Integrand size = 22, antiderivative size = 166 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{3 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}{5 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{7/2}}{7 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{9/2}}{9 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5} \]
2/3*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(3/2)/e^5-4/5*(-b*e+2*c*d)*(a*e^2-b*d*e+ c*d^2)*(e*x+d)^(5/2)/e^5+2/7*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(e*x+d )^(7/2)/e^5-4/9*c*(-b*e+2*c*d)*(e*x+d)^(9/2)/e^5+2/11*c^2*(e*x+d)^(11/2)/e ^5
Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.04 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{3/2} \left (c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+33 e^2 \left (35 a^2 e^2+14 a b e (-2 d+3 e x)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-22 c e \left (-3 a e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )\right )\right )}{3465 e^5} \]
(2*(d + e*x)^(3/2)*(c^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e ^3*x^3 + 315*e^4*x^4) + 33*e^2*(35*a^2*e^2 + 14*a*b*e*(-2*d + 3*e*x) + b^2 *(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 22*c*e*(-3*a*e*(8*d^2 - 12*d*e*x + 15* e^2*x^2) + b*(16*d^3 - 24*d^2*e*x + 30*d*e^2*x^2 - 35*e^3*x^3))))/(3465*e^ 5)
Time = 0.29 (sec) , antiderivative size = 166, 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.
| \(\displaystyle \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^{5/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac {2 (d+e x)^{3/2} (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^4}+\frac {\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2}{e^4}-\frac {2 c (d+e x)^{7/2} (2 c d-b e)}{e^4}+\frac {c^2 (d+e x)^{9/2}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (d+e x)^{7/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^5}-\frac {4 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5}+\frac {2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}{3 e^5}-\frac {4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5}\) |
(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2))/(3*e^5) - (4*(2*c*d - b*e)*( c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2))/(5*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(7/2))/(7*e^5) - (4*c*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^5) + (2*c^2*(d + e*x)^(11/2))/(11*e^5)
3.23.77.3.1 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 = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\) | \(136\) |
| default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\) | \(136\) |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (\left (\frac {3 c^{2} x^{4}}{11}+\frac {6 \left (\frac {7 b x}{9}+a \right ) x^{2} c}{7}+\frac {3 b^{2} x^{2}}{7}+\frac {6 a b x}{5}+a^{2}\right ) e^{4}-\frac {4 \left (\frac {10 c^{2} x^{3}}{33}+\frac {6 \left (\frac {5 b x}{6}+a \right ) x c}{7}+b \left (\frac {3 b x}{7}+a \right )\right ) d \,e^{3}}{5}+\frac {16 \left (\frac {5 c^{2} x^{2}}{11}+\left (b x +a \right ) c +\frac {b^{2}}{2}\right ) d^{2} e^{2}}{35}-\frac {32 \left (\frac {6 c x}{11}+b \right ) c \,d^{3} e}{105}+\frac {128 c^{2} d^{4}}{1155}\right )}{3 e^{5}}\) | \(138\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 c^{2} x^{4} e^{4}+770 b c \,e^{4} x^{3}-280 c^{2} d \,e^{3} x^{3}+990 a c \,e^{4} x^{2}+495 b^{2} e^{4} x^{2}-660 b c d \,e^{3} x^{2}+240 c^{2} d^{2} e^{2} x^{2}+1386 a b \,e^{4} x -792 a c d \,e^{3} x -396 b^{2} d \,e^{3} x +528 b c \,d^{2} e^{2} x -192 c^{2} d^{3} e x +1155 a^{2} e^{4}-924 a b d \,e^{3}+528 a c \,d^{2} e^{2}+264 b^{2} d^{2} e^{2}-352 b c \,d^{3} e +128 c^{2} d^{4}\right )}{3465 e^{5}}\) | \(194\) |
| trager | \(\frac {2 \left (315 c^{2} e^{5} x^{5}+770 b c \,e^{5} x^{4}+35 c^{2} d \,e^{4} x^{4}+990 a c \,e^{5} x^{3}+495 b^{2} e^{5} x^{3}+110 b c d \,e^{4} x^{3}-40 c^{2} d^{2} e^{3} x^{3}+1386 a b \,e^{5} x^{2}+198 a c d \,e^{4} x^{2}+99 b^{2} d \,e^{4} x^{2}-132 b c \,d^{2} e^{3} x^{2}+48 c^{2} d^{3} e^{2} x^{2}+1155 a^{2} e^{5} x +462 a b d \,e^{4} x -264 a c \,d^{2} e^{3} x -132 b^{2} d^{2} e^{3} x +176 b c \,d^{3} e^{2} x -64 c^{2} d^{4} e x +1155 a^{2} d \,e^{4}-924 a b \,d^{2} e^{3}+528 a c \,d^{3} e^{2}+264 b^{2} d^{3} e^{2}-352 b c \,d^{4} e +128 c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\) | \(271\) |
| risch | \(\frac {2 \left (315 c^{2} e^{5} x^{5}+770 b c \,e^{5} x^{4}+35 c^{2} d \,e^{4} x^{4}+990 a c \,e^{5} x^{3}+495 b^{2} e^{5} x^{3}+110 b c d \,e^{4} x^{3}-40 c^{2} d^{2} e^{3} x^{3}+1386 a b \,e^{5} x^{2}+198 a c d \,e^{4} x^{2}+99 b^{2} d \,e^{4} x^{2}-132 b c \,d^{2} e^{3} x^{2}+48 c^{2} d^{3} e^{2} x^{2}+1155 a^{2} e^{5} x +462 a b d \,e^{4} x -264 a c \,d^{2} e^{3} x -132 b^{2} d^{2} e^{3} x +176 b c \,d^{3} e^{2} x -64 c^{2} d^{4} e x +1155 a^{2} d \,e^{4}-924 a b \,d^{2} e^{3}+528 a c \,d^{3} e^{2}+264 b^{2} d^{3} e^{2}-352 b c \,d^{4} e +128 c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\) | \(271\) |
2/e^5*(1/11*c^2*(e*x+d)^(11/2)+2/9*c*(b*e-2*c*d)*(e*x+d)^(9/2)+1/7*(2*(a*e ^2-b*d*e+c*d^2)*c+(b*e-2*c*d)^2)*(e*x+d)^(7/2)+2/5*(a*e^2-b*d*e+c*d^2)*(b* e-2*c*d)*(e*x+d)^(5/2)+1/3*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(3/2))
Time = 0.27 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.43 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (315 \, c^{2} e^{5} x^{5} + 128 \, c^{2} d^{5} - 352 \, b c d^{4} e - 924 \, a b d^{2} e^{3} + 1155 \, a^{2} d e^{4} + 264 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} + 35 \, {\left (c^{2} d e^{4} + 22 \, b c e^{5}\right )} x^{4} - 5 \, {\left (8 \, c^{2} d^{2} e^{3} - 22 \, b c d e^{4} - 99 \, {\left (b^{2} + 2 \, a c\right )} e^{5}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{3} e^{2} - 44 \, b c d^{2} e^{3} + 462 \, a b e^{5} + 33 \, {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{2} - {\left (64 \, c^{2} d^{4} e - 176 \, b c d^{3} e^{2} - 462 \, a b d e^{4} - 1155 \, a^{2} e^{5} + 132 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \]
2/3465*(315*c^2*e^5*x^5 + 128*c^2*d^5 - 352*b*c*d^4*e - 924*a*b*d^2*e^3 + 1155*a^2*d*e^4 + 264*(b^2 + 2*a*c)*d^3*e^2 + 35*(c^2*d*e^4 + 22*b*c*e^5)*x ^4 - 5*(8*c^2*d^2*e^3 - 22*b*c*d*e^4 - 99*(b^2 + 2*a*c)*e^5)*x^3 + 3*(16*c ^2*d^3*e^2 - 44*b*c*d^2*e^3 + 462*a*b*e^5 + 33*(b^2 + 2*a*c)*d*e^4)*x^2 - (64*c^2*d^4*e - 176*b*c*d^3*e^2 - 462*a*b*d*e^4 - 1155*a^2*e^5 + 132*(b^2 + 2*a*c)*d^2*e^3)*x)*sqrt(e*x + d)/e^5
Time = 0.90 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.67 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 a b e^{3} - 4 a c d e^{2} - 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (a^{2} x + a b x^{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 a c + b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(c**2*(d + e*x)**(11/2)/(11*e**4) + (d + e*x)**(9/2)*(2*b*c*e - 4*c**2*d)/(9*e**4) + (d + e*x)**(7/2)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d *e + 6*c**2*d**2)/(7*e**4) + (d + e*x)**(5/2)*(2*a*b*e**3 - 4*a*c*d*e**2 - 2*b**2*d*e**2 + 6*b*c*d**2*e - 4*c**2*d**3)/(5*e**4) + (d + e*x)**(3/2)*( a**2*e**4 - 2*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 - 2*b*c*d**3*e + c**2*d**4)/(3*e**4))/e, Ne(e, 0)), (sqrt(d)*(a**2*x + a*b*x**2 + b*c*x* *4/2 + c**2*x**5/5 + x**3*(2*a*c + b**2)/3), True))
Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.06 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{2} - 770 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 1386 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{3465 \, e^{5}} \]
2/3465*(315*(e*x + d)^(11/2)*c^2 - 770*(2*c^2*d - b*c*e)*(e*x + d)^(9/2) + 495*(6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a*c)*e^2)*(e*x + d)^(7/2) - 1386*(2 *c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d)^(5/2) + 1155*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^ 2)*(e*x + d)^(3/2))/e^5
Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (146) = 292\).
Time = 0.28 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.30 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, \sqrt {e x + d} a^{2} d + 1155 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} + \frac {2310 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b d}{e} + \frac {231 \, {\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} d}{e^{2}} + \frac {462 \, {\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 c d}{e^{2}} + \frac {462 \, {\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}{e} + \frac {198 \, {\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 d}{e^{3}} + \frac {99 \, {\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^{2}}{e^{2}} + \frac {198 \, {\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 c}{e^{2}} + \frac {11 \, {\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} d}{e^{4}} + \frac {22 \, {\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 c}{e^{3}} + \frac {5 \, {\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 )} c^{2}}{e^{4}}\right )}}{3465 \, e} \]
2/3465*(3465*sqrt(e*x + d)*a^2*d + 1155*((e*x + d)^(3/2) - 3*sqrt(e*x + d) *d)*a^2 + 2310*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b*d/e + 231*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2*d/e^2 + 462 *(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*c*d/e ^2 + 462*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2) *a*b/e + 198*(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*d/e^3 + 99*(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^2/e^2 + 19 8*(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*c/e^2 + 11*(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*d/e^4 + 22*(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*c/e^3 + 5*(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)*c^2/e^4)/e
Time = 0.05 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{7\,e^5}+\frac {2\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{3\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{5\,e^5} \]