Integrand size = 22, antiderivative size = 282 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}{e^7}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{e^7}+\frac {6 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^7}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^7}+\frac {2 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{3 e^7}-\frac {6 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7} \]
-2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(3/2)/e^7+6/5*(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)^(5/2)/e^7-2/7*(-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)^(7/2)/e^7+2/3*c*(5*c ^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^(9/2)/e^7-6/11*c^2*(-b*e+2*c*d)*( e*x+d)^(11/2)/e^7+2/13*c^3*(e*x+d)^(13/2)/e^7+2*(a*e^2-b*d*e+c*d^2)^3*(e*x +d)^(1/2)/e^7
Time = 0.22 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (5 c^3 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )+429 e^3 \left (35 a^3 e^3+35 a^2 b e^2 (-2 d+e x)+7 a b^2 e \left (8 d^2-4 d e x+3 e^2 x^2\right )+b^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )\right )+143 c e^2 \left (21 a^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+18 a b e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+b^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )-13 c^2 e \left (-11 a e \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+5 b \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{15015 e^7} \]
(2*Sqrt[d + e*x]*(5*c^3*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^ 3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6) + 429*e^3*(35*a ^3*e^3 + 35*a^2*b*e^2*(-2*d + e*x) + 7*a*b^2*e*(8*d^2 - 4*d*e*x + 3*e^2*x^ 2) + b^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3)) + 143*c*e^2*(21* a^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 18*a*b*e*(-16*d^3 + 8*d^2*e*x - 6* d*e^2*x^2 + 5*e^3*x^3) + b^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d *e^3*x^3 + 35*e^4*x^4)) - 13*c^2*e*(-11*a*e*(128*d^4 - 64*d^3*e*x + 48*d^2 *e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 5*b*(256*d^5 - 128*d^4*e*x + 96*d^ 3*e^2*x^2 - 80*d^2*e^3*x^3 + 70*d*e^4*x^4 - 63*e^5*x^5))))/(15015*e^7)
Time = 0.42 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1140, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^{5/2} (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6}+\frac {3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {3 \sqrt {d+e x} (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6}+\frac {\left (a e^2-b d e+c d^2\right )^3}{e^6 \sqrt {d+e x}}-\frac {3 c^2 (d+e x)^{9/2} (2 c d-b e)}{e^6}+\frac {c^3 (d+e x)^{11/2}}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^7}+\frac {6 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}+\frac {2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^3}{e^7}-\frac {6 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^7}+\frac {2 c^3 (d+e x)^{13/2}}{13 e^7}\) |
(2*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x])/e^7 - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2))/e^7 + (6*(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)^(5/2))/(5*e^7) - (2*(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)^(7/2))/(7* e^7) + (2*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(3* e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^7) + (2*c^3*(d + e*x)^ (13/2))/(13*e^7)
3.23.85.3.1 Defintions of rubi rules used
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] && IGtQ[p, 0]
Time = 0.29 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {e x +d}\, \left (\left (\frac {c^{3} x^{6}}{13}+\frac {\left (\frac {9 b x}{11}+a \right ) x^{4} c^{2}}{3}+\left (\frac {1}{3} b^{2} x^{4}+\frac {3}{5} a^{2} x^{2}+\frac {6}{7} a b \,x^{3}\right ) c +a^{3}+\frac {3 a \,b^{2} x^{2}}{5}+a^{2} b x +\frac {b^{3} x^{3}}{7}\right ) e^{6}-2 \left (\frac {6 c^{3} x^{5}}{143}+\left (\frac {5}{33} b \,x^{4}+\frac {4}{21} a \,x^{3}\right ) c^{2}+\left (\frac {4}{21} b^{2} x^{3}+\frac {18}{35} a b \,x^{2}+\frac {2}{5} a^{2} x \right ) c +b \left (\frac {3}{35} b^{2} x^{2}+\frac {2}{5} a b x +a^{2}\right )\right ) d \,e^{5}+\frac {8 \left (\frac {25 c^{3} x^{4}}{429}+\frac {2 x^{2} \left (\frac {25 b x}{33}+a \right ) c^{2}}{7}+\left (\frac {2}{7} b^{2} x^{2}+\frac {6}{7} a b x +a^{2}\right ) c +b^{2} \left (\frac {b x}{7}+a \right )\right ) d^{2} e^{4}}{5}-\frac {96 \left (\frac {50 c^{3} x^{3}}{1287}+\left (\frac {5}{33} b \,x^{2}+\frac {2}{9} a x \right ) c^{2}+b \left (\frac {2 b x}{9}+a \right ) c +\frac {b^{3}}{6}\right ) d^{3} e^{3}}{35}+\frac {128 c \left (\frac {15 c^{2} x^{2}}{143}+\left (\frac {5 b x}{11}+a \right ) c +b^{2}\right ) d^{4} e^{2}}{105}-\frac {256 \left (\frac {2 c x}{13}+b \right ) c^{2} d^{5} e}{231}+\frac {1024 c^{3} d^{6}}{3003}\right )}{e^{7}}\) | \(326\) |
| derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}+2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {e x +d}}{e^{7}}\) | \(355\) |
| default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (b e -2 c d \right ) c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) c^{2}+2 \left (b e -2 c d \right )^{2} c +c \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \left (b e -2 c d \right )+\left (b e -2 c d \right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right )+2 \left (b e -2 c d \right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )+c \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {3}{2}}+2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {e x +d}}{e^{7}}\) | \(355\) |
| gosper | \(\frac {2 \sqrt {e x +d}\, \left (1155 c^{3} x^{6} e^{6}+4095 b \,c^{2} e^{6} x^{5}-1260 c^{3} d \,e^{5} x^{5}+5005 a \,c^{2} e^{6} x^{4}+5005 b^{2} c \,e^{6} x^{4}-4550 b \,c^{2} d \,e^{5} x^{4}+1400 c^{3} d^{2} e^{4} x^{4}+12870 a b c \,e^{6} x^{3}-5720 a \,c^{2} d \,e^{5} x^{3}+2145 b^{3} e^{6} x^{3}-5720 b^{2} c d \,e^{5} x^{3}+5200 b \,c^{2} d^{2} e^{4} x^{3}-1600 c^{3} d^{3} e^{3} x^{3}+9009 a^{2} c \,e^{6} x^{2}+9009 a \,b^{2} e^{6} x^{2}-15444 a b c d \,e^{5} x^{2}+6864 a \,c^{2} d^{2} e^{4} x^{2}-2574 b^{3} d \,e^{5} x^{2}+6864 b^{2} c \,d^{2} e^{4} x^{2}-6240 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+15015 a^{2} b \,e^{6} x -12012 a^{2} c d \,e^{5} x -12012 a \,b^{2} d \,e^{5} x +20592 a b c \,d^{2} e^{4} x -9152 a \,c^{2} d^{3} e^{3} x +3432 b^{3} d^{2} e^{4} x -9152 b^{2} c \,d^{3} e^{3} x +8320 b \,c^{2} d^{4} e^{2} x -2560 c^{3} d^{5} e x +15015 a^{3} e^{6}-30030 a^{2} b d \,e^{5}+24024 a^{2} c \,d^{2} e^{4}+24024 a \,b^{2} d^{2} e^{4}-41184 a b c \,d^{3} e^{3}+18304 a \,c^{2} d^{4} e^{2}-6864 b^{3} d^{3} e^{3}+18304 b^{2} c \,d^{4} e^{2}-16640 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{15015 e^{7}}\) | \(495\) |
| trager | \(\frac {2 \sqrt {e x +d}\, \left (1155 c^{3} x^{6} e^{6}+4095 b \,c^{2} e^{6} x^{5}-1260 c^{3} d \,e^{5} x^{5}+5005 a \,c^{2} e^{6} x^{4}+5005 b^{2} c \,e^{6} x^{4}-4550 b \,c^{2} d \,e^{5} x^{4}+1400 c^{3} d^{2} e^{4} x^{4}+12870 a b c \,e^{6} x^{3}-5720 a \,c^{2} d \,e^{5} x^{3}+2145 b^{3} e^{6} x^{3}-5720 b^{2} c d \,e^{5} x^{3}+5200 b \,c^{2} d^{2} e^{4} x^{3}-1600 c^{3} d^{3} e^{3} x^{3}+9009 a^{2} c \,e^{6} x^{2}+9009 a \,b^{2} e^{6} x^{2}-15444 a b c d \,e^{5} x^{2}+6864 a \,c^{2} d^{2} e^{4} x^{2}-2574 b^{3} d \,e^{5} x^{2}+6864 b^{2} c \,d^{2} e^{4} x^{2}-6240 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+15015 a^{2} b \,e^{6} x -12012 a^{2} c d \,e^{5} x -12012 a \,b^{2} d \,e^{5} x +20592 a b c \,d^{2} e^{4} x -9152 a \,c^{2} d^{3} e^{3} x +3432 b^{3} d^{2} e^{4} x -9152 b^{2} c \,d^{3} e^{3} x +8320 b \,c^{2} d^{4} e^{2} x -2560 c^{3} d^{5} e x +15015 a^{3} e^{6}-30030 a^{2} b d \,e^{5}+24024 a^{2} c \,d^{2} e^{4}+24024 a \,b^{2} d^{2} e^{4}-41184 a b c \,d^{3} e^{3}+18304 a \,c^{2} d^{4} e^{2}-6864 b^{3} d^{3} e^{3}+18304 b^{2} c \,d^{4} e^{2}-16640 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{15015 e^{7}}\) | \(495\) |
| risch | \(\frac {2 \sqrt {e x +d}\, \left (1155 c^{3} x^{6} e^{6}+4095 b \,c^{2} e^{6} x^{5}-1260 c^{3} d \,e^{5} x^{5}+5005 a \,c^{2} e^{6} x^{4}+5005 b^{2} c \,e^{6} x^{4}-4550 b \,c^{2} d \,e^{5} x^{4}+1400 c^{3} d^{2} e^{4} x^{4}+12870 a b c \,e^{6} x^{3}-5720 a \,c^{2} d \,e^{5} x^{3}+2145 b^{3} e^{6} x^{3}-5720 b^{2} c d \,e^{5} x^{3}+5200 b \,c^{2} d^{2} e^{4} x^{3}-1600 c^{3} d^{3} e^{3} x^{3}+9009 a^{2} c \,e^{6} x^{2}+9009 a \,b^{2} e^{6} x^{2}-15444 a b c d \,e^{5} x^{2}+6864 a \,c^{2} d^{2} e^{4} x^{2}-2574 b^{3} d \,e^{5} x^{2}+6864 b^{2} c \,d^{2} e^{4} x^{2}-6240 b \,c^{2} d^{3} e^{3} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+15015 a^{2} b \,e^{6} x -12012 a^{2} c d \,e^{5} x -12012 a \,b^{2} d \,e^{5} x +20592 a b c \,d^{2} e^{4} x -9152 a \,c^{2} d^{3} e^{3} x +3432 b^{3} d^{2} e^{4} x -9152 b^{2} c \,d^{3} e^{3} x +8320 b \,c^{2} d^{4} e^{2} x -2560 c^{3} d^{5} e x +15015 a^{3} e^{6}-30030 a^{2} b d \,e^{5}+24024 a^{2} c \,d^{2} e^{4}+24024 a \,b^{2} d^{2} e^{4}-41184 a b c \,d^{3} e^{3}+18304 a \,c^{2} d^{4} e^{2}-6864 b^{3} d^{3} e^{3}+18304 b^{2} c \,d^{4} e^{2}-16640 b \,c^{2} d^{5} e +5120 c^{3} d^{6}\right )}{15015 e^{7}}\) | \(495\) |
2*(e*x+d)^(1/2)*((1/13*c^3*x^6+1/3*(9/11*b*x+a)*x^4*c^2+(1/3*b^2*x^4+3/5*a ^2*x^2+6/7*a*b*x^3)*c+a^3+3/5*a*b^2*x^2+a^2*b*x+1/7*b^3*x^3)*e^6-2*(6/143* c^3*x^5+(5/33*b*x^4+4/21*a*x^3)*c^2+(4/21*b^2*x^3+18/35*a*b*x^2+2/5*a^2*x) *c+b*(3/35*b^2*x^2+2/5*a*b*x+a^2))*d*e^5+8/5*(25/429*c^3*x^4+2/7*x^2*(25/3 3*b*x+a)*c^2+(2/7*b^2*x^2+6/7*a*b*x+a^2)*c+b^2*(1/7*b*x+a))*d^2*e^4-96/35* (50/1287*c^3*x^3+(5/33*b*x^2+2/9*a*x)*c^2+b*(2/9*b*x+a)*c+1/6*b^3)*d^3*e^3 +128/105*c*(15/143*c^2*x^2+(5/11*b*x+a)*c+b^2)*d^4*e^2-256/231*(2/13*c*x+b )*c^2*d^5*e+1024/3003*c^3*d^6)/e^7
Time = 0.28 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (1155 \, c^{3} e^{6} x^{6} + 5120 \, c^{3} d^{6} - 16640 \, b c^{2} d^{5} e - 30030 \, a^{2} b d e^{5} + 15015 \, a^{3} e^{6} + 18304 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 6864 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 24024 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 315 \, {\left (4 \, c^{3} d e^{5} - 13 \, b c^{2} e^{6}\right )} x^{5} + 35 \, {\left (40 \, c^{3} d^{2} e^{4} - 130 \, b c^{2} d e^{5} + 143 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 5 \, {\left (320 \, c^{3} d^{3} e^{3} - 1040 \, b c^{2} d^{2} e^{4} + 1144 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - 429 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \, {\left (640 \, c^{3} d^{4} e^{2} - 2080 \, b c^{2} d^{3} e^{3} + 2288 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 858 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 3003 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - {\left (2560 \, c^{3} d^{5} e - 8320 \, b c^{2} d^{4} e^{2} - 15015 \, a^{2} b e^{6} + 9152 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 3432 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 12012 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{7}} \]
2/15015*(1155*c^3*e^6*x^6 + 5120*c^3*d^6 - 16640*b*c^2*d^5*e - 30030*a^2*b *d*e^5 + 15015*a^3*e^6 + 18304*(b^2*c + a*c^2)*d^4*e^2 - 6864*(b^3 + 6*a*b *c)*d^3*e^3 + 24024*(a*b^2 + a^2*c)*d^2*e^4 - 315*(4*c^3*d*e^5 - 13*b*c^2* e^6)*x^5 + 35*(40*c^3*d^2*e^4 - 130*b*c^2*d*e^5 + 143*(b^2*c + a*c^2)*e^6) *x^4 - 5*(320*c^3*d^3*e^3 - 1040*b*c^2*d^2*e^4 + 1144*(b^2*c + a*c^2)*d*e^ 5 - 429*(b^3 + 6*a*b*c)*e^6)*x^3 + 3*(640*c^3*d^4*e^2 - 2080*b*c^2*d^3*e^3 + 2288*(b^2*c + a*c^2)*d^2*e^4 - 858*(b^3 + 6*a*b*c)*d*e^5 + 3003*(a*b^2 + a^2*c)*e^6)*x^2 - (2560*c^3*d^5*e - 8320*b*c^2*d^4*e^2 - 15015*a^2*b*e^6 + 9152*(b^2*c + a*c^2)*d^3*e^3 - 3432*(b^3 + 6*a*b*c)*d^2*e^4 + 12012*(a* b^2 + a^2*c)*d*e^5)*x)*sqrt(e*x + d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (279) = 558\).
Time = 1.15 (sec) , antiderivative size = 631, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{6}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 b c^{2} e - 6 c^{3} d\right )}{11 e^{6}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a c^{2} e^{2} + 3 b^{2} c e^{2} - 15 b c^{2} d e + 15 c^{3} d^{2}\right )}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} c e^{4} + 3 a b^{2} e^{4} - 18 a b c d e^{3} + 18 a c^{2} d^{2} e^{2} - 3 b^{3} d e^{3} + 18 b^{2} c d^{2} e^{2} - 30 b c^{2} d^{3} e + 15 c^{3} d^{4}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 a^{2} b e^{5} - 6 a^{2} c d e^{4} - 6 a b^{2} d e^{4} + 18 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} + 3 b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 15 b c^{2} d^{4} e - 6 c^{3} d^{5}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (a^{3} e^{6} - 3 a^{2} b d e^{5} + 3 a^{2} c d^{2} e^{4} + 3 a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 3 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 3 b^{2} c d^{4} e^{2} - 3 b c^{2} d^{5} e + c^{3} d^{6}\right )}{e^{6}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
Piecewise((2*(c**3*(d + e*x)**(13/2)/(13*e**6) + (d + e*x)**(11/2)*(3*b*c* *2*e - 6*c**3*d)/(11*e**6) + (d + e*x)**(9/2)*(3*a*c**2*e**2 + 3*b**2*c*e* *2 - 15*b*c**2*d*e + 15*c**3*d**2)/(9*e**6) + (d + e*x)**(7/2)*(6*a*b*c*e* *3 - 12*a*c**2*d*e**2 + b**3*e**3 - 12*b**2*c*d*e**2 + 30*b*c**2*d**2*e - 20*c**3*d**3)/(7*e**6) + (d + e*x)**(5/2)*(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)/(5*e**6) + (d + e*x)**(3/2)*(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)/(3*e**6) + sqrt(d + e*x)*(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** 6)/e, Ne(e, 0)), ((a**3*x + 3*a**2*b*x**2/2 + b*c**2*x**6/2 + c**3*x**7/7 + x**5*(3*a*c**2 + 3*b**2*c)/5 + x**4*(6*a*b*c + b**3)/4 + x**3*(3*a**2*c + 3*a*b**2)/3)/sqrt(d), True))
Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (258) = 516\).
Time = 0.20 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{3} + 3003 \, a^{2} {\left (\frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} b}{e} + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c}{e^{2}}\right )} + 143 \, a {\left (\frac {21 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2}}{e^{2}} + \frac {18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b c}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )} + \frac {429 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{3}}{e^{3}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{2} c}{e^{4}} + \frac {65 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b c^{2}}{e^{5}} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \]
2/15015*(15015*sqrt(e*x + d)*a^3 + 3003*a^2*(5*((e*x + d)^(3/2) - 3*sqrt(e *x + d)*d)*b/e + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c/e^2) + 143*a*(21*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15 *sqrt(e*x + d)*d^2)*b^2/e^2 + 18*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b*c/e^3 + (35*(e*x + d)^ (9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3 /2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2/e^4) + 429*(5*(e*x + d)^(7/2) - 21*(e *x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b^3/e^3 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^2*c/e^4 + 65*(63*(e* x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e* x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b*c^2 /e^5 + 5*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^ (9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e* x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^3/e^6)/e
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (258) = 516\).
Time = 0.27 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{3} + \frac {15015 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} b}{e} + \frac {3003 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a b^{2}}{e^{2}} + \frac {3003 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} c}{e^{2}} + \frac {429 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{3}}{e^{3}} + \frac {2574 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a b c}{e^{3}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{2} c}{e^{4}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a c^{2}}{e^{4}} + \frac {65 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b c^{2}}{e^{5}} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3}}{e^{6}}\right )}}{15015 \, e} \]
2/15015*(15015*sqrt(e*x + d)*a^3 + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d )*d)*a^2*b/e + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e* x + d)*d^2)*a*b^2/e^2 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 1 5*sqrt(e*x + d)*d^2)*a^2*c/e^2 + 429*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/ 2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b^3/e^3 + 2574*(5*(e *x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e* x + d)*d^3)*a*b*c/e^3 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4) *b^2*c/e^4 + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*c^2/e^4 + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^ 2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d )*d^5)*b*c^2/e^5 + 5*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 500 5*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^ 4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*c^3/e^6)/e
Time = 9.78 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {{\left (d+e\,x\right )}^{5/2}\,\left (6\,a^2\,c\,e^4+6\,a\,b^2\,e^4-36\,a\,b\,c\,d\,e^3+36\,a\,c^2\,d^2\,e^2-6\,b^3\,d\,e^3+36\,b^2\,c\,d^2\,e^2-60\,b\,c^2\,d^3\,e+30\,c^3\,d^4\right )}{5\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^7}-\frac {\left (12\,c^3\,d-6\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (6\,b^2\,c\,e^2-30\,b\,c^2\,d\,e+30\,c^3\,d^2+6\,a\,c^2\,e^2\right )}{9\,e^7}+\frac {2\,\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{7\,e^7}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{e^7} \]
((d + e*x)^(5/2)*(30*c^3*d^4 + 6*a*b^2*e^4 + 6*a^2*c*e^4 - 6*b^3*d*e^3 + 3 6*a*c^2*d^2*e^2 + 36*b^2*c*d^2*e^2 - 60*b*c^2*d^3*e - 36*a*b*c*d*e^3))/(5* e^7) + (2*c^3*(d + e*x)^(13/2))/(13*e^7) - ((12*c^3*d - 6*b*c^2*e)*(d + e* x)^(11/2))/(11*e^7) + ((d + e*x)^(9/2)*(30*c^3*d^2 + 6*a*c^2*e^2 + 6*b^2*c *e^2 - 30*b*c^2*d*e))/(9*e^7) + (2*(d + e*x)^(1/2)*(a*e^2 + c*d^2 - b*d*e) ^3)/e^7 + (2*(b*e - 2*c*d)*(d + e*x)^(7/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e ^2 - 10*b*c*d*e))/(7*e^7) + (2*(b*e - 2*c*d)*(d + e*x)^(3/2)*(a*e^2 + c*d^ 2 - b*d*e)^2)/e^7