Integrand size = 39, antiderivative size = 295 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {256 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{45045 c^5 d^5 (d+e x)^{7/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\frac {32 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d} \]
256/45045*(-a*e^2+c*d^2)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^5/d^5 /(e*x+d)^(7/2)+128/6435*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2) ^(7/2)/c^4/d^4/(e*x+d)^(5/2)+32/715*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)* x+c*d*e*x^2)^(7/2)/c^3/d^3/(e*x+d)^(3/2)+16/195*(-a*e^2+c*d^2)*(a*d*e+(a*e ^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2/(e*x+d)^(1/2)+2/15*(a*d*e+(a*e^2+c*d^ 2)*x+c*d*e*x^2)^(7/2)*(e*x+d)^(1/2)/c/d
Time = 0.13 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.67 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (128 a^4 e^8-64 a^3 c d e^6 (15 d+7 e x)+48 a^2 c^2 d^2 e^4 \left (65 d^2+70 d e x+21 e^2 x^2\right )-8 a c^3 d^3 e^2 \left (715 d^3+1365 d^2 e x+945 d e^2 x^2+231 e^3 x^3\right )+c^4 d^4 \left (6435 d^4+20020 d^3 e x+24570 d^2 e^2 x^2+13860 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 c^5 d^5 \sqrt {d+e x}} \]
(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(128*a^4*e^8 - 64*a^3*c*d *e^6*(15*d + 7*e*x) + 48*a^2*c^2*d^2*e^4*(65*d^2 + 70*d*e*x + 21*e^2*x^2) - 8*a*c^3*d^3*e^2*(715*d^3 + 1365*d^2*e*x + 945*d*e^2*x^2 + 231*e^3*x^3) + c^4*d^4*(6435*d^4 + 20020*d^3*e*x + 24570*d^2*e^2*x^2 + 13860*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*c^5*d^5*Sqrt[d + e*x])
Time = 0.47 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1128, 1128, 1128, 1128, 1122}
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.
| \(\displaystyle \int (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {8 \left (d^2-\frac {a e^2}{c}\right ) \int \sqrt {d+e x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}dx}{15 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {8 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{\sqrt {d+e x}}dx}{13 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}\right )}{15 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {8 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^{3/2}}dx}{11 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}}\right )}{13 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}\right )}{15 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {8 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^{5/2}}dx}{9 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}}\right )}{11 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}}\right )}{13 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}\right )}{15 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d}+\frac {8 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}+\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 c d^2 (d+e x)^{7/2}}\right )}{11 d}\right )}{13 d}\right )}{15 d}\) |
(2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(15*c*d) + (8*(d^2 - (a*e^2)/c)*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/( 13*c*d*Sqrt[d + e*x]) + (6*(d^2 - (a*e^2)/c)*((2*(a*d*e + (c*d^2 + a*e^2)* x + c*d*e*x^2)^(7/2))/(11*c*d*(d + e*x)^(3/2)) + (4*(d^2 - (a*e^2)/c)*((4* (d^2 - (a*e^2)/c)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(63*c*d^2 *(d + e*x)^(7/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c *d*(d + e*x)^(5/2))))/(11*d)))/(13*d)))/(15*d)
3.21.46.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[m + p], 0]
Time = 2.87 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.80
| method | result | size |
| default | \(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{3} \left (3003 c^{4} d^{4} e^{4} x^{4}-1848 a \,c^{3} d^{3} e^{5} x^{3}+13860 c^{4} d^{5} e^{3} x^{3}+1008 a^{2} c^{2} d^{2} e^{6} x^{2}-7560 a \,c^{3} d^{4} e^{4} x^{2}+24570 c^{4} d^{6} e^{2} x^{2}-448 a^{3} c d \,e^{7} x +3360 a^{2} c^{2} d^{3} e^{5} x -10920 a \,c^{3} d^{5} e^{3} x +20020 c^{4} d^{7} e x +128 a^{4} e^{8}-960 a^{3} c \,d^{2} e^{6}+3120 a^{2} c^{2} d^{4} e^{4}-5720 a \,c^{3} d^{6} e^{2}+6435 c^{4} d^{8}\right )}{45045 \sqrt {e x +d}\, c^{5} d^{5}}\) | \(235\) |
| gosper | \(\frac {2 \left (c d x +a e \right ) \left (3003 c^{4} d^{4} e^{4} x^{4}-1848 a \,c^{3} d^{3} e^{5} x^{3}+13860 c^{4} d^{5} e^{3} x^{3}+1008 a^{2} c^{2} d^{2} e^{6} x^{2}-7560 a \,c^{3} d^{4} e^{4} x^{2}+24570 c^{4} d^{6} e^{2} x^{2}-448 a^{3} c d \,e^{7} x +3360 a^{2} c^{2} d^{3} e^{5} x -10920 a \,c^{3} d^{5} e^{3} x +20020 c^{4} d^{7} e x +128 a^{4} e^{8}-960 a^{3} c \,d^{2} e^{6}+3120 a^{2} c^{2} d^{4} e^{4}-5720 a \,c^{3} d^{6} e^{2}+6435 c^{4} d^{8}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{45045 c^{5} d^{5} \left (e x +d \right )^{\frac {5}{2}}}\) | \(243\) |
2/45045/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(c*d*x+a*e)^3*(3003*c^4* d^4*e^4*x^4-1848*a*c^3*d^3*e^5*x^3+13860*c^4*d^5*e^3*x^3+1008*a^2*c^2*d^2* e^6*x^2-7560*a*c^3*d^4*e^4*x^2+24570*c^4*d^6*e^2*x^2-448*a^3*c*d*e^7*x+336 0*a^2*c^2*d^3*e^5*x-10920*a*c^3*d^5*e^3*x+20020*c^4*d^7*e*x+128*a^4*e^8-96 0*a^3*c*d^2*e^6+3120*a^2*c^2*d^4*e^4-5720*a*c^3*d^6*e^2+6435*c^4*d^8)/c^5/ d^5
Time = 0.29 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.58 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, c^{7} d^{7} e^{4} x^{7} + 6435 \, a^{3} c^{4} d^{8} e^{3} - 5720 \, a^{4} c^{3} d^{6} e^{5} + 3120 \, a^{5} c^{2} d^{4} e^{7} - 960 \, a^{6} c d^{2} e^{9} + 128 \, a^{7} e^{11} + 231 \, {\left (60 \, c^{7} d^{8} e^{3} + 31 \, a c^{6} d^{6} e^{5}\right )} x^{6} + 63 \, {\left (390 \, c^{7} d^{9} e^{2} + 540 \, a c^{6} d^{7} e^{4} + 71 \, a^{2} c^{5} d^{5} e^{6}\right )} x^{5} + 35 \, {\left (572 \, c^{7} d^{10} e + 1794 \, a c^{6} d^{8} e^{3} + 636 \, a^{2} c^{5} d^{6} e^{5} + a^{3} c^{4} d^{4} e^{7}\right )} x^{4} + 5 \, {\left (1287 \, c^{7} d^{11} + 10868 \, a c^{6} d^{9} e^{2} + 8814 \, a^{2} c^{5} d^{7} e^{4} + 60 \, a^{3} c^{4} d^{5} e^{6} - 8 \, a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \, {\left (6435 \, a c^{6} d^{10} e + 14300 \, a^{2} c^{5} d^{8} e^{3} + 390 \, a^{3} c^{4} d^{6} e^{5} - 120 \, a^{4} c^{3} d^{4} e^{7} + 16 \, a^{5} c^{2} d^{2} e^{9}\right )} x^{2} + {\left (19305 \, a^{2} c^{5} d^{9} e^{2} + 2860 \, a^{3} c^{4} d^{7} e^{4} - 1560 \, a^{4} c^{3} d^{5} e^{6} + 480 \, a^{5} c^{2} d^{3} e^{8} - 64 \, a^{6} c d e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{45045 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
2/45045*(3003*c^7*d^7*e^4*x^7 + 6435*a^3*c^4*d^8*e^3 - 5720*a^4*c^3*d^6*e^ 5 + 3120*a^5*c^2*d^4*e^7 - 960*a^6*c*d^2*e^9 + 128*a^7*e^11 + 231*(60*c^7* d^8*e^3 + 31*a*c^6*d^6*e^5)*x^6 + 63*(390*c^7*d^9*e^2 + 540*a*c^6*d^7*e^4 + 71*a^2*c^5*d^5*e^6)*x^5 + 35*(572*c^7*d^10*e + 1794*a*c^6*d^8*e^3 + 636* a^2*c^5*d^6*e^5 + a^3*c^4*d^4*e^7)*x^4 + 5*(1287*c^7*d^11 + 10868*a*c^6*d^ 9*e^2 + 8814*a^2*c^5*d^7*e^4 + 60*a^3*c^4*d^5*e^6 - 8*a^4*c^3*d^3*e^8)*x^3 + 3*(6435*a*c^6*d^10*e + 14300*a^2*c^5*d^8*e^3 + 390*a^3*c^4*d^6*e^5 - 12 0*a^4*c^3*d^4*e^7 + 16*a^5*c^2*d^2*e^9)*x^2 + (19305*a^2*c^5*d^9*e^2 + 286 0*a^3*c^4*d^7*e^4 - 1560*a^4*c^3*d^5*e^6 + 480*a^5*c^2*d^3*e^8 - 64*a^6*c* d*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^5* d^5*e*x + c^5*d^6)
Timed out. \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.52 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, c^{7} d^{7} e^{4} x^{7} + 6435 \, a^{3} c^{4} d^{8} e^{3} - 5720 \, a^{4} c^{3} d^{6} e^{5} + 3120 \, a^{5} c^{2} d^{4} e^{7} - 960 \, a^{6} c d^{2} e^{9} + 128 \, a^{7} e^{11} + 231 \, {\left (60 \, c^{7} d^{8} e^{3} + 31 \, a c^{6} d^{6} e^{5}\right )} x^{6} + 63 \, {\left (390 \, c^{7} d^{9} e^{2} + 540 \, a c^{6} d^{7} e^{4} + 71 \, a^{2} c^{5} d^{5} e^{6}\right )} x^{5} + 35 \, {\left (572 \, c^{7} d^{10} e + 1794 \, a c^{6} d^{8} e^{3} + 636 \, a^{2} c^{5} d^{6} e^{5} + a^{3} c^{4} d^{4} e^{7}\right )} x^{4} + 5 \, {\left (1287 \, c^{7} d^{11} + 10868 \, a c^{6} d^{9} e^{2} + 8814 \, a^{2} c^{5} d^{7} e^{4} + 60 \, a^{3} c^{4} d^{5} e^{6} - 8 \, a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \, {\left (6435 \, a c^{6} d^{10} e + 14300 \, a^{2} c^{5} d^{8} e^{3} + 390 \, a^{3} c^{4} d^{6} e^{5} - 120 \, a^{4} c^{3} d^{4} e^{7} + 16 \, a^{5} c^{2} d^{2} e^{9}\right )} x^{2} + {\left (19305 \, a^{2} c^{5} d^{9} e^{2} + 2860 \, a^{3} c^{4} d^{7} e^{4} - 1560 \, a^{4} c^{3} d^{5} e^{6} + 480 \, a^{5} c^{2} d^{3} e^{8} - 64 \, a^{6} c d e^{10}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{45045 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
2/45045*(3003*c^7*d^7*e^4*x^7 + 6435*a^3*c^4*d^8*e^3 - 5720*a^4*c^3*d^6*e^ 5 + 3120*a^5*c^2*d^4*e^7 - 960*a^6*c*d^2*e^9 + 128*a^7*e^11 + 231*(60*c^7* d^8*e^3 + 31*a*c^6*d^6*e^5)*x^6 + 63*(390*c^7*d^9*e^2 + 540*a*c^6*d^7*e^4 + 71*a^2*c^5*d^5*e^6)*x^5 + 35*(572*c^7*d^10*e + 1794*a*c^6*d^8*e^3 + 636* a^2*c^5*d^6*e^5 + a^3*c^4*d^4*e^7)*x^4 + 5*(1287*c^7*d^11 + 10868*a*c^6*d^ 9*e^2 + 8814*a^2*c^5*d^7*e^4 + 60*a^3*c^4*d^5*e^6 - 8*a^4*c^3*d^3*e^8)*x^3 + 3*(6435*a*c^6*d^10*e + 14300*a^2*c^5*d^8*e^3 + 390*a^3*c^4*d^6*e^5 - 12 0*a^4*c^3*d^4*e^7 + 16*a^5*c^2*d^2*e^9)*x^2 + (19305*a^2*c^5*d^9*e^2 + 286 0*a^3*c^4*d^7*e^4 - 1560*a^4*c^3*d^5*e^6 + 480*a^5*c^2*d^3*e^8 - 64*a^6*c* d*e^10)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^5*d^5*e*x + c^5*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 4241 vs. \(2 (265) = 530\).
Time = 0.47 (sec) , antiderivative size = 4241, normalized size of antiderivative = 14.38 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]
2/45045*(15015*a^2*d^4*((sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^2*e + a* e^3)*a*e^2)/(c*d) + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)/(c*d*e))*abs (e) + 3432*a*c*d^4*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c *d^2*e + a*e^3)*a^3*e^6)/(c^3*d^3*e^2) + (35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))/(c^3*d^3*e^5))*abs(e) + 42 9*c^2*d^6*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a *c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)/(c^3*d^3*e^2) + (35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3 /2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((e* x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))/(c^3*d^3*e^5))*abs(e)/e^2 + 2574*a^ 2*d^2*e^2*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a *c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*e^6)/(c^3*d^3*e^2) + (35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3 /2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((e* x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))/(c^3*d^3*e^5))*abs(e) - 572*c^2*d^5 *((35*sqrt(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6* e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sqrt(-c*d^2*e + a*e^3)* a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)/(c^4*d^4*e^3) + (105...
Time = 11.12 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.70 \[ \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e\,x^5\,\sqrt {d+e\,x}\,\left (71\,a^2\,e^4+540\,a\,c\,d^2\,e^2+390\,c^2\,d^4\right )}{715}+\frac {2\,x^4\,\sqrt {d+e\,x}\,\left (a^3\,e^6+636\,a^2\,c\,d^2\,e^4+1794\,a\,c^2\,d^4\,e^2+572\,c^3\,d^6\right )}{1287\,c\,d}+\frac {\sqrt {d+e\,x}\,\left (256\,a^7\,e^{11}-1920\,a^6\,c\,d^2\,e^9+6240\,a^5\,c^2\,d^4\,e^7-11440\,a^4\,c^3\,d^6\,e^5+12870\,a^3\,c^4\,d^8\,e^3\right )}{45045\,c^5\,d^5\,e}+\frac {2\,c^2\,d^2\,e^3\,x^7\,\sqrt {d+e\,x}}{15}+\frac {2\,a\,x^2\,\sqrt {d+e\,x}\,\left (16\,a^4\,e^8-120\,a^3\,c\,d^2\,e^6+390\,a^2\,c^2\,d^4\,e^4+14300\,a\,c^3\,d^6\,e^2+6435\,c^4\,d^8\right )}{15015\,c^3\,d^3}+\frac {x^3\,\sqrt {d+e\,x}\,\left (-80\,a^4\,c^3\,d^3\,e^8+600\,a^3\,c^4\,d^5\,e^6+88140\,a^2\,c^5\,d^7\,e^4+108680\,a\,c^6\,d^9\,e^2+12870\,c^7\,d^{11}\right )}{45045\,c^5\,d^5\,e}+\frac {2\,c\,d\,e^2\,x^6\,\left (60\,c\,d^2+31\,a\,e^2\right )\,\sqrt {d+e\,x}}{195}+\frac {2\,a^2\,e\,x\,\sqrt {d+e\,x}\,\left (-64\,a^4\,e^8+480\,a^3\,c\,d^2\,e^6-1560\,a^2\,c^2\,d^4\,e^4+2860\,a\,c^3\,d^6\,e^2+19305\,c^4\,d^8\right )}{45045\,c^4\,d^4}\right )}{x+\frac {d}{e}} \]
((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*e*x^5*(d + e*x)^(1/2)*( 71*a^2*e^4 + 390*c^2*d^4 + 540*a*c*d^2*e^2))/715 + (2*x^4*(d + e*x)^(1/2)* (a^3*e^6 + 572*c^3*d^6 + 1794*a*c^2*d^4*e^2 + 636*a^2*c*d^2*e^4))/(1287*c* d) + ((d + e*x)^(1/2)*(256*a^7*e^11 - 1920*a^6*c*d^2*e^9 + 12870*a^3*c^4*d ^8*e^3 - 11440*a^4*c^3*d^6*e^5 + 6240*a^5*c^2*d^4*e^7))/(45045*c^5*d^5*e) + (2*c^2*d^2*e^3*x^7*(d + e*x)^(1/2))/15 + (2*a*x^2*(d + e*x)^(1/2)*(16*a^ 4*e^8 + 6435*c^4*d^8 + 14300*a*c^3*d^6*e^2 - 120*a^3*c*d^2*e^6 + 390*a^2*c ^2*d^4*e^4))/(15015*c^3*d^3) + (x^3*(d + e*x)^(1/2)*(12870*c^7*d^11 + 1086 80*a*c^6*d^9*e^2 + 88140*a^2*c^5*d^7*e^4 + 600*a^3*c^4*d^5*e^6 - 80*a^4*c^ 3*d^3*e^8))/(45045*c^5*d^5*e) + (2*c*d*e^2*x^6*(31*a*e^2 + 60*c*d^2)*(d + e*x)^(1/2))/195 + (2*a^2*e*x*(d + e*x)^(1/2)*(19305*c^4*d^8 - 64*a^4*e^8 + 2860*a*c^3*d^6*e^2 + 480*a^3*c*d^2*e^6 - 1560*a^2*c^2*d^4*e^4))/(45045*c^ 4*d^4)))/(x + d/e)