\(\int \frac {(a+b x+c x^2)^3}{d+e x} \, dx\) [2136]

Optimal result

Integrand size = 20, antiderivative size = 260 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x} \, dx=-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 x}{e^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^2}{2 e^7}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^3}{3 e^7}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^4}{4 e^7}-\frac {3 c^2 (2 c d-b e) (d+e x)^5}{5 e^7}+\frac {c^3 (d+e x)^6}{6 e^7}+\frac {\left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 260, 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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {3 c (d+e x)^4 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7}-\frac {(d+e x)^3 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7}+\frac {3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {3 x (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac {3 c^2 (d+e x)^5 (2 c d-b e)}{5 e^7}+\frac {c^3 (d+e x)^6}{6 e^7} \]

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {e x \left (c^3 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+10 b e^3 \left (18 a^2 e^2+9 a b e (-2 d+e x)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+15 c e^2 \left (6 a^2 e^2 (-2 d+e x)+4 a b e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+3 c^2 e \left (5 a e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+b \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )\right )+60 \left (c d^2+e (-b d+a e)\right )^3 \log (d+e x)}{60 e^7} \]

[In]

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

[Out]

(e*x*(c^3*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + 10*b*e^3*(18*
a^2*e^2 + 9*a*b*e*(-2*d + e*x) + b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 15*c*e^2*(6*a^2*e^2*(-2*d + e*x) + 4*a*b
*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + b^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 3*c^2*e*(5*a*e*(-12*
d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x
^4))) + 60*(c*d^2 + e*(-(b*d) + a*e))^3*Log[d + e*x])/(60*e^7)

Maple [A] (verified)

Time = 3.17 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.66

method	result	size
norman	\(\frac {\left (3 a^{2} b \,e^{5}-3 d \,e^{4} a^{2} c -3 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-3 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}-3 b^{2} c \,d^{3} e^{2}+3 b \,c^{2} d^{4} e -d^{5} c^{3}\right ) x}{e^{6}}+\frac {c^{3} x^{6}}{6 e}+\frac {\left (3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-6 a b c d \,e^{3}+3 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+3 b^{2} c \,d^{2} e^{2}-3 d^{3} e b \,c^{2}+d^{4} c^{3}\right ) x^{2}}{2 e^{5}}+\frac {\left (6 a b c \,e^{3}-3 c^{2} a d \,e^{2}+b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right ) x^{3}}{3 e^{4}}+\frac {c^{2} \left (3 b e -c d \right ) x^{5}}{5 e^{2}}+\frac {c \left (3 a c \,e^{2}+3 b^{2} e^{2}-3 b c d e +c^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {\left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(432\)
default	\(\frac {\frac {c^{3} x^{6} e^{5}}{6}+\frac {\left (\left (b e -c d \right ) e^{4} c^{2}+2 c^{2} e^{5} b \right ) x^{5}}{5}+\frac {\left (2 \left (b e -c d \right ) e^{4} b c +c e \left (3 a c \,e^{4}+b^{2} e^{4}-d \,e^{3} b c +c^{2} d^{2} e^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (b e -c d \right ) \left (3 a c \,e^{4}+b^{2} e^{4}-d \,e^{3} b c +c^{2} d^{2} e^{2}\right )+c e \left (3 a b \,e^{4}-b^{2} d \,e^{3}+b c \,d^{2} e^{2}\right )\right ) x^{3}}{3}+\frac {\left (\left (b e -c d \right ) \left (3 a b \,e^{4}-b^{2} d \,e^{3}+b c \,d^{2} e^{2}\right )+c e \left (3 a^{2} e^{4}-3 a b d \,e^{3}+3 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}\right )\right ) x^{2}}{2}+\left (b e -c d \right ) \left (3 a^{2} e^{4}-3 a b d \,e^{3}+3 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}\right ) x}{e^{6}}+\frac {\left (e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(455\)
risch	\(\frac {c^{3} x^{6}}{6 e}+\frac {3 \ln \left (e x +d \right ) a \,b^{2} d^{2}}{e^{3}}+\frac {3 \ln \left (e x +d \right ) d^{4} c^{2} a}{e^{5}}+\frac {3 \ln \left (e x +d \right ) b^{2} c \,d^{4}}{e^{5}}-\frac {3 \ln \left (e x +d \right ) b \,c^{2} d^{5}}{e^{6}}+\frac {b^{3} x^{3}}{3 e}-\frac {3 x^{2} a b c d}{e^{2}}-\frac {6 \ln \left (e x +d \right ) a b c \,d^{3}}{e^{4}}+\frac {6 a b c \,d^{2} x}{e^{3}}+\frac {x^{3} b \,c^{2} d^{2}}{e^{3}}+\frac {3 x^{2} a \,c^{2} d^{2}}{2 e^{3}}-\frac {3 \ln \left (e x +d \right ) a^{2} b d}{e^{2}}+\frac {3 \ln \left (e x +d \right ) d^{2} a^{2} c}{e^{3}}-\frac {c^{3} d^{5} x}{e^{6}}+\frac {3 a^{2} b x}{e}+\frac {b^{3} d^{2} x}{e^{3}}+\frac {3 x^{5} c^{2} b}{5 e}+\frac {3 x^{4} b^{2} c}{4 e}+\frac {3 x^{4} a \,c^{2}}{4 e}+\frac {x^{4} c^{3} d^{2}}{4 e^{3}}-\frac {x^{3} c^{3} d^{3}}{3 e^{4}}+\frac {3 x^{2} a \,b^{2}}{2 e}-\frac {x^{2} b^{3} d}{2 e^{2}}+\frac {3 x^{2} a^{2} c}{2 e}+\frac {x^{2} c^{3} d^{4}}{2 e^{5}}-\frac {\ln \left (e x +d \right ) b^{3} d^{3}}{e^{4}}+\frac {\ln \left (e x +d \right ) c^{3} d^{6}}{e^{7}}+\frac {\ln \left (e x +d \right ) a^{3}}{e}-\frac {3 a^{2} c d x}{e^{2}}-\frac {3 a \,b^{2} d x}{e^{2}}-\frac {3 a \,c^{2} d^{3} x}{e^{4}}-\frac {3 b^{2} c \,d^{3} x}{e^{4}}+\frac {3 b \,c^{2} d^{4} x}{e^{5}}+\frac {3 x^{2} b^{2} c \,d^{2}}{2 e^{3}}-\frac {3 x^{2} b \,c^{2} d^{3}}{2 e^{4}}-\frac {3 x^{4} b \,c^{2} d}{4 e^{2}}+\frac {2 x^{3} a b c}{e}-\frac {x^{3} a \,c^{2} d}{e^{2}}-\frac {x^{3} b^{2} c d}{e^{2}}-\frac {c^{3} d \,x^{5}}{5 e^{2}}\)	\(546\)
parallelrisch	\(\frac {45 a \,c^{2} e^{6} x^{4}+15 c^{3} d^{2} e^{4} x^{4}-20 x^{3} c^{3} d^{3} e^{3}+30 x^{2} c^{3} d^{4} e^{2}-360 \ln \left (e x +d \right ) a b c \,d^{3} e^{3}-180 x^{2} a b c d \,e^{5}+360 x a b c \,d^{2} e^{4}-180 x \,a^{2} c d \,e^{5}-180 x a \,c^{2} d^{3} e^{3}+10 x^{6} c^{3} e^{6}+60 \ln \left (e x +d \right ) c^{3} d^{6}-60 x^{3} a \,c^{2} d \,e^{5}+90 x^{2} a \,c^{2} d^{2} e^{4}-12 x^{5} c^{3} d \,e^{5}+90 x^{2} a^{2} c \,e^{6}-60 x \,c^{3} d^{5} e +36 x^{5} b \,c^{2} e^{6}+45 x^{4} b^{2} c \,e^{6}+90 x^{2} a \,b^{2} e^{6}-30 x^{2} b^{3} d \,e^{5}+180 x \,a^{2} b \,e^{6}+60 x \,b^{3} d^{2} e^{4}-60 \ln \left (e x +d \right ) b^{3} d^{3} e^{3}+20 x^{3} b^{3} e^{6}+60 \ln \left (e x +d \right ) a^{3} e^{6}+180 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}+180 \ln \left (e x +d \right ) b^{2} c \,d^{4} e^{2}-180 \ln \left (e x +d \right ) b \,c^{2} d^{5} e +90 x^{2} b^{2} c \,d^{2} e^{4}-90 x^{2} b \,c^{2} d^{3} e^{3}-180 x a \,b^{2} d \,e^{5}-180 x \,b^{2} c \,d^{3} e^{3}+180 x b \,c^{2} d^{4} e^{2}-45 x^{4} b \,c^{2} d \,e^{5}+120 x^{3} a b c \,e^{6}-60 x^{3} b^{2} c d \,e^{5}+60 x^{3} b \,c^{2} d^{2} e^{4}-180 \ln \left (e x +d \right ) a^{2} b d \,e^{5}+180 \ln \left (e x +d \right ) a^{2} c \,d^{2} e^{4}+180 \ln \left (e x +d \right ) a \,b^{2} d^{2} e^{4}}{60 e^{7}}\)	\(548\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {10 \, c^{3} e^{6} x^{6} - 12 \, {\left (c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 15 \, {\left (c^{3} d^{2} e^{4} - 3 \, b c^{2} d e^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 20 \, {\left (c^{3} d^{3} e^{3} - 3 \, b c^{2} d^{2} e^{4} + 3 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 30 \, {\left (c^{3} d^{4} e^{2} - 3 \, b c^{2} d^{3} e^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 60 \, {\left (c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} - 3 \, a^{2} b e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 60 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} \log \left (e x + d\right )}{60 \, e^{7}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {c^{3} x^{6}}{6 e} + x^{5} \cdot \left (\frac {3 b c^{2}}{5 e} - \frac {c^{3} d}{5 e^{2}}\right ) + x^{4} \cdot \left (\frac {3 a c^{2}}{4 e} + \frac {3 b^{2} c}{4 e} - \frac {3 b c^{2} d}{4 e^{2}} + \frac {c^{3} d^{2}}{4 e^{3}}\right ) + x^{3} \cdot \left (\frac {2 a b c}{e} - \frac {a c^{2} d}{e^{2}} + \frac {b^{3}}{3 e} - \frac {b^{2} c d}{e^{2}} + \frac {b c^{2} d^{2}}{e^{3}} - \frac {c^{3} d^{3}}{3 e^{4}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c}{2 e} + \frac {3 a b^{2}}{2 e} - \frac {3 a b c d}{e^{2}} + \frac {3 a c^{2} d^{2}}{2 e^{3}} - \frac {b^{3} d}{2 e^{2}} + \frac {3 b^{2} c d^{2}}{2 e^{3}} - \frac {3 b c^{2} d^{3}}{2 e^{4}} + \frac {c^{3} d^{4}}{2 e^{5}}\right ) + x \left (\frac {3 a^{2} b}{e} - \frac {3 a^{2} c d}{e^{2}} - \frac {3 a b^{2} d}{e^{2}} + \frac {6 a b c d^{2}}{e^{3}} - \frac {3 a c^{2} d^{3}}{e^{4}} + \frac {b^{3} d^{2}}{e^{3}} - \frac {3 b^{2} c d^{3}}{e^{4}} + \frac {3 b c^{2} d^{4}}{e^{5}} - \frac {c^{3} d^{5}}{e^{6}}\right ) + \frac {\left (a e^{2} - b d e + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{7}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {10 \, c^{3} e^{5} x^{6} - 12 \, {\left (c^{3} d e^{4} - 3 \, b c^{2} e^{5}\right )} x^{5} + 15 \, {\left (c^{3} d^{2} e^{3} - 3 \, b c^{2} d e^{4} + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{4} - 20 \, {\left (c^{3} d^{3} e^{2} - 3 \, b c^{2} d^{2} e^{3} + 3 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} - {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{3} + 30 \, {\left (c^{3} d^{4} e - 3 \, b c^{2} d^{3} e^{2} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x^{2} - 60 \, {\left (c^{3} d^{5} - 3 \, b c^{2} d^{4} e - 3 \, a^{2} b e^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} x}{60 \, e^{6}} + \frac {{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x} \, dx=\frac {10 \, c^{3} e^{5} x^{6} - 12 \, c^{3} d e^{4} x^{5} + 36 \, b c^{2} e^{5} x^{5} + 15 \, c^{3} d^{2} e^{3} x^{4} - 45 \, b c^{2} d e^{4} x^{4} + 45 \, b^{2} c e^{5} x^{4} + 45 \, a c^{2} e^{5} x^{4} - 20 \, c^{3} d^{3} e^{2} x^{3} + 60 \, b c^{2} d^{2} e^{3} x^{3} - 60 \, b^{2} c d e^{4} x^{3} - 60 \, a c^{2} d e^{4} x^{3} + 20 \, b^{3} e^{5} x^{3} + 120 \, a b c e^{5} x^{3} + 30 \, c^{3} d^{4} e x^{2} - 90 \, b c^{2} d^{3} e^{2} x^{2} + 90 \, b^{2} c d^{2} e^{3} x^{2} + 90 \, a c^{2} d^{2} e^{3} x^{2} - 30 \, b^{3} d e^{4} x^{2} - 180 \, a b c d e^{4} x^{2} + 90 \, a b^{2} e^{5} x^{2} + 90 \, a^{2} c e^{5} x^{2} - 60 \, c^{3} d^{5} x + 180 \, b c^{2} d^{4} e x - 180 \, b^{2} c d^{3} e^{2} x - 180 \, a c^{2} d^{3} e^{2} x + 60 \, b^{3} d^{2} e^{3} x + 360 \, a b c d^{2} e^{3} x - 180 \, a b^{2} d e^{4} x - 180 \, a^{2} c d e^{4} x + 180 \, a^{2} b e^{5} x}{60 \, e^{6}} + \frac {{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} \]

[In]

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

[Out]

1/60*(10*c^3*e^5*x^6 - 12*c^3*d*e^4*x^5 + 36*b*c^2*e^5*x^5 + 15*c^3*d^2*e^3*x^4 - 45*b*c^2*d*e^4*x^4 + 45*b^2*
c*e^5*x^4 + 45*a*c^2*e^5*x^4 - 20*c^3*d^3*e^2*x^3 + 60*b*c^2*d^2*e^3*x^3 - 60*b^2*c*d*e^4*x^3 - 60*a*c^2*d*e^4
*x^3 + 20*b^3*e^5*x^3 + 120*a*b*c*e^5*x^3 + 30*c^3*d^4*e*x^2 - 90*b*c^2*d^3*e^2*x^2 + 90*b^2*c*d^2*e^3*x^2 + 9
0*a*c^2*d^2*e^3*x^2 - 30*b^3*d*e^4*x^2 - 180*a*b*c*d*e^4*x^2 + 90*a*b^2*e^5*x^2 + 90*a^2*c*e^5*x^2 - 60*c^3*d^
5*x + 180*b*c^2*d^4*e*x - 180*b^2*c*d^3*e^2*x - 180*a*c^2*d^3*e^2*x + 60*b^3*d^2*e^3*x + 360*a*b*c*d^2*e^3*x -
180*a*b^2*d*e^4*x - 180*a^2*c*d*e^4*x + 180*a^2*b*e^5*x)/e^6 + (c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3
*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*
log(abs(e*x + d))/e^7

Mupad [B] (verification not implemented)

Time = 10.00 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b x+c x^2\right )^3}{d+e x} \, dx=x\,\left (\frac {3\,a^2\,b}{e}-\frac {d\,\left (\frac {3\,a\,\left (b^2+a\,c\right )}{e}-\frac {d\,\left (\frac {b^3+6\,a\,c\,b}{e}+\frac {d\,\left (\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{e}-\frac {3\,c\,\left (b^2+a\,c\right )}{e}\right )}{e}\right )}{e}\right )}{e}\right )+x^5\,\left (\frac {3\,b\,c^2}{5\,e}-\frac {c^3\,d}{5\,e^2}\right )+x^2\,\left (\frac {3\,a\,\left (b^2+a\,c\right )}{2\,e}-\frac {d\,\left (\frac {b^3+6\,a\,c\,b}{e}+\frac {d\,\left (\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{e}-\frac {3\,c\,\left (b^2+a\,c\right )}{e}\right )}{e}\right )}{2\,e}\right )+x^3\,\left (\frac {b^3+6\,a\,c\,b}{3\,e}+\frac {d\,\left (\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{e}-\frac {3\,c\,\left (b^2+a\,c\right )}{e}\right )}{3\,e}\right )-x^4\,\left (\frac {d\,\left (\frac {3\,b\,c^2}{e}-\frac {c^3\,d}{e^2}\right )}{4\,e}-\frac {3\,c\,\left (b^2+a\,c\right )}{4\,e}\right )+\frac {c^3\,x^6}{6\,e}+\frac {\ln \left (d+e\,x\right )\,\left (a^3\,e^6-3\,a^2\,b\,d\,e^5+3\,a^2\,c\,d^2\,e^4+3\,a\,b^2\,d^2\,e^4-6\,a\,b\,c\,d^3\,e^3+3\,a\,c^2\,d^4\,e^2-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6\right )}{e^7} \]

[In]

int((a + b*x + c*x^2)^3/(d + e*x),x)

[Out]

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