{2 e^7 (d+e x)^2}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \log (d+e x)}{e^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 255, 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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=-\frac {x \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )}{e^6}+\frac {3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac {3 c x^2 \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )}{2 e^5}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^3}{2 e^7 (d+e x)^2}-\frac {c^2 x^3 (c d-b e)}{e^4}+\frac {c^3 x^4}{4 e^3} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {4 e \left (-10 c^3 d^3+b^3 e^3+9 c^2 d e (2 b d-a e)+3 b c e^2 (-3 b d+2 a e)\right ) x+6 c e^2 \left (2 c^2 d^2+b^2 e^2+c e (-3 b d+a e)\right ) x^2+4 c^2 e^3 (-c d+b e) x^3+c^3 e^4 x^4-\frac {2 \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^2}+\frac {12 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^2}{d+e x}+12 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{4 e^7} \]

[In]

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

[Out]

(4*e*(-10*c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c*e^2*(-3*b*d + 2*a*e))*x + 6*c*e^2*(2*c^2*d^2 + b
^2*e^2 + c*e*(-3*b*d + a*e))*x^2 + 4*c^2*e^3*(-(c*d) + b*e)*x^3 + c^3*e^4*x^4 - (2*(c*d^2 + e*(-(b*d) + a*e))^
3)/(d + e*x)^2 + (12*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x) + 12*(5*c^3*d^4 + b^2*e^3*(-(b*d) +
a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*Log[d + e*x])/(4*e^7)

Maple [A] (verified)

Time = 3.00 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.73

method	result	size
norman	\(\frac {\frac {\left (6 a b c \,e^{3}-6 c^{2} a d \,e^{2}+b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right ) x^{3}}{e^{4}}-\frac {e^{6} a^{3}+3 a^{2} b d \,e^{5}-9 d^{2} e^{4} a^{2} c -9 a \,b^{2} d^{2} e^{4}+54 a b c \,d^{3} e^{3}-54 d^{4} e^{2} c^{2} a +9 b^{3} d^{3} e^{3}-54 b^{2} c \,d^{4} e^{2}+90 b \,c^{2} d^{5} e -45 c^{3} d^{6}}{2 e^{7}}+\frac {c^{3} x^{6}}{4 e}-\frac {\left (3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+36 a b c \,d^{2} e^{3}-36 d^{3} e^{2} c^{2} a +6 b^{3} d^{2} e^{3}-36 b^{2} c \,d^{3} e^{2}+60 b \,c^{2} d^{4} e -30 d^{5} c^{3}\right ) x}{e^{6}}+\frac {c \left (6 a c \,e^{2}+6 b^{2} e^{2}-10 b c d e +5 c^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {c^{2} \left (2 b e -c d \right ) x^{5}}{2 e^{2}}}{\left (e x +d \right )^{2}}+\frac {3 \left (e^{4} a^{2} c +a \,b^{2} e^{4}-6 a b c d \,e^{3}+6 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+6 b^{2} c \,d^{2} e^{2}-10 d^{3} e b \,c^{2}+5 d^{4} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(441\)
default	\(\frac {\frac {1}{4} c^{3} x^{4} e^{3}+b \,c^{2} e^{3} x^{3}-c^{3} d \,e^{2} x^{3}+\frac {3}{2} a \,c^{2} e^{3} x^{2}+\frac {3}{2} b^{2} c \,e^{3} x^{2}-\frac {9}{2} b \,c^{2} d \,e^{2} x^{2}+3 c^{3} d^{2} e \,x^{2}+6 a b c \,e^{3} x -9 a \,c^{2} d \,e^{2} x +b^{3} e^{3} x -9 b^{2} d \,e^{2} c x +18 b \,c^{2} d^{2} e x -10 c^{3} d^{3} x}{e^{6}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{e^{7} \left (e x +d \right )}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}}{2 e^{7} \left (e x +d \right )^{2}}+\frac {\left (3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(461\)
risch	\(\frac {c^{3} x^{4}}{4 e^{3}}+\frac {b \,c^{2} x^{3}}{e^{3}}-\frac {c^{3} d \,x^{3}}{e^{4}}+\frac {3 a \,c^{2} x^{2}}{2 e^{3}}+\frac {3 b^{2} c \,x^{2}}{2 e^{3}}-\frac {9 b \,c^{2} d \,x^{2}}{2 e^{4}}+\frac {3 c^{3} d^{2} x^{2}}{e^{5}}+\frac {6 a b c x}{e^{3}}-\frac {9 a \,c^{2} d x}{e^{4}}+\frac {b^{3} x}{e^{3}}-\frac {9 b^{2} d c x}{e^{4}}+\frac {18 b \,c^{2} d^{2} x}{e^{5}}-\frac {10 c^{3} d^{3} x}{e^{6}}+\frac {\left (-3 a^{2} b \,e^{5}+6 d \,e^{4} a^{2} c +6 a \,b^{2} d \,e^{4}-18 a b c \,d^{2} e^{3}+12 d^{3} e^{2} c^{2} a -3 b^{3} d^{2} e^{3}+12 b^{2} c \,d^{3} e^{2}-15 b \,c^{2} d^{4} e +6 d^{5} c^{3}\right ) x -\frac {e^{6} a^{3}+3 a^{2} b d \,e^{5}-9 d^{2} e^{4} a^{2} c -9 a \,b^{2} d^{2} e^{4}+30 a b c \,d^{3} e^{3}-21 d^{4} e^{2} c^{2} a +5 b^{3} d^{3} e^{3}-21 b^{2} c \,d^{4} e^{2}+27 b \,c^{2} d^{5} e -11 c^{3} d^{6}}{2 e}}{e^{6} \left (e x +d \right )^{2}}+\frac {3 \ln \left (e x +d \right ) a^{2} c}{e^{3}}+\frac {3 \ln \left (e x +d \right ) a \,b^{2}}{e^{3}}-\frac {18 \ln \left (e x +d \right ) a b c d}{e^{4}}+\frac {18 \ln \left (e x +d \right ) d^{2} c^{2} a}{e^{5}}-\frac {3 \ln \left (e x +d \right ) b^{3} d}{e^{4}}+\frac {18 \ln \left (e x +d \right ) b^{2} c \,d^{2}}{e^{5}}-\frac {30 \ln \left (e x +d \right ) d^{3} b \,c^{2}}{e^{6}}+\frac {15 \ln \left (e x +d \right ) d^{4} c^{3}}{e^{7}}\)	\(501\)
parallelrisch	\(\frac {6 a \,c^{2} e^{6} x^{4}+5 c^{3} d^{2} e^{4} x^{4}+144 \ln \left (e x +d \right ) x \,b^{2} c \,d^{3} e^{3}-108 a b c \,d^{3} e^{3}-18 b^{3} d^{3} e^{3}+108 d^{4} e^{2} c^{2} a +18 d^{2} e^{4} a^{2} c -72 \ln \left (e x +d \right ) x^{2} a b c d \,e^{5}-144 \ln \left (e x +d \right ) x a b c \,d^{2} e^{4}-20 x^{3} c^{3} d^{3} e^{3}-72 \ln \left (e x +d \right ) a b c \,d^{3} e^{3}+90 c^{3} d^{6}-144 x a b c \,d^{2} e^{4}+24 x \,a^{2} c d \,e^{5}+144 x a \,c^{2} d^{3} e^{3}+x^{6} c^{3} e^{6}+108 b^{2} c \,d^{4} e^{2}-6 a^{2} b d \,e^{5}+18 a \,b^{2} d^{2} e^{4}+12 \ln \left (e x +d \right ) x^{2} a \,b^{2} e^{6}-12 \ln \left (e x +d \right ) x^{2} b^{3} d \,e^{5}-120 \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{3} e^{3}-2 e^{6} a^{3}+60 \ln \left (e x +d \right ) c^{3} d^{6}-24 x^{3} a \,c^{2} d \,e^{5}-2 x^{5} c^{3} d \,e^{5}-180 b \,c^{2} d^{5} e +120 x \,c^{3} d^{5} e +4 x^{5} b \,c^{2} e^{6}+6 x^{4} b^{2} c \,e^{6}-12 x \,a^{2} b \,e^{6}-24 x \,b^{3} d^{2} e^{4}-12 \ln \left (e x +d \right ) b^{3} d^{3} e^{3}+4 x^{3} b^{3} e^{6}+12 \ln \left (e x +d \right ) x^{2} a^{2} c \,e^{6}-24 \ln \left (e x +d \right ) x \,b^{3} d^{2} e^{4}+72 \ln \left (e x +d \right ) x^{2} b^{2} c \,d^{2} e^{4}+72 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}+72 \ln \left (e x +d \right ) b^{2} c \,d^{4} e^{2}-120 \ln \left (e x +d \right ) b \,c^{2} d^{5} e +24 x a \,b^{2} d \,e^{5}+144 x \,b^{2} c \,d^{3} e^{3}-240 x b \,c^{2} d^{4} e^{2}-10 x^{4} b \,c^{2} d \,e^{5}+24 x^{3} a b c \,e^{6}-24 x^{3} b^{2} c d \,e^{5}+40 x^{3} b \,c^{2} d^{2} e^{4}+12 \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{4}+12 \ln \left (e x +d \right ) a \,b^{2} d^{2} e^{4}+60 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}+24 \ln \left (e x +d \right ) x \,a^{2} c d \,e^{5}+24 \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 -240 \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{2}+72 \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{2} e^{4}+144 \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{3}}{4 e^{7} \left (e x +d \right )^{2}}\)	\(821\)

[In]

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

[Out]

((6*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-6*b^2*c*d*e^2+10*b*c^2*d^2*e-5*c^3*d^3)/e^4*x^3-1/2*(a^3*e^6+3*a^2*b*d*e^5
-9*a^2*c*d^2*e^4-9*a*b^2*d^2*e^4+54*a*b*c*d^3*e^3-54*a*c^2*d^4*e^2+9*b^3*d^3*e^3-54*b^2*c*d^4*e^2+90*b*c^2*d^5
*e-45*c^3*d^6)/e^7+1/4*c^3*x^6/e-(3*a^2*b*e^5-6*a^2*c*d*e^4-6*a*b^2*d*e^4+36*a*b*c*d^2*e^3-36*a*c^2*d^3*e^2+6*
b^3*d^2*e^3-36*b^2*c*d^3*e^2+60*b*c^2*d^4*e-30*c^3*d^5)/e^6*x+1/4*c*(6*a*c*e^2+6*b^2*e^2-10*b*c*d*e+5*c^2*d^2)
/e^3*x^4+1/2*c^2*(2*b*e-c*d)/e^2*x^5)/(e*x+d)^2+3*(a^2*c*e^4+a*b^2*e^4-6*a*b*c*d*e^3+6*a*c^2*d^2*e^2-b^3*d*e^3
+6*b^2*c*d^2*e^2-10*b*c^2*d^3*e+5*c^3*d^4)/e^7*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (249) = 498\).

Time = 0.36 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.50 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {c^{3} e^{6} x^{6} + 22 \, c^{3} d^{6} - 54 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 2 \, a^{3} e^{6} + 42 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 10 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 18 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 2 \, {\left (c^{3} d e^{5} - 2 \, b c^{2} e^{6}\right )} x^{5} + {\left (5 \, c^{3} d^{2} e^{4} - 10 \, b c^{2} d e^{5} + 6 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 4 \, {\left (5 \, c^{3} d^{3} e^{3} - 10 \, b c^{2} d^{2} e^{4} + 6 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 2 \, {\left (34 \, c^{3} d^{4} e^{2} - 63 \, b c^{2} d^{3} e^{3} + 33 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 4 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5}\right )} x^{2} - 4 \, {\left (4 \, c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} - 3 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 6 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 12 \, {\left (5 \, c^{3} d^{6} - 10 \, b c^{2} d^{5} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} + {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 2 \, {\left (5 \, c^{3} d^{5} e - 10 \, b c^{2} d^{4} e^{2} + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

[In]

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

[Out]

1/4*(c^3*e^6*x^6 + 22*c^3*d^6 - 54*b*c^2*d^5*e - 6*a^2*b*d*e^5 - 2*a^3*e^6 + 42*(b^2*c + a*c^2)*d^4*e^2 - 10*(
b^3 + 6*a*b*c)*d^3*e^3 + 18*(a*b^2 + a^2*c)*d^2*e^4 - 2*(c^3*d*e^5 - 2*b*c^2*e^6)*x^5 + (5*c^3*d^2*e^4 - 10*b*
c^2*d*e^5 + 6*(b^2*c + a*c^2)*e^6)*x^4 - 4*(5*c^3*d^3*e^3 - 10*b*c^2*d^2*e^4 + 6*(b^2*c + a*c^2)*d*e^5 - (b^3
+ 6*a*b*c)*e^6)*x^3 - 2*(34*c^3*d^4*e^2 - 63*b*c^2*d^3*e^3 + 33*(b^2*c + a*c^2)*d^2*e^4 - 4*(b^3 + 6*a*b*c)*d*
e^5)*x^2 - 4*(4*c^3*d^5*e - 3*b*c^2*d^4*e^2 + 3*a^2*b*e^6 - 3*(b^2*c + a*c^2)*d^3*e^3 + 2*(b^3 + 6*a*b*c)*d^2*
e^4 - 6*(a*b^2 + a^2*c)*d*e^5)*x + 12*(5*c^3*d^6 - 10*b*c^2*d^5*e + 6*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c
)*d^3*e^3 + (a*b^2 + a^2*c)*d^2*e^4 + (5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6
*a*b*c)*d*e^5 + (a*b^2 + a^2*c)*e^6)*x^2 + 2*(5*c^3*d^5*e - 10*b*c^2*d^4*e^2 + 6*(b^2*c + a*c^2)*d^3*e^3 - (b^
3 + 6*a*b*c)*d^2*e^4 + (a*b^2 + a^2*c)*d*e^5)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e^7)

Sympy [A] (verification not implemented)

Time = 3.44 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {c^{3} x^{4}}{4 e^{3}} + x^{3} \left (\frac {b c^{2}}{e^{3}} - \frac {c^{3} d}{e^{4}}\right ) + x^{2} \cdot \left (\frac {3 a c^{2}}{2 e^{3}} + \frac {3 b^{2} c}{2 e^{3}} - \frac {9 b c^{2} d}{2 e^{4}} + \frac {3 c^{3} d^{2}}{e^{5}}\right ) + x \left (\frac {6 a b c}{e^{3}} - \frac {9 a c^{2} d}{e^{4}} + \frac {b^{3}}{e^{3}} - \frac {9 b^{2} c d}{e^{4}} + \frac {18 b c^{2} d^{2}}{e^{5}} - \frac {10 c^{3} d^{3}}{e^{6}}\right ) + \frac {- a^{3} e^{6} - 3 a^{2} b d e^{5} + 9 a^{2} c d^{2} e^{4} + 9 a b^{2} d^{2} e^{4} - 30 a b c d^{3} e^{3} + 21 a c^{2} d^{4} e^{2} - 5 b^{3} d^{3} e^{3} + 21 b^{2} c d^{4} e^{2} - 27 b c^{2} d^{5} e + 11 c^{3} d^{6} + x \left (- 6 a^{2} b e^{6} + 12 a^{2} c d e^{5} + 12 a b^{2} d e^{5} - 36 a b c d^{2} e^{4} + 24 a c^{2} d^{3} e^{3} - 6 b^{3} d^{2} e^{4} + 24 b^{2} c d^{3} e^{3} - 30 b c^{2} d^{4} e^{2} + 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac {3 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} \]

[In]

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

[Out]

c**3*x**4/(4*e**3) + x**3*(b*c**2/e**3 - c**3*d/e**4) + x**2*(3*a*c**2/(2*e**3) + 3*b**2*c/(2*e**3) - 9*b*c**2
*d/(2*e**4) + 3*c**3*d**2/e**5) + x*(6*a*b*c/e**3 - 9*a*c**2*d/e**4 + b**3/e**3 - 9*b**2*c*d/e**4 + 18*b*c**2*
d**2/e**5 - 10*c**3*d**3/e**6) + (-a**3*e**6 - 3*a**2*b*d*e**5 + 9*a**2*c*d**2*e**4 + 9*a*b**2*d**2*e**4 - 30*
a*b*c*d**3*e**3 + 21*a*c**2*d**4*e**2 - 5*b**3*d**3*e**3 + 21*b**2*c*d**4*e**2 - 27*b*c**2*d**5*e + 11*c**3*d*
*6 + x*(-6*a**2*b*e**6 + 12*a**2*c*d*e**5 + 12*a*b**2*d*e**5 - 36*a*b*c*d**2*e**4 + 24*a*c**2*d**3*e**3 - 6*b*
*3*d**2*e**4 + 24*b**2*c*d**3*e**3 - 30*b*c**2*d**4*e**2 + 12*c**3*d**5*e))/(2*d**2*e**7 + 4*d*e**8*x + 2*e**9
*x**2) + 3*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(d + e*x)/e**7

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - a^{3} e^{6} + 21 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 5 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 9 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 6 \, {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{2 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac {c^{3} e^{3} x^{4} - 4 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 6 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{2} - 4 \, {\left (10 \, c^{3} d^{3} - 18 \, b c^{2} d^{2} e + 9 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x}{4 \, e^{6}} + \frac {3 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]

[In]

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

[Out]

1/2*(11*c^3*d^6 - 27*b*c^2*d^5*e - 3*a^2*b*d*e^5 - a^3*e^6 + 21*(b^2*c + a*c^2)*d^4*e^2 - 5*(b^3 + 6*a*b*c)*d^
3*e^3 + 9*(a*b^2 + a^2*c)*d^2*e^4 + 6*(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)/(e^9*x^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(c^3*e^3*x^4 - 4*
(c^3*d*e^2 - b*c^2*e^3)*x^3 + 6*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + (b^2*c + a*c^2)*e^3)*x^2 - 4*(10*c^3*d^3 - 18*b
*c^2*d^2*e + 9*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a*b*c)*e^3)*x)/e^6 + 3*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c
+ a*c^2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*log(e*x + d)/e^7

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.82 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {3 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} + 6 \, a c^{2} d^{2} e^{2} - b^{3} d e^{3} - 6 \, a b c d e^{3} + a b^{2} e^{4} + a^{2} c e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} + 21 \, a c^{2} d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} - 30 \, a b c d^{3} e^{3} + 9 \, a b^{2} d^{2} e^{4} + 9 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} - a^{3} e^{6} + 6 \, {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + 4 \, a c^{2} d^{3} e^{3} - b^{3} d^{2} e^{4} - 6 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} - a^{2} b e^{6}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{7}} + \frac {c^{3} e^{9} x^{4} - 4 \, c^{3} d e^{8} x^{3} + 4 \, b c^{2} e^{9} x^{3} + 12 \, c^{3} d^{2} e^{7} x^{2} - 18 \, b c^{2} d e^{8} x^{2} + 6 \, b^{2} c e^{9} x^{2} + 6 \, a c^{2} e^{9} x^{2} - 40 \, c^{3} d^{3} e^{6} x + 72 \, b c^{2} d^{2} e^{7} x - 36 \, b^{2} c d e^{8} x - 36 \, a c^{2} d e^{8} x + 4 \, b^{3} e^{9} x + 24 \, a b c e^{9} x}{4 \, e^{12}} \]

[In]

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

[Out]

3*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 + 6*a*c^2*d^2*e^2 - b^3*d*e^3 - 6*a*b*c*d*e^3 + a*b^2*e^4 + a^
2*c*e^4)*log(abs(e*x + d))/e^7 + 1/2*(11*c^3*d^6 - 27*b*c^2*d^5*e + 21*b^2*c*d^4*e^2 + 21*a*c^2*d^4*e^2 - 5*b^
3*d^3*e^3 - 30*a*b*c*d^3*e^3 + 9*a*b^2*d^2*e^4 + 9*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 - a^3*e^6 + 6*(2*c^3*d^5*e -
5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 + 4*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 - a^2*b*e^6)*x)/((e*x + d)^2*e^7) + 1/4*(c^3*e^9*x^4 - 4*c^3*d*e^8*x^3 + 4*b*c^2*e^9*x^3 + 12*c^3*d^2*e^
7*x^2 - 18*b*c^2*d*e^8*x^2 + 6*b^2*c*e^9*x^2 + 6*a*c^2*e^9*x^2 - 40*c^3*d^3*e^6*x + 72*b*c^2*d^2*e^7*x - 36*b^
2*c*d*e^8*x - 36*a*c^2*d*e^8*x + 4*b^3*e^9*x + 24*a*b*c*e^9*x)/e^12

Mupad [B] (verification not implemented)

Time = 10.16 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx=\frac {x\,\left (-3\,a^2\,b\,e^5+6\,a^2\,c\,d\,e^4+6\,a\,b^2\,d\,e^4-18\,a\,b\,c\,d^2\,e^3+12\,a\,c^2\,d^3\,e^2-3\,b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2-15\,b\,c^2\,d^4\,e+6\,c^3\,d^5\right )-\frac {a^3\,e^6+3\,a^2\,b\,d\,e^5-9\,a^2\,c\,d^2\,e^4-9\,a\,b^2\,d^2\,e^4+30\,a\,b\,c\,d^3\,e^3-21\,a\,c^2\,d^4\,e^2+5\,b^3\,d^3\,e^3-21\,b^2\,c\,d^4\,e^2+27\,b\,c^2\,d^5\,e-11\,c^3\,d^6}{2\,e}}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+x^3\,\left (\frac {b\,c^2}{e^3}-\frac {c^3\,d}{e^4}\right )+x\,\left (\frac {b^3+6\,a\,c\,b}{e^3}-\frac {c^3\,d^3}{e^6}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{e}+\frac {3\,c^3\,d^2}{e^5}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^3}\right )}{e}-\frac {3\,d^2\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{e^2}\right )-x^2\,\left (\frac {3\,d\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{2\,e}+\frac {3\,c^3\,d^2}{2\,e^5}-\frac {3\,c\,\left (b^2+a\,c\right )}{2\,e^3}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,a^2\,c\,e^4+3\,a\,b^2\,e^4-18\,a\,b\,c\,d\,e^3+18\,a\,c^2\,d^2\,e^2-3\,b^3\,d\,e^3+18\,b^2\,c\,d^2\,e^2-30\,b\,c^2\,d^3\,e+15\,c^3\,d^4\right )}{e^7}+\frac {c^3\,x^4}{4\,e^3} \]

[In]

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

[Out]

(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) - (a^3*e^6 - 11*c^3*d^6 + 5*b^3*d^3*e^3 - 9*a*b^2*d^2*e^4 - 21*a*c^2*
d^4*e^2 - 9*a^2*c*d^2*e^4 - 21*b^2*c*d^4*e^2 + 3*a^2*b*d*e^5 + 27*b*c^2*d^5*e + 30*a*b*c*d^3*e^3)/(2*e))/(d^2*
e^6 + e^8*x^2 + 2*d*e^7*x) + x^3*((b*c^2)/e^3 - (c^3*d)/e^4) + x*((b^3 + 6*a*b*c)/e^3 - (c^3*d^3)/e^6 + (3*d*(
(3*d*((3*b*c^2)/e^3 - (3*c^3*d)/e^4))/e + (3*c^3*d^2)/e^5 - (3*c*(a*c + b^2))/e^3))/e - (3*d^2*((3*b*c^2)/e^3
- (3*c^3*d)/e^4))/e^2) - x^2*((3*d*((3*b*c^2)/e^3 - (3*c^3*d)/e^4))/(2*e) + (3*c^3*d^2)/(2*e^5) - (3*c*(a*c +
b^2))/(2*e^3)) + (log(d + e*x)*(15*c^3*d^4 + 3*a*b^2*e^4 + 3*a^2*c*e^4 - 3*b^3*d*e^3 + 18*a*c^2*d^2*e^2 + 18*b
^2*c*d^2*e^2 - 30*b*c^2*d^3*e - 18*a*b*c*d*e^3))/e^7 + (c^3*x^4)/(4*e^3)