Integrand size = 39, antiderivative size = 305 \[ \int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {2 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c^5 d^5 (d+e x)^{5/2}}+\frac {8 e \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^5 d^5 (d+e x)^{7/2}}+\frac {4 e^2 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{3 c^5 d^5 (d+e x)^{9/2}}+\frac {8 e^3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{11/2}}{11 c^5 d^5 (d+e x)^{11/2}}+\frac {2 e^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{13/2}}{13 c^5 d^5 (d+e x)^{13/2}} \] Output:
2/5*(-a*e^2+c*d^2)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^5/d^5/(e*x+ d)^(5/2)+8/7*e*(-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^ 5/d^5/(e*x+d)^(7/2)+4/3*e^2*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e* x^2)^(9/2)/c^5/d^5/(e*x+d)^(9/2)+8/11*e^3*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d ^2)*x+c*d*e*x^2)^(11/2)/c^5/d^5/(e*x+d)^(11/2)+2/13*e^4*(a*d*e+(a*e^2+c*d^ 2)*x+c*d*e*x^2)^(13/2)/c^5/d^5/(e*x+d)^(13/2)
Time = 0.18 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.61 \[ \int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2} \left (128 a^4 e^8-64 a^3 c d e^6 (13 d+5 e x)+16 a^2 c^2 d^2 e^4 \left (143 d^2+130 d e x+35 e^2 x^2\right )-8 a c^3 d^3 e^2 \left (429 d^3+715 d^2 e x+455 d e^2 x^2+105 e^3 x^3\right )+c^4 d^4 \left (3003 d^4+8580 d^3 e x+10010 d^2 e^2 x^2+5460 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 c^5 d^5 (d+e x)^{5/2}} \] Input:
Integrate[(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
Output:
(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(128*a^4*e^8 - 64*a^3*c*d*e^6*(13*d + 5 *e*x) + 16*a^2*c^2*d^2*e^4*(143*d^2 + 130*d*e*x + 35*e^2*x^2) - 8*a*c^3*d^ 3*e^2*(429*d^3 + 715*d^2*e*x + 455*d*e^2*x^2 + 105*e^3*x^3) + c^4*d^4*(300 3*d^4 + 8580*d^3*e*x + 10010*d^2*e^2*x^2 + 5460*d*e^3*x^3 + 1155*e^4*x^4)) )/(15015*c^5*d^5*(d + e*x)^(5/2))
Time = 0.79 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.04, 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.
| \(\displaystyle \int (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {8 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x)^{3/2} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{13 d}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {8 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \int \sqrt {d+e x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{11 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d}\right )}{13 d}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {8 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{\sqrt {d+e x}}dx}{9 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}}\right )}{11 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d}\right )}{13 d}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {8 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2}}dx}{7 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}}\right )}{9 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}}\right )}{11 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d}\right )}{13 d}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d}+\frac {8 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d}+\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d \sqrt {d+e x}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 c d^2 (d+e x)^{5/2}}\right )}{9 d}\right )}{11 d}\right )}{13 d}\) |
Input:
Int[(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
Output:
(2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d) + (8*(d^2 - (a*e^2)/c)*((2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d *e*x^2)^(5/2))/(11*c*d) + (6*(d^2 - (a*e^2)/c)*((2*(a*d*e + (c*d^2 + a*e^2 )*x + c*d*e*x^2)^(5/2))/(9*c*d*Sqrt[d + e*x]) + (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)^(5/2))/(35*c*d^2* (d + e*x)^(5/2)) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c* d*(d + e*x)^(3/2))))/(9*d)))/(11*d)))/(13*d)
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 = 1.20 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (c d x +a e \right )^{2} \left (1155 c^{4} d^{4} e^{4} x^{4}-840 a \,c^{3} d^{3} e^{5} x^{3}+5460 c^{4} d^{5} e^{3} x^{3}+560 a^{2} c^{2} d^{2} e^{6} x^{2}-3640 a \,c^{3} d^{4} e^{4} x^{2}+10010 c^{4} d^{6} e^{2} x^{2}-320 a^{3} c d \,e^{7} x +2080 a^{2} c^{2} d^{3} e^{5} x -5720 a \,c^{3} d^{5} e^{3} x +8580 c^{4} d^{7} e x +128 a^{4} e^{8}-832 a^{3} c \,d^{2} e^{6}+2288 a^{2} c^{2} d^{4} e^{4}-3432 a \,c^{3} d^{6} e^{2}+3003 c^{4} d^{8}\right )}{15015 \sqrt {e x +d}\, d^{5} c^{5}}\) | \(235\) |
| gosper | \(\frac {2 \left (c d x +a e \right ) \left (1155 c^{4} d^{4} e^{4} x^{4}-840 a \,c^{3} d^{3} e^{5} x^{3}+5460 c^{4} d^{5} e^{3} x^{3}+560 a^{2} c^{2} d^{2} e^{6} x^{2}-3640 a \,c^{3} d^{4} e^{4} x^{2}+10010 c^{4} d^{6} e^{2} x^{2}-320 a^{3} c d \,e^{7} x +2080 a^{2} c^{2} d^{3} e^{5} x -5720 a \,c^{3} d^{5} e^{3} x +8580 c^{4} d^{7} e x +128 a^{4} e^{8}-832 a^{3} c \,d^{2} e^{6}+2288 a^{2} c^{2} d^{4} e^{4}-3432 a \,c^{3} d^{6} e^{2}+3003 c^{4} d^{8}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{15015 d^{5} c^{5} \left (e x +d \right )^{\frac {3}{2}}}\) | \(243\) |
| orering | \(\frac {2 \left (1155 c^{4} d^{4} e^{4} x^{4}-840 a \,c^{3} d^{3} e^{5} x^{3}+5460 c^{4} d^{5} e^{3} x^{3}+560 a^{2} c^{2} d^{2} e^{6} x^{2}-3640 a \,c^{3} d^{4} e^{4} x^{2}+10010 c^{4} d^{6} e^{2} x^{2}-320 a^{3} c d \,e^{7} x +2080 a^{2} c^{2} d^{3} e^{5} x -5720 a \,c^{3} d^{5} e^{3} x +8580 c^{4} d^{7} e x +128 a^{4} e^{8}-832 a^{3} c \,d^{2} e^{6}+2288 a^{2} c^{2} d^{4} e^{4}-3432 a \,c^{3} d^{6} e^{2}+3003 c^{4} d^{8}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{15015 d^{5} c^{5} \left (e x +d \right )^{\frac {3}{2}}}\) | \(244\) |
Input:
int((e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURN VERBOSE)
Output:
2/15015/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(c*d*x+a*e)^2*(1155*c^4* d^4*e^4*x^4-840*a*c^3*d^3*e^5*x^3+5460*c^4*d^5*e^3*x^3+560*a^2*c^2*d^2*e^6 *x^2-3640*a*c^3*d^4*e^4*x^2+10010*c^4*d^6*e^2*x^2-320*a^3*c*d*e^7*x+2080*a ^2*c^2*d^3*e^5*x-5720*a*c^3*d^5*e^3*x+8580*c^4*d^7*e*x+128*a^4*e^8-832*a^3 *c*d^2*e^6+2288*a^2*c^2*d^4*e^4-3432*a*c^3*d^6*e^2+3003*c^4*d^8)/d^5/c^5
Time = 0.09 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.29 \[ \int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {2 \, {\left (1155 \, c^{6} d^{6} e^{4} x^{6} + 3003 \, a^{2} c^{4} d^{8} e^{2} - 3432 \, a^{3} c^{3} d^{6} e^{4} + 2288 \, a^{4} c^{2} d^{4} e^{6} - 832 \, a^{5} c d^{2} e^{8} + 128 \, a^{6} e^{10} + 210 \, {\left (26 \, c^{6} d^{7} e^{3} + 7 \, a c^{5} d^{5} e^{5}\right )} x^{5} + 35 \, {\left (286 \, c^{6} d^{8} e^{2} + 208 \, a c^{5} d^{6} e^{4} + a^{2} c^{4} d^{4} e^{6}\right )} x^{4} + 20 \, {\left (429 \, c^{6} d^{9} e + 715 \, a c^{5} d^{7} e^{3} + 13 \, a^{2} c^{4} d^{5} e^{5} - 2 \, a^{3} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \, {\left (1001 \, c^{6} d^{10} + 4576 \, a c^{5} d^{8} e^{2} + 286 \, a^{2} c^{4} d^{6} e^{4} - 104 \, a^{3} c^{3} d^{4} e^{6} + 16 \, a^{4} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \, {\left (3003 \, a c^{5} d^{9} e + 858 \, a^{2} c^{4} d^{7} e^{3} - 572 \, a^{3} c^{3} d^{5} e^{5} + 208 \, a^{4} c^{2} d^{3} e^{7} - 32 \, a^{5} c d e^{9}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{15015 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \] Input:
integrate((e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="fricas")
Output:
2/15015*(1155*c^6*d^6*e^4*x^6 + 3003*a^2*c^4*d^8*e^2 - 3432*a^3*c^3*d^6*e^ 4 + 2288*a^4*c^2*d^4*e^6 - 832*a^5*c*d^2*e^8 + 128*a^6*e^10 + 210*(26*c^6* d^7*e^3 + 7*a*c^5*d^5*e^5)*x^5 + 35*(286*c^6*d^8*e^2 + 208*a*c^5*d^6*e^4 + a^2*c^4*d^4*e^6)*x^4 + 20*(429*c^6*d^9*e + 715*a*c^5*d^7*e^3 + 13*a^2*c^4 *d^5*e^5 - 2*a^3*c^3*d^3*e^7)*x^3 + 3*(1001*c^6*d^10 + 4576*a*c^5*d^8*e^2 + 286*a^2*c^4*d^6*e^4 - 104*a^3*c^3*d^4*e^6 + 16*a^4*c^2*d^2*e^8)*x^2 + 2* (3003*a*c^5*d^9*e + 858*a^2*c^4*d^7*e^3 - 572*a^3*c^3*d^5*e^5 + 208*a^4*c^ 2*d^3*e^7 - 32*a^5*c*d*e^9)*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. \[ \int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(5/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
Output:
Timed out
Time = 0.07 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.22 \[ \int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {2 \, {\left (1155 \, c^{6} d^{6} e^{4} x^{6} + 3003 \, a^{2} c^{4} d^{8} e^{2} - 3432 \, a^{3} c^{3} d^{6} e^{4} + 2288 \, a^{4} c^{2} d^{4} e^{6} - 832 \, a^{5} c d^{2} e^{8} + 128 \, a^{6} e^{10} + 210 \, {\left (26 \, c^{6} d^{7} e^{3} + 7 \, a c^{5} d^{5} e^{5}\right )} x^{5} + 35 \, {\left (286 \, c^{6} d^{8} e^{2} + 208 \, a c^{5} d^{6} e^{4} + a^{2} c^{4} d^{4} e^{6}\right )} x^{4} + 20 \, {\left (429 \, c^{6} d^{9} e + 715 \, a c^{5} d^{7} e^{3} + 13 \, a^{2} c^{4} d^{5} e^{5} - 2 \, a^{3} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \, {\left (1001 \, c^{6} d^{10} + 4576 \, a c^{5} d^{8} e^{2} + 286 \, a^{2} c^{4} d^{6} e^{4} - 104 \, a^{3} c^{3} d^{4} e^{6} + 16 \, a^{4} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \, {\left (3003 \, a c^{5} d^{9} e + 858 \, a^{2} c^{4} d^{7} e^{3} - 572 \, a^{3} c^{3} d^{5} e^{5} + 208 \, a^{4} c^{2} d^{3} e^{7} - 32 \, a^{5} c d e^{9}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{15015 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \] Input:
integrate((e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="maxima")
Output:
2/15015*(1155*c^6*d^6*e^4*x^6 + 3003*a^2*c^4*d^8*e^2 - 3432*a^3*c^3*d^6*e^ 4 + 2288*a^4*c^2*d^4*e^6 - 832*a^5*c*d^2*e^8 + 128*a^6*e^10 + 210*(26*c^6* d^7*e^3 + 7*a*c^5*d^5*e^5)*x^5 + 35*(286*c^6*d^8*e^2 + 208*a*c^5*d^6*e^4 + a^2*c^4*d^4*e^6)*x^4 + 20*(429*c^6*d^9*e + 715*a*c^5*d^7*e^3 + 13*a^2*c^4 *d^5*e^5 - 2*a^3*c^3*d^3*e^7)*x^3 + 3*(1001*c^6*d^10 + 4576*a*c^5*d^8*e^2 + 286*a^2*c^4*d^6*e^4 - 104*a^3*c^3*d^4*e^6 + 16*a^4*c^2*d^2*e^8)*x^2 + 2* (3003*a*c^5*d^9*e + 858*a^2*c^4*d^7*e^3 - 572*a^3*c^3*d^5*e^5 + 208*a^4*c^ 2*d^3*e^7 - 32*a^5*c*d*e^9)*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. 1115 vs. \(2 (275) = 550\).
Time = 0.18 (sec) , antiderivative size = 1115, normalized size of antiderivative = 3.66 \[ \int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit hm="giac")
Output:
2/45045*(45045*sqrt(c*d*x + a*e)*a^2*d^4*e^2 - 30030*(3*sqrt(c*d*x + a*e)* a*e - (c*d*x + a*e)^(3/2))*a*d^4*e - 60060*(3*sqrt(c*d*x + a*e)*a*e - (c*d *x + a*e)^(3/2))*a^2*d^2*e^3/c + 3003*(15*sqrt(c*d*x + a*e)*a^2*e^2 - 10*( c*d*x + a*e)^(3/2)*a*e + 3*(c*d*x + a*e)^(5/2))*d^4 + 24024*(15*sqrt(c*d*x + a*e)*a^2*e^2 - 10*(c*d*x + a*e)^(3/2)*a*e + 3*(c*d*x + a*e)^(5/2))*a*d^ 2*e^2/c + 18018*(15*sqrt(c*d*x + a*e)*a^2*e^2 - 10*(c*d*x + a*e)^(3/2)*a*e + 3*(c*d*x + a*e)^(5/2))*a^2*e^4/c^2 - 5148*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c*d*x + a*e)^(3/2)*a^2*e^2 + 21*(c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + a*e)^(7/2))*d^2*e/c - 15444*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c*d*x + a*e)^(3/2)*a^2*e^2 + 21*(c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + a*e)^(7/2))*a *e^3/c^2 - 5148*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c*d*x + a*e)^(3/2)*a^2 *e^2 + 21*(c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + a*e)^(7/2))*a^2*e^5/(c^3*d^ 2) + 858*(315*sqrt(c*d*x + a*e)*a^4*e^4 - 420*(c*d*x + a*e)^(3/2)*a^3*e^3 + 378*(c*d*x + a*e)^(5/2)*a^2*e^2 - 180*(c*d*x + a*e)^(7/2)*a*e + 35*(c*d* x + a*e)^(9/2))*e^2/c^2 + 1144*(315*sqrt(c*d*x + a*e)*a^4*e^4 - 420*(c*d*x + a*e)^(3/2)*a^3*e^3 + 378*(c*d*x + a*e)^(5/2)*a^2*e^2 - 180*(c*d*x + a*e )^(7/2)*a*e + 35*(c*d*x + a*e)^(9/2))*a*e^4/(c^3*d^2) + 143*(315*sqrt(c*d* x + a*e)*a^4*e^4 - 420*(c*d*x + a*e)^(3/2)*a^3*e^3 + 378*(c*d*x + a*e)^(5/ 2)*a^2*e^2 - 180*(c*d*x + a*e)^(7/2)*a*e + 35*(c*d*x + a*e)^(9/2))*a^2*e^6 /(c^4*d^4) - 260*(693*sqrt(c*d*x + a*e)*a^5*e^5 - 1155*(c*d*x + a*e)^(3...
Time = 5.63 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.39 \[ \int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,e^2\,x^5\,\left (26\,c\,d^2+7\,a\,e^2\right )\,\sqrt {d+e\,x}}{143}+\frac {2\,c\,d\,e^3\,x^6\,\sqrt {d+e\,x}}{13}+\frac {8\,x^3\,\sqrt {d+e\,x}\,\left (-2\,a^3\,e^6+13\,a^2\,c\,d^2\,e^4+715\,a\,c^2\,d^4\,e^2+429\,c^3\,d^6\right )}{3003\,c^2\,d^2}+\frac {\sqrt {d+e\,x}\,\left (256\,a^6\,e^{10}-1664\,a^5\,c\,d^2\,e^8+4576\,a^4\,c^2\,d^4\,e^6-6864\,a^3\,c^3\,d^6\,e^4+6006\,a^2\,c^4\,d^8\,e^2\right )}{15015\,c^5\,d^5\,e}+\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (a^2\,e^4+208\,a\,c\,d^2\,e^2+286\,c^2\,d^4\right )}{429\,c\,d}+\frac {x^2\,\sqrt {d+e\,x}\,\left (96\,a^4\,c^2\,d^2\,e^8-624\,a^3\,c^3\,d^4\,e^6+1716\,a^2\,c^4\,d^6\,e^4+27456\,a\,c^5\,d^8\,e^2+6006\,c^6\,d^{10}\right )}{15015\,c^5\,d^5\,e}+\frac {4\,a\,x\,\sqrt {d+e\,x}\,\left (-32\,a^4\,e^8+208\,a^3\,c\,d^2\,e^6-572\,a^2\,c^2\,d^4\,e^4+858\,a\,c^3\,d^6\,e^2+3003\,c^4\,d^8\right )}{15015\,c^4\,d^4}\right )}{x+\frac {d}{e}} \] Input:
int((d + e*x)^(5/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2),x)
Output:
((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((4*e^2*x^5*(7*a*e^2 + 26*c *d^2)*(d + e*x)^(1/2))/143 + (2*c*d*e^3*x^6*(d + e*x)^(1/2))/13 + (8*x^3*( d + e*x)^(1/2)*(429*c^3*d^6 - 2*a^3*e^6 + 715*a*c^2*d^4*e^2 + 13*a^2*c*d^2 *e^4))/(3003*c^2*d^2) + ((d + e*x)^(1/2)*(256*a^6*e^10 - 1664*a^5*c*d^2*e^ 8 + 6006*a^2*c^4*d^8*e^2 - 6864*a^3*c^3*d^6*e^4 + 4576*a^4*c^2*d^4*e^6))/( 15015*c^5*d^5*e) + (2*e*x^4*(d + e*x)^(1/2)*(a^2*e^4 + 286*c^2*d^4 + 208*a *c*d^2*e^2))/(429*c*d) + (x^2*(d + e*x)^(1/2)*(6006*c^6*d^10 + 27456*a*c^5 *d^8*e^2 + 1716*a^2*c^4*d^6*e^4 - 624*a^3*c^3*d^4*e^6 + 96*a^4*c^2*d^2*e^8 ))/(15015*c^5*d^5*e) + (4*a*x*(d + e*x)^(1/2)*(3003*c^4*d^8 - 32*a^4*e^8 + 858*a*c^3*d^6*e^2 + 208*a^3*c*d^2*e^6 - 572*a^2*c^2*d^4*e^4))/(15015*c^4* d^4)))/(x + d/e)
Time = 0.24 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.23 \[ \int (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (1155 c^{6} d^{6} e^{4} x^{6}+1470 a \,c^{5} d^{5} e^{5} x^{5}+5460 c^{6} d^{7} e^{3} x^{5}+35 a^{2} c^{4} d^{4} e^{6} x^{4}+7280 a \,c^{5} d^{6} e^{4} x^{4}+10010 c^{6} d^{8} e^{2} x^{4}-40 a^{3} c^{3} d^{3} e^{7} x^{3}+260 a^{2} c^{4} d^{5} e^{5} x^{3}+14300 a \,c^{5} d^{7} e^{3} x^{3}+8580 c^{6} d^{9} e \,x^{3}+48 a^{4} c^{2} d^{2} e^{8} x^{2}-312 a^{3} c^{3} d^{4} e^{6} x^{2}+858 a^{2} c^{4} d^{6} e^{4} x^{2}+13728 a \,c^{5} d^{8} e^{2} x^{2}+3003 c^{6} d^{10} x^{2}-64 a^{5} c d \,e^{9} x +416 a^{4} c^{2} d^{3} e^{7} x -1144 a^{3} c^{3} d^{5} e^{5} x +1716 a^{2} c^{4} d^{7} e^{3} x +6006 a \,c^{5} d^{9} e x +128 a^{6} e^{10}-832 a^{5} c \,d^{2} e^{8}+2288 a^{4} c^{2} d^{4} e^{6}-3432 a^{3} c^{3} d^{6} e^{4}+3003 a^{2} c^{4} d^{8} e^{2}\right )}{15015 c^{5} d^{5}} \] Input:
int((e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
Output:
(2*sqrt(a*e + c*d*x)*(128*a**6*e**10 - 832*a**5*c*d**2*e**8 - 64*a**5*c*d* e**9*x + 2288*a**4*c**2*d**4*e**6 + 416*a**4*c**2*d**3*e**7*x + 48*a**4*c* *2*d**2*e**8*x**2 - 3432*a**3*c**3*d**6*e**4 - 1144*a**3*c**3*d**5*e**5*x - 312*a**3*c**3*d**4*e**6*x**2 - 40*a**3*c**3*d**3*e**7*x**3 + 3003*a**2*c **4*d**8*e**2 + 1716*a**2*c**4*d**7*e**3*x + 858*a**2*c**4*d**6*e**4*x**2 + 260*a**2*c**4*d**5*e**5*x**3 + 35*a**2*c**4*d**4*e**6*x**4 + 6006*a*c**5 *d**9*e*x + 13728*a*c**5*d**8*e**2*x**2 + 14300*a*c**5*d**7*e**3*x**3 + 72 80*a*c**5*d**6*e**4*x**4 + 1470*a*c**5*d**5*e**5*x**5 + 3003*c**6*d**10*x* *2 + 8580*c**6*d**9*e*x**3 + 10010*c**6*d**8*e**2*x**4 + 5460*c**6*d**7*e* *3*x**5 + 1155*c**6*d**6*e**4*x**6))/(15015*c**5*d**5)