Integrand size = 22, antiderivative size = 274 \[ \int \frac {\left (c+d x+e x^2\right )^3}{\sqrt {a+b x}} \, dx=\frac {2 \left (b^2 c-a b d+a^2 e\right )^3 \sqrt {a+b x}}{b^7}+\frac {2 (b d-2 a e) \left (b^2 c-a b d+a^2 e\right )^2 (a+b x)^{3/2}}{b^7}-\frac {6 \left (b^2 c-a b d+a^2 e\right ) \left (5 a b d e-5 a^2 e^2-b^2 \left (d^2+c e\right )\right ) (a+b x)^{5/2}}{5 b^7}-\frac {2 (b d-2 a e) \left (10 a b d e-10 a^2 e^2-b^2 \left (d^2+6 c e\right )\right ) (a+b x)^{7/2}}{7 b^7}-\frac {2 e \left (5 a b d e-5 a^2 e^2-b^2 \left (d^2+c e\right )\right ) (a+b x)^{9/2}}{3 b^7}+\frac {6 e^2 (b d-2 a e) (a+b x)^{11/2}}{11 b^7}+\frac {2 e^3 (a+b x)^{13/2}}{13 b^7} \] Output:
2*(a^2*e-a*b*d+b^2*c)^3*(b*x+a)^(1/2)/b^7+2*(-2*a*e+b*d)*(a^2*e-a*b*d+b^2* c)^2*(b*x+a)^(3/2)/b^7-6/5*(a^2*e-a*b*d+b^2*c)*(5*a*b*d*e-5*a^2*e^2-b^2*(c *e+d^2))*(b*x+a)^(5/2)/b^7-2/7*(-2*a*e+b*d)*(10*a*b*d*e-10*a^2*e^2-b^2*(6* c*e+d^2))*(b*x+a)^(7/2)/b^7-2/3*e*(5*a*b*d*e-5*a^2*e^2-b^2*(c*e+d^2))*(b*x +a)^(9/2)/b^7+6/11*e^2*(-2*a*e+b*d)*(b*x+a)^(11/2)/b^7+2/13*e^3*(b*x+a)^(1 3/2)/b^7
Time = 0.19 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.30 \[ \int \frac {\left (c+d x+e x^2\right )^3}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \left (5120 a^6 e^3-1280 a^5 b e^2 (13 d+2 e x)+128 a^4 b^2 e \left (143 d^2+65 d e x+e \left (143 c+15 e x^2\right )\right )-16 a^3 b^3 \left (429 d^3+572 d^2 e x+78 d e \left (33 c+5 e x^2\right )+4 e^2 x \left (143 c+25 e x^2\right )\right )+8 a^2 b^4 \left (3003 c^2 e+429 c \left (7 d^2+6 d e x+2 e^2 x^2\right )+x \left (429 d^3+858 d^2 e x+650 d e^2 x^2+175 e^3 x^3\right )\right )+b^6 \left (15015 c^3+3003 c^2 x (5 d+3 e x)+143 c x^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )+5 x^3 \left (429 d^3+1001 d^2 e x+819 d e^2 x^2+231 e^3 x^3\right )\right )-2 a b^5 \left (3003 c^2 (5 d+2 e x)+286 c x \left (21 d^2+27 d e x+10 e^2 x^2\right )+x^2 \left (1287 d^3+2860 d^2 e x+2275 d e^2 x^2+630 e^3 x^3\right )\right )\right )}{15015 b^7} \] Input:
Integrate[(c + d*x + e*x^2)^3/Sqrt[a + b*x],x]
Output:
(2*Sqrt[a + b*x]*(5120*a^6*e^3 - 1280*a^5*b*e^2*(13*d + 2*e*x) + 128*a^4*b ^2*e*(143*d^2 + 65*d*e*x + e*(143*c + 15*e*x^2)) - 16*a^3*b^3*(429*d^3 + 5 72*d^2*e*x + 78*d*e*(33*c + 5*e*x^2) + 4*e^2*x*(143*c + 25*e*x^2)) + 8*a^2 *b^4*(3003*c^2*e + 429*c*(7*d^2 + 6*d*e*x + 2*e^2*x^2) + x*(429*d^3 + 858* d^2*e*x + 650*d*e^2*x^2 + 175*e^3*x^3)) + b^6*(15015*c^3 + 3003*c^2*x*(5*d + 3*e*x) + 143*c*x^2*(63*d^2 + 90*d*e*x + 35*e^2*x^2) + 5*x^3*(429*d^3 + 1001*d^2*e*x + 819*d*e^2*x^2 + 231*e^3*x^3)) - 2*a*b^5*(3003*c^2*(5*d + 2* e*x) + 286*c*x*(21*d^2 + 27*d*e*x + 10*e^2*x^2) + x^2*(1287*d^3 + 2860*d^2 *e*x + 2275*d*e^2*x^2 + 630*e^3*x^3))))/(15015*b^7)
Time = 0.50 (sec) , antiderivative size = 274, 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.
| \(\displaystyle \int \frac {\left (c+d x+e x^2\right )^3}{\sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 e (a+b x)^{7/2} \left (5 a^2 e^2-5 a b d e+b^2 \left (c e+d^2\right )\right )}{b^6}+\frac {(a+b x)^{5/2} (b d-2 a e) \left (10 a^2 e^2-10 a b d e+b^2 \left (6 c e+d^2\right )\right )}{b^6}+\frac {3 (a+b x)^{3/2} \left (a^2 e-a b d+b^2 c\right ) \left (5 a^2 e^2-5 a b d e+b^2 c e+b^2 d^2\right )}{b^6}+\frac {3 \sqrt {a+b x} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )^2}{b^6}+\frac {\left (a^2 e-a b d+b^2 c\right )^3}{b^6 \sqrt {a+b x}}+\frac {3 e^2 (a+b x)^{9/2} (b d-2 a e)}{b^6}+\frac {e^3 (a+b x)^{11/2}}{b^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 e (a+b x)^{9/2} \left (-5 a^2 e^2+5 a b d e-\left (b^2 \left (c e+d^2\right )\right )\right )}{3 b^7}-\frac {2 (a+b x)^{7/2} (b d-2 a e) \left (-10 a^2 e^2+10 a b d e-\left (b^2 \left (6 c e+d^2\right )\right )\right )}{7 b^7}-\frac {6 (a+b x)^{5/2} \left (a^2 e-a b d+b^2 c\right ) \left (-5 a^2 e^2+5 a b d e-\left (b^2 \left (c e+d^2\right )\right )\right )}{5 b^7}+\frac {2 (a+b x)^{3/2} (b d-2 a e) \left (a^2 e-a b d+b^2 c\right )^2}{b^7}+\frac {2 \sqrt {a+b x} \left (a^2 e-a b d+b^2 c\right )^3}{b^7}+\frac {6 e^2 (a+b x)^{11/2} (b d-2 a e)}{11 b^7}+\frac {2 e^3 (a+b x)^{13/2}}{13 b^7}\) |
Input:
Int[(c + d*x + e*x^2)^3/Sqrt[a + b*x],x]
Output:
(2*(b^2*c - a*b*d + a^2*e)^3*Sqrt[a + b*x])/b^7 + (2*(b*d - 2*a*e)*(b^2*c - a*b*d + a^2*e)^2*(a + b*x)^(3/2))/b^7 - (6*(b^2*c - a*b*d + a^2*e)*(5*a* b*d*e - 5*a^2*e^2 - b^2*(d^2 + c*e))*(a + b*x)^(5/2))/(5*b^7) - (2*(b*d - 2*a*e)*(10*a*b*d*e - 10*a^2*e^2 - b^2*(d^2 + 6*c*e))*(a + b*x)^(7/2))/(7*b ^7) - (2*e*(5*a*b*d*e - 5*a^2*e^2 - b^2*(d^2 + c*e))*(a + b*x)^(9/2))/(3*b ^7) + (6*e^2*(b*d - 2*a*e)*(a + b*x)^(11/2))/(11*b^7) + (2*e^3*(a + b*x)^( 13/2))/(13*b^7)
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 = 0.78 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.19
| method | result | size |
| pseudoelliptic | \(\frac {2048 \sqrt {b x +a}\, \left (\left (\frac {231 e^{3} x^{6}}{1024}+\frac {1001 x^{4} \left (\frac {9 x d}{11}+c \right ) e^{2}}{1024}+\frac {9009 x^{2} \left (\frac {5}{9} d^{2} x^{2}+\frac {10}{7} c d x +c^{2}\right ) e}{5120}+\frac {3003 c^{2} d x}{1024}+\frac {9009 c \,d^{2} x^{2}}{5120}+\frac {3003 c^{3}}{1024}+\frac {429 x^{3} d^{3}}{1024}\right ) b^{6}-\frac {3003 a \left (\frac {6 e^{3} x^{5}}{143}+\left (\frac {5}{33} d \,x^{4}+\frac {4}{21} c \,x^{3}\right ) e^{2}+\left (\frac {4}{21} d^{2} x^{3}+\frac {18}{35} c d \,x^{2}+\frac {2}{5} c^{2} x \right ) e +d \left (\frac {3}{35} d^{2} x^{2}+\frac {2}{5} c d x +c^{2}\right )\right ) b^{5}}{512}+\frac {3003 a^{2} \left (\frac {25 e^{3} x^{4}}{429}+\frac {2 \left (\frac {25 x d}{33}+c \right ) x^{2} e^{2}}{7}+\left (\frac {2}{7} d^{2} x^{2}+\frac {6}{7} c d x +c^{2}\right ) e +d^{2} \left (\frac {x d}{7}+c \right )\right ) b^{4}}{640}-\frac {1287 a^{3} \left (\frac {50 e^{3} x^{3}}{1287}+\left (\frac {5}{33} x^{2} d +\frac {2}{9} c x \right ) e^{2}+d \left (\frac {2 x d}{9}+c \right ) e +\frac {d^{3}}{6}\right ) b^{3}}{160}+\frac {143 a^{4} \left (\frac {15 e^{2} x^{2}}{143}+\left (\frac {5 x d}{11}+c \right ) e +d^{2}\right ) e \,b^{2}}{40}-\frac {13 a^{5} e^{2} \left (\frac {2 e x}{13}+d \right ) b}{4}+a^{6} e^{3}\right )}{3003 b^{7}}\) | \(325\) |
| derivativedivides | \(\frac {\frac {2 e^{3} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {6 \left (-2 a e +d b \right ) e^{2} \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a^{2} e -a b d +b^{2} c \right ) e^{2}+2 \left (-2 a e +d b \right )^{2} e +e \left (2 \left (a^{2} e -a b d +b^{2} c \right ) e +\left (-2 a e +d b \right )^{2}\right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (4 \left (a^{2} e -a b d +b^{2} c \right ) \left (-2 a e +d b \right ) e +\left (-2 a e +d b \right ) \left (2 \left (a^{2} e -a b d +b^{2} c \right ) e +\left (-2 a e +d b \right )^{2}\right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a^{2} e -a b d +b^{2} c \right ) \left (2 \left (a^{2} e -a b d +b^{2} c \right ) e +\left (-2 a e +d b \right )^{2}\right )+2 \left (-2 a e +d b \right )^{2} \left (a^{2} e -a b d +b^{2} c \right )+e \left (a^{2} e -a b d +b^{2} c \right )^{2}\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+2 \left (a^{2} e -a b d +b^{2} c \right )^{2} \left (-2 a e +d b \right ) \left (b x +a \right )^{\frac {3}{2}}+2 \left (a^{2} e -a b d +b^{2} c \right )^{3} \sqrt {b x +a}}{b^{7}}\) | \(355\) |
| default | \(\frac {\frac {2 e^{3} \left (b x +a \right )^{\frac {13}{2}}}{13}+\frac {6 \left (-2 a e +d b \right ) e^{2} \left (b x +a \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a^{2} e -a b d +b^{2} c \right ) e^{2}+2 \left (-2 a e +d b \right )^{2} e +e \left (2 \left (a^{2} e -a b d +b^{2} c \right ) e +\left (-2 a e +d b \right )^{2}\right )\right ) \left (b x +a \right )^{\frac {9}{2}}}{9}+\frac {2 \left (4 \left (a^{2} e -a b d +b^{2} c \right ) \left (-2 a e +d b \right ) e +\left (-2 a e +d b \right ) \left (2 \left (a^{2} e -a b d +b^{2} c \right ) e +\left (-2 a e +d b \right )^{2}\right )\right ) \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a^{2} e -a b d +b^{2} c \right ) \left (2 \left (a^{2} e -a b d +b^{2} c \right ) e +\left (-2 a e +d b \right )^{2}\right )+2 \left (-2 a e +d b \right )^{2} \left (a^{2} e -a b d +b^{2} c \right )+e \left (a^{2} e -a b d +b^{2} c \right )^{2}\right ) \left (b x +a \right )^{\frac {5}{2}}}{5}+2 \left (a^{2} e -a b d +b^{2} c \right )^{2} \left (-2 a e +d b \right ) \left (b x +a \right )^{\frac {3}{2}}+2 \left (a^{2} e -a b d +b^{2} c \right )^{3} \sqrt {b x +a}}{b^{7}}\) | \(355\) |
| gosper | \(\frac {2 \sqrt {b x +a}\, \left (1155 e^{3} x^{6} b^{6}-1260 a \,b^{5} e^{3} x^{5}+4095 b^{6} d \,e^{2} x^{5}+1400 a^{2} b^{4} e^{3} x^{4}-4550 a \,b^{5} d \,e^{2} x^{4}+5005 b^{6} c \,e^{2} x^{4}+5005 b^{6} d^{2} e \,x^{4}-1600 a^{3} b^{3} e^{3} x^{3}+5200 a^{2} b^{4} d \,e^{2} x^{3}-5720 a \,b^{5} c \,e^{2} x^{3}-5720 a \,b^{5} d^{2} e \,x^{3}+12870 b^{6} c d e \,x^{3}+2145 b^{6} d^{3} x^{3}+1920 a^{4} b^{2} e^{3} x^{2}-6240 a^{3} b^{3} d \,e^{2} x^{2}+6864 a^{2} b^{4} c \,e^{2} x^{2}+6864 a^{2} b^{4} d^{2} e \,x^{2}-15444 a \,b^{5} c d e \,x^{2}-2574 a \,b^{5} d^{3} x^{2}+9009 b^{6} c^{2} e \,x^{2}+9009 b^{6} c \,d^{2} x^{2}-2560 a^{5} b \,e^{3} x +8320 a^{4} b^{2} d \,e^{2} x -9152 a^{3} b^{3} c \,e^{2} x -9152 a^{3} b^{3} d^{2} e x +20592 a^{2} b^{4} c d e x +3432 a^{2} b^{4} d^{3} x -12012 a \,b^{5} c^{2} e x -12012 a \,b^{5} c \,d^{2} x +15015 b^{6} c^{2} d x +5120 a^{6} e^{3}-16640 a^{5} b d \,e^{2}+18304 a^{4} b^{2} c \,e^{2}+18304 a^{4} b^{2} d^{2} e -41184 a^{3} b^{3} c d e -6864 a^{3} b^{3} d^{3}+24024 a^{2} b^{4} c^{2} e +24024 a^{2} b^{4} c \,d^{2}-30030 a \,b^{5} c^{2} d +15015 c^{3} b^{6}\right )}{15015 b^{7}}\) | \(495\) |
| trager | \(\frac {2 \sqrt {b x +a}\, \left (1155 e^{3} x^{6} b^{6}-1260 a \,b^{5} e^{3} x^{5}+4095 b^{6} d \,e^{2} x^{5}+1400 a^{2} b^{4} e^{3} x^{4}-4550 a \,b^{5} d \,e^{2} x^{4}+5005 b^{6} c \,e^{2} x^{4}+5005 b^{6} d^{2} e \,x^{4}-1600 a^{3} b^{3} e^{3} x^{3}+5200 a^{2} b^{4} d \,e^{2} x^{3}-5720 a \,b^{5} c \,e^{2} x^{3}-5720 a \,b^{5} d^{2} e \,x^{3}+12870 b^{6} c d e \,x^{3}+2145 b^{6} d^{3} x^{3}+1920 a^{4} b^{2} e^{3} x^{2}-6240 a^{3} b^{3} d \,e^{2} x^{2}+6864 a^{2} b^{4} c \,e^{2} x^{2}+6864 a^{2} b^{4} d^{2} e \,x^{2}-15444 a \,b^{5} c d e \,x^{2}-2574 a \,b^{5} d^{3} x^{2}+9009 b^{6} c^{2} e \,x^{2}+9009 b^{6} c \,d^{2} x^{2}-2560 a^{5} b \,e^{3} x +8320 a^{4} b^{2} d \,e^{2} x -9152 a^{3} b^{3} c \,e^{2} x -9152 a^{3} b^{3} d^{2} e x +20592 a^{2} b^{4} c d e x +3432 a^{2} b^{4} d^{3} x -12012 a \,b^{5} c^{2} e x -12012 a \,b^{5} c \,d^{2} x +15015 b^{6} c^{2} d x +5120 a^{6} e^{3}-16640 a^{5} b d \,e^{2}+18304 a^{4} b^{2} c \,e^{2}+18304 a^{4} b^{2} d^{2} e -41184 a^{3} b^{3} c d e -6864 a^{3} b^{3} d^{3}+24024 a^{2} b^{4} c^{2} e +24024 a^{2} b^{4} c \,d^{2}-30030 a \,b^{5} c^{2} d +15015 c^{3} b^{6}\right )}{15015 b^{7}}\) | \(495\) |
| risch | \(\frac {2 \sqrt {b x +a}\, \left (1155 e^{3} x^{6} b^{6}-1260 a \,b^{5} e^{3} x^{5}+4095 b^{6} d \,e^{2} x^{5}+1400 a^{2} b^{4} e^{3} x^{4}-4550 a \,b^{5} d \,e^{2} x^{4}+5005 b^{6} c \,e^{2} x^{4}+5005 b^{6} d^{2} e \,x^{4}-1600 a^{3} b^{3} e^{3} x^{3}+5200 a^{2} b^{4} d \,e^{2} x^{3}-5720 a \,b^{5} c \,e^{2} x^{3}-5720 a \,b^{5} d^{2} e \,x^{3}+12870 b^{6} c d e \,x^{3}+2145 b^{6} d^{3} x^{3}+1920 a^{4} b^{2} e^{3} x^{2}-6240 a^{3} b^{3} d \,e^{2} x^{2}+6864 a^{2} b^{4} c \,e^{2} x^{2}+6864 a^{2} b^{4} d^{2} e \,x^{2}-15444 a \,b^{5} c d e \,x^{2}-2574 a \,b^{5} d^{3} x^{2}+9009 b^{6} c^{2} e \,x^{2}+9009 b^{6} c \,d^{2} x^{2}-2560 a^{5} b \,e^{3} x +8320 a^{4} b^{2} d \,e^{2} x -9152 a^{3} b^{3} c \,e^{2} x -9152 a^{3} b^{3} d^{2} e x +20592 a^{2} b^{4} c d e x +3432 a^{2} b^{4} d^{3} x -12012 a \,b^{5} c^{2} e x -12012 a \,b^{5} c \,d^{2} x +15015 b^{6} c^{2} d x +5120 a^{6} e^{3}-16640 a^{5} b d \,e^{2}+18304 a^{4} b^{2} c \,e^{2}+18304 a^{4} b^{2} d^{2} e -41184 a^{3} b^{3} c d e -6864 a^{3} b^{3} d^{3}+24024 a^{2} b^{4} c^{2} e +24024 a^{2} b^{4} c \,d^{2}-30030 a \,b^{5} c^{2} d +15015 c^{3} b^{6}\right )}{15015 b^{7}}\) | \(495\) |
| orering | \(\frac {2 \sqrt {b x +a}\, \left (1155 e^{3} x^{6} b^{6}-1260 a \,b^{5} e^{3} x^{5}+4095 b^{6} d \,e^{2} x^{5}+1400 a^{2} b^{4} e^{3} x^{4}-4550 a \,b^{5} d \,e^{2} x^{4}+5005 b^{6} c \,e^{2} x^{4}+5005 b^{6} d^{2} e \,x^{4}-1600 a^{3} b^{3} e^{3} x^{3}+5200 a^{2} b^{4} d \,e^{2} x^{3}-5720 a \,b^{5} c \,e^{2} x^{3}-5720 a \,b^{5} d^{2} e \,x^{3}+12870 b^{6} c d e \,x^{3}+2145 b^{6} d^{3} x^{3}+1920 a^{4} b^{2} e^{3} x^{2}-6240 a^{3} b^{3} d \,e^{2} x^{2}+6864 a^{2} b^{4} c \,e^{2} x^{2}+6864 a^{2} b^{4} d^{2} e \,x^{2}-15444 a \,b^{5} c d e \,x^{2}-2574 a \,b^{5} d^{3} x^{2}+9009 b^{6} c^{2} e \,x^{2}+9009 b^{6} c \,d^{2} x^{2}-2560 a^{5} b \,e^{3} x +8320 a^{4} b^{2} d \,e^{2} x -9152 a^{3} b^{3} c \,e^{2} x -9152 a^{3} b^{3} d^{2} e x +20592 a^{2} b^{4} c d e x +3432 a^{2} b^{4} d^{3} x -12012 a \,b^{5} c^{2} e x -12012 a \,b^{5} c \,d^{2} x +15015 b^{6} c^{2} d x +5120 a^{6} e^{3}-16640 a^{5} b d \,e^{2}+18304 a^{4} b^{2} c \,e^{2}+18304 a^{4} b^{2} d^{2} e -41184 a^{3} b^{3} c d e -6864 a^{3} b^{3} d^{3}+24024 a^{2} b^{4} c^{2} e +24024 a^{2} b^{4} c \,d^{2}-30030 a \,b^{5} c^{2} d +15015 c^{3} b^{6}\right )}{15015 b^{7}}\) | \(495\) |
Input:
int((e*x^2+d*x+c)^3/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2048/3003*(b*x+a)^(1/2)*((231/1024*e^3*x^6+1001/1024*x^4*(9/11*x*d+c)*e^2+ 9009/5120*x^2*(5/9*d^2*x^2+10/7*c*d*x+c^2)*e+3003/1024*c^2*d*x+9009/5120*c *d^2*x^2+3003/1024*c^3+429/1024*x^3*d^3)*b^6-3003/512*a*(6/143*e^3*x^5+(5/ 33*d*x^4+4/21*c*x^3)*e^2+(4/21*d^2*x^3+18/35*c*d*x^2+2/5*c^2*x)*e+d*(3/35* d^2*x^2+2/5*c*d*x+c^2))*b^5+3003/640*a^2*(25/429*e^3*x^4+2/7*(25/33*x*d+c) *x^2*e^2+(2/7*d^2*x^2+6/7*c*d*x+c^2)*e+d^2*(1/7*x*d+c))*b^4-1287/160*a^3*( 50/1287*e^3*x^3+(5/33*x^2*d+2/9*c*x)*e^2+d*(2/9*x*d+c)*e+1/6*d^3)*b^3+143/ 40*a^4*(15/143*e^2*x^2+(5/11*x*d+c)*e+d^2)*e*b^2-13/4*a^5*e^2*(2/13*e*x+d) *b+a^6*e^3)/b^7
Time = 0.08 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.67 \[ \int \frac {\left (c+d x+e x^2\right )^3}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (1155 \, b^{6} e^{3} x^{6} + 15015 \, b^{6} c^{3} - 30030 \, a b^{5} c^{2} d + 24024 \, a^{2} b^{4} c d^{2} - 6864 \, a^{3} b^{3} d^{3} + 5120 \, a^{6} e^{3} + 315 \, {\left (13 \, b^{6} d e^{2} - 4 \, a b^{5} e^{3}\right )} x^{5} + 35 \, {\left (143 \, b^{6} d^{2} e + 40 \, a^{2} b^{4} e^{3} + 13 \, {\left (11 \, b^{6} c - 10 \, a b^{5} d\right )} e^{2}\right )} x^{4} + 5 \, {\left (429 \, b^{6} d^{3} - 320 \, a^{3} b^{3} e^{3} - 104 \, {\left (11 \, a b^{5} c - 10 \, a^{2} b^{4} d\right )} e^{2} + 286 \, {\left (9 \, b^{6} c d - 4 \, a b^{5} d^{2}\right )} e\right )} x^{3} + 1664 \, {\left (11 \, a^{4} b^{2} c - 10 \, a^{5} b d\right )} e^{2} + 3 \, {\left (3003 \, b^{6} c d^{2} - 858 \, a b^{5} d^{3} + 640 \, a^{4} b^{2} e^{3} + 208 \, {\left (11 \, a^{2} b^{4} c - 10 \, a^{3} b^{3} d\right )} e^{2} + 143 \, {\left (21 \, b^{6} c^{2} - 36 \, a b^{5} c d + 16 \, a^{2} b^{4} d^{2}\right )} e\right )} x^{2} + 1144 \, {\left (21 \, a^{2} b^{4} c^{2} - 36 \, a^{3} b^{3} c d + 16 \, a^{4} b^{2} d^{2}\right )} e + {\left (15015 \, b^{6} c^{2} d - 12012 \, a b^{5} c d^{2} + 3432 \, a^{2} b^{4} d^{3} - 2560 \, a^{5} b e^{3} - 832 \, {\left (11 \, a^{3} b^{3} c - 10 \, a^{4} b^{2} d\right )} e^{2} - 572 \, {\left (21 \, a b^{5} c^{2} - 36 \, a^{2} b^{4} c d + 16 \, a^{3} b^{3} d^{2}\right )} e\right )} x\right )} \sqrt {b x + a}}{15015 \, b^{7}} \] Input:
integrate((e*x^2+d*x+c)^3/(b*x+a)^(1/2),x, algorithm="fricas")
Output:
2/15015*(1155*b^6*e^3*x^6 + 15015*b^6*c^3 - 30030*a*b^5*c^2*d + 24024*a^2* b^4*c*d^2 - 6864*a^3*b^3*d^3 + 5120*a^6*e^3 + 315*(13*b^6*d*e^2 - 4*a*b^5* e^3)*x^5 + 35*(143*b^6*d^2*e + 40*a^2*b^4*e^3 + 13*(11*b^6*c - 10*a*b^5*d) *e^2)*x^4 + 5*(429*b^6*d^3 - 320*a^3*b^3*e^3 - 104*(11*a*b^5*c - 10*a^2*b^ 4*d)*e^2 + 286*(9*b^6*c*d - 4*a*b^5*d^2)*e)*x^3 + 1664*(11*a^4*b^2*c - 10* a^5*b*d)*e^2 + 3*(3003*b^6*c*d^2 - 858*a*b^5*d^3 + 640*a^4*b^2*e^3 + 208*( 11*a^2*b^4*c - 10*a^3*b^3*d)*e^2 + 143*(21*b^6*c^2 - 36*a*b^5*c*d + 16*a^2 *b^4*d^2)*e)*x^2 + 1144*(21*a^2*b^4*c^2 - 36*a^3*b^3*c*d + 16*a^4*b^2*d^2) *e + (15015*b^6*c^2*d - 12012*a*b^5*c*d^2 + 3432*a^2*b^4*d^3 - 2560*a^5*b* e^3 - 832*(11*a^3*b^3*c - 10*a^4*b^2*d)*e^2 - 572*(21*a*b^5*c^2 - 36*a^2*b ^4*c*d + 16*a^3*b^3*d^2)*e)*x)*sqrt(b*x + a)/b^7
Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (277) = 554\).
Time = 1.02 (sec) , antiderivative size = 631, normalized size of antiderivative = 2.30 \[ \int \frac {\left (c+d x+e x^2\right )^3}{\sqrt {a+b x}} \, dx=\begin {cases} \frac {2 \left (\frac {e^{3} \left (a + b x\right )^{\frac {13}{2}}}{13 b^{6}} + \frac {\left (a + b x\right )^{\frac {11}{2}} \left (- 6 a e^{3} + 3 b d e^{2}\right )}{11 b^{6}} + \frac {\left (a + b x\right )^{\frac {9}{2}} \cdot \left (15 a^{2} e^{3} - 15 a b d e^{2} + 3 b^{2} c e^{2} + 3 b^{2} d^{2} e\right )}{9 b^{6}} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 20 a^{3} e^{3} + 30 a^{2} b d e^{2} - 12 a b^{2} c e^{2} - 12 a b^{2} d^{2} e + 6 b^{3} c d e + b^{3} d^{3}\right )}{7 b^{6}} + \frac {\left (a + b x\right )^{\frac {5}{2}} \cdot \left (15 a^{4} e^{3} - 30 a^{3} b d e^{2} + 18 a^{2} b^{2} c e^{2} + 18 a^{2} b^{2} d^{2} e - 18 a b^{3} c d e - 3 a b^{3} d^{3} + 3 b^{4} c^{2} e + 3 b^{4} c d^{2}\right )}{5 b^{6}} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- 6 a^{5} e^{3} + 15 a^{4} b d e^{2} - 12 a^{3} b^{2} c e^{2} - 12 a^{3} b^{2} d^{2} e + 18 a^{2} b^{3} c d e + 3 a^{2} b^{3} d^{3} - 6 a b^{4} c^{2} e - 6 a b^{4} c d^{2} + 3 b^{5} c^{2} d\right )}{3 b^{6}} + \frac {\sqrt {a + b x} \left (a^{6} e^{3} - 3 a^{5} b d e^{2} + 3 a^{4} b^{2} c e^{2} + 3 a^{4} b^{2} d^{2} e - 6 a^{3} b^{3} c d e - a^{3} b^{3} d^{3} + 3 a^{2} b^{4} c^{2} e + 3 a^{2} b^{4} c d^{2} - 3 a b^{5} c^{2} d + b^{6} c^{3}\right )}{b^{6}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {c^{3} x + \frac {3 c^{2} d x^{2}}{2} + \frac {d e^{2} x^{6}}{2} + \frac {e^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 c e^{2} + 3 d^{2} e\right )}{5} + \frac {x^{4} \cdot \left (6 c d e + d^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 c^{2} e + 3 c d^{2}\right )}{3}}{\sqrt {a}} & \text {otherwise} \end {cases} \] Input:
integrate((e*x**2+d*x+c)**3/(b*x+a)**(1/2),x)
Output:
Piecewise((2*(e**3*(a + b*x)**(13/2)/(13*b**6) + (a + b*x)**(11/2)*(-6*a*e **3 + 3*b*d*e**2)/(11*b**6) + (a + b*x)**(9/2)*(15*a**2*e**3 - 15*a*b*d*e* *2 + 3*b**2*c*e**2 + 3*b**2*d**2*e)/(9*b**6) + (a + b*x)**(7/2)*(-20*a**3* e**3 + 30*a**2*b*d*e**2 - 12*a*b**2*c*e**2 - 12*a*b**2*d**2*e + 6*b**3*c*d *e + b**3*d**3)/(7*b**6) + (a + b*x)**(5/2)*(15*a**4*e**3 - 30*a**3*b*d*e* *2 + 18*a**2*b**2*c*e**2 + 18*a**2*b**2*d**2*e - 18*a*b**3*c*d*e - 3*a*b** 3*d**3 + 3*b**4*c**2*e + 3*b**4*c*d**2)/(5*b**6) + (a + b*x)**(3/2)*(-6*a* *5*e**3 + 15*a**4*b*d*e**2 - 12*a**3*b**2*c*e**2 - 12*a**3*b**2*d**2*e + 1 8*a**2*b**3*c*d*e + 3*a**2*b**3*d**3 - 6*a*b**4*c**2*e - 6*a*b**4*c*d**2 + 3*b**5*c**2*d)/(3*b**6) + sqrt(a + b*x)*(a**6*e**3 - 3*a**5*b*d*e**2 + 3* a**4*b**2*c*e**2 + 3*a**4*b**2*d**2*e - 6*a**3*b**3*c*d*e - a**3*b**3*d**3 + 3*a**2*b**4*c**2*e + 3*a**2*b**4*c*d**2 - 3*a*b**5*c**2*d + b**6*c**3)/ b**6)/b, Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/2 + d*e**2*x**6/2 + e**3*x**7 /7 + x**5*(3*c*e**2 + 3*d**2*e)/5 + x**4*(6*c*d*e + d**3)/4 + x**3*(3*c**2 *e + 3*c*d**2)/3)/sqrt(a), True))
Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (250) = 500\).
Time = 0.04 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.92 \[ \int \frac {\left (c+d x+e x^2\right )^3}{\sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:
integrate((e*x^2+d*x+c)^3/(b*x+a)^(1/2),x, algorithm="maxima")
Output:
2/15015*(15015*sqrt(b*x + a)*c^3 + 3003*c^2*(5*((b*x + a)^(3/2) - 3*sqrt(b *x + a)*a)*d/b + (3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*e/b^2) + 143*c*(21*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15 *sqrt(b*x + a)*a^2)*d^2/b^2 + 18*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*d*e/b^3 + (35*(b*x + a)^ (9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3 /2)*a^3 + 315*sqrt(b*x + a)*a^4)*e^2/b^4) + 429*(5*(b*x + a)^(7/2) - 21*(b *x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*d^3/b^3 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*d^2*e/b^4 + 65*(63*(b* x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b* x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*d*e^2 /b^5 + 5*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 5005*(b*x + a)^ (9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b* x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*e^3/b^6)/b
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (250) = 500\).
Time = 0.12 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.92 \[ \int \frac {\left (c+d x+e x^2\right )^3}{\sqrt {a+b x}} \, dx =\text {Too large to display} \] Input:
integrate((e*x^2+d*x+c)^3/(b*x+a)^(1/2),x, algorithm="giac")
Output:
2/15015*(15015*sqrt(b*x + a)*c^3 + 15015*((b*x + a)^(3/2) - 3*sqrt(b*x + a )*a)*c^2*d/b + 3003*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b* x + a)*a^2)*c*d^2/b^2 + 3003*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 1 5*sqrt(b*x + a)*a^2)*c^2*e/b^2 + 429*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/ 2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*d^3/b^3 + 2574*(5*(b *x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b* x + a)*a^3)*c*d*e/b^3 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4) *d^2*e/b^4 + 143*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*c*e^2/b^4 + 65*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^ 2 - 1386*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a )*a^5)*d*e^2/b^5 + 5*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^(11/2)*a + 500 5*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^ 4 - 6006*(b*x + a)^(3/2)*a^5 + 3003*sqrt(b*x + a)*a^6)*e^3/b^6)/b
Time = 3.09 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+d x+e x^2\right )^3}{\sqrt {a+b x}} \, dx=\frac {2\,e^3\,{\left (a+b\,x\right )}^{13/2}}{13\,b^7}-\frac {\left (12\,a\,e^3-6\,b\,d\,e^2\right )\,{\left (a+b\,x\right )}^{11/2}}{11\,b^7}+\frac {{\left (a+b\,x\right )}^{9/2}\,\left (30\,a^2\,e^3-30\,a\,b\,d\,e^2+6\,b^2\,d^2\,e+6\,c\,b^2\,e^2\right )}{9\,b^7}+\frac {2\,\sqrt {a+b\,x}\,{\left (e\,a^2-d\,a\,b+c\,b^2\right )}^3}{b^7}+\frac {{\left (a+b\,x\right )}^{5/2}\,\left (30\,a^4\,e^3-60\,a^3\,b\,d\,e^2+36\,a^2\,b^2\,c\,e^2+36\,a^2\,b^2\,d^2\,e-36\,a\,b^3\,c\,d\,e-6\,a\,b^3\,d^3+6\,b^4\,c^2\,e+6\,b^4\,c\,d^2\right )}{5\,b^7}-\frac {2\,\left (2\,a\,e-b\,d\right )\,{\left (a+b\,x\right )}^{7/2}\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+b^2\,d^2+6\,c\,b^2\,e\right )}{7\,b^7}-\frac {2\,\left (2\,a\,e-b\,d\right )\,{\left (a+b\,x\right )}^{3/2}\,{\left (e\,a^2-d\,a\,b+c\,b^2\right )}^2}{b^7} \] Input:
int((c + d*x + e*x^2)^3/(a + b*x)^(1/2),x)
Output:
(2*e^3*(a + b*x)^(13/2))/(13*b^7) - ((12*a*e^3 - 6*b*d*e^2)*(a + b*x)^(11/ 2))/(11*b^7) + ((a + b*x)^(9/2)*(30*a^2*e^3 + 6*b^2*c*e^2 + 6*b^2*d^2*e - 30*a*b*d*e^2))/(9*b^7) + (2*(a + b*x)^(1/2)*(b^2*c + a^2*e - a*b*d)^3)/b^7 + ((a + b*x)^(5/2)*(30*a^4*e^3 - 6*a*b^3*d^3 + 6*b^4*c*d^2 + 6*b^4*c^2*e + 36*a^2*b^2*c*e^2 + 36*a^2*b^2*d^2*e - 60*a^3*b*d*e^2 - 36*a*b^3*c*d*e))/ (5*b^7) - (2*(2*a*e - b*d)*(a + b*x)^(7/2)*(10*a^2*e^2 + b^2*d^2 + 6*b^2*c *e - 10*a*b*d*e))/(7*b^7) - (2*(2*a*e - b*d)*(a + b*x)^(3/2)*(b^2*c + a^2* e - a*b*d)^2)/b^7
Time = 0.14 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.80 \[ \int \frac {\left (c+d x+e x^2\right )^3}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {b x +a}\, \left (1155 b^{6} e^{3} x^{6}-1260 a \,b^{5} e^{3} x^{5}+4095 b^{6} d \,e^{2} x^{5}+1400 a^{2} b^{4} e^{3} x^{4}-4550 a \,b^{5} d \,e^{2} x^{4}+5005 b^{6} c \,e^{2} x^{4}+5005 b^{6} d^{2} e \,x^{4}-1600 a^{3} b^{3} e^{3} x^{3}+5200 a^{2} b^{4} d \,e^{2} x^{3}-5720 a \,b^{5} c \,e^{2} x^{3}-5720 a \,b^{5} d^{2} e \,x^{3}+12870 b^{6} c d e \,x^{3}+2145 b^{6} d^{3} x^{3}+1920 a^{4} b^{2} e^{3} x^{2}-6240 a^{3} b^{3} d \,e^{2} x^{2}+6864 a^{2} b^{4} c \,e^{2} x^{2}+6864 a^{2} b^{4} d^{2} e \,x^{2}-15444 a \,b^{5} c d e \,x^{2}-2574 a \,b^{5} d^{3} x^{2}+9009 b^{6} c^{2} e \,x^{2}+9009 b^{6} c \,d^{2} x^{2}-2560 a^{5} b \,e^{3} x +8320 a^{4} b^{2} d \,e^{2} x -9152 a^{3} b^{3} c \,e^{2} x -9152 a^{3} b^{3} d^{2} e x +20592 a^{2} b^{4} c d e x +3432 a^{2} b^{4} d^{3} x -12012 a \,b^{5} c^{2} e x -12012 a \,b^{5} c \,d^{2} x +15015 b^{6} c^{2} d x +5120 a^{6} e^{3}-16640 a^{5} b d \,e^{2}+18304 a^{4} b^{2} c \,e^{2}+18304 a^{4} b^{2} d^{2} e -41184 a^{3} b^{3} c d e -6864 a^{3} b^{3} d^{3}+24024 a^{2} b^{4} c^{2} e +24024 a^{2} b^{4} c \,d^{2}-30030 a \,b^{5} c^{2} d +15015 b^{6} c^{3}\right )}{15015 b^{7}} \] Input:
int((e*x^2+d*x+c)^3/(b*x+a)^(1/2),x)
Output:
(2*sqrt(a + b*x)*(5120*a**6*e**3 - 16640*a**5*b*d*e**2 - 2560*a**5*b*e**3* x + 18304*a**4*b**2*c*e**2 + 18304*a**4*b**2*d**2*e + 8320*a**4*b**2*d*e** 2*x + 1920*a**4*b**2*e**3*x**2 - 41184*a**3*b**3*c*d*e - 9152*a**3*b**3*c* e**2*x - 6864*a**3*b**3*d**3 - 9152*a**3*b**3*d**2*e*x - 6240*a**3*b**3*d* e**2*x**2 - 1600*a**3*b**3*e**3*x**3 + 24024*a**2*b**4*c**2*e + 24024*a**2 *b**4*c*d**2 + 20592*a**2*b**4*c*d*e*x + 6864*a**2*b**4*c*e**2*x**2 + 3432 *a**2*b**4*d**3*x + 6864*a**2*b**4*d**2*e*x**2 + 5200*a**2*b**4*d*e**2*x** 3 + 1400*a**2*b**4*e**3*x**4 - 30030*a*b**5*c**2*d - 12012*a*b**5*c**2*e*x - 12012*a*b**5*c*d**2*x - 15444*a*b**5*c*d*e*x**2 - 5720*a*b**5*c*e**2*x* *3 - 2574*a*b**5*d**3*x**2 - 5720*a*b**5*d**2*e*x**3 - 4550*a*b**5*d*e**2* x**4 - 1260*a*b**5*e**3*x**5 + 15015*b**6*c**3 + 15015*b**6*c**2*d*x + 900 9*b**6*c**2*e*x**2 + 9009*b**6*c*d**2*x**2 + 12870*b**6*c*d*e*x**3 + 5005* b**6*c*e**2*x**4 + 2145*b**6*d**3*x**3 + 5005*b**6*d**2*e*x**4 + 4095*b**6 *d*e**2*x**5 + 1155*b**6*e**3*x**6))/(15015*b**7)