Integrand size = 37, antiderivative size = 119 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=-\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{9/2}}{9 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{11/2}}{11 e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{13/2}}{13 e^4}+\frac {2 c^3 d^3 (d+e x)^{15/2}}{15 e^4} \] Output:
-2/9*(-a*e^2+c*d^2)^3*(e*x+d)^(9/2)/e^4+6/11*c*d*(-a*e^2+c*d^2)^2*(e*x+d)^ (11/2)/e^4-6/13*c^2*d^2*(-a*e^2+c*d^2)*(e*x+d)^(13/2)/e^4+2/15*c^3*d^3*(e* x+d)^(15/2)/e^4
Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {2 (d+e x)^{9/2} \left (715 a^3 e^6-195 a^2 c d e^4 (2 d-9 e x)+15 a c^2 d^2 e^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )}{6435 e^4} \] Input:
Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
(2*(d + e*x)^(9/2)*(715*a^3*e^6 - 195*a^2*c*d*e^4*(2*d - 9*e*x) + 15*a*c^2 *d^2*e^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + c^3*d^3*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3)))/(6435*e^4)
Time = 0.44 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {1121, 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.
| \(\displaystyle \int \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \left (-\frac {3 c^2 d^2 (d+e x)^{11/2} \left (c d^2-a e^2\right )}{e^3}+\frac {3 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )^2}{e^3}+\frac {(d+e x)^{7/2} \left (a e^2-c d^2\right )^3}{e^3}+\frac {c^3 d^3 (d+e x)^{13/2}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 c^2 d^2 (d+e x)^{13/2} \left (c d^2-a e^2\right )}{13 e^4}+\frac {6 c d (d+e x)^{11/2} \left (c d^2-a e^2\right )^2}{11 e^4}-\frac {2 (d+e x)^{9/2} \left (c d^2-a e^2\right )^3}{9 e^4}+\frac {2 c^3 d^3 (d+e x)^{15/2}}{15 e^4}\) |
Input:
Int[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(9/2))/(9*e^4) + (6*c*d*(c*d^2 - a*e^2)^2* (d + e*x)^(11/2))/(11*e^4) - (6*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x)^(13/2))/ (13*e^4) + (2*c^3*d^3*(d + e*x)^(15/2))/(15*e^4)
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 2.56 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {6 \left (a \,e^{2}-c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a \,e^{2}-c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\) | \(97\) |
| default | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {6 \left (a \,e^{2}-c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a \,e^{2}-c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\) | \(97\) |
| pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (e^{6} a^{3}+\frac {27 x \,a^{2} c d \,e^{5}}{11}-\frac {6 c a \,d^{2} \left (-\frac {99 c \,x^{2}}{26}+a \right ) e^{4}}{11}-\frac {108 x \,c^{2} d^{3} \left (-\frac {143 c \,x^{2}}{180}+a \right ) e^{3}}{143}+\frac {24 c^{2} d^{4} \left (-\frac {33 c \,x^{2}}{20}+a \right ) e^{2}}{143}+\frac {72 c^{3} d^{5} e x}{715}-\frac {16 d^{6} c^{3}}{715}\right )}{9 e^{4}}\) | \(107\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (429 c^{3} d^{3} e^{3} x^{3}+1485 x^{2} a \,c^{2} d^{2} e^{4}-198 c^{3} d^{4} e^{2} x^{2}+1755 x \,a^{2} c d \,e^{5}-540 x a \,c^{2} d^{3} e^{3}+72 c^{3} d^{5} e x +715 e^{6} a^{3}-390 d^{2} e^{4} a^{2} c +120 d^{4} e^{2} a \,c^{2}-16 d^{6} c^{3}\right )}{6435 e^{4}}\) | \(131\) |
| orering | \(\frac {2 \left (429 c^{3} d^{3} e^{3} x^{3}+1485 x^{2} a \,c^{2} d^{2} e^{4}-198 c^{3} d^{4} e^{2} x^{2}+1755 x \,a^{2} c d \,e^{5}-540 x a \,c^{2} d^{3} e^{3}+72 c^{3} d^{5} e x +715 e^{6} a^{3}-390 d^{2} e^{4} a^{2} c +120 d^{4} e^{2} a \,c^{2}-16 d^{6} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{3}}{6435 e^{4} \left (c d x +a e \right )^{3}}\) | \(168\) |
| trager | \(\frac {2 \left (429 c^{3} d^{3} e^{7} x^{7}+1485 a \,c^{2} d^{2} e^{8} x^{6}+1518 c^{3} d^{4} e^{6} x^{6}+1755 a^{2} c d \,e^{9} x^{5}+5400 a \,c^{2} d^{3} e^{7} x^{5}+1854 c^{3} d^{5} e^{5} x^{5}+715 a^{3} e^{10} x^{4}+6630 a^{2} c \,d^{2} e^{8} x^{4}+6870 d^{4} a \,c^{2} e^{6} x^{4}+800 c^{3} d^{6} e^{4} x^{4}+2860 a^{3} d \,e^{9} x^{3}+8970 a^{2} c \,d^{3} e^{7} x^{3}+3180 a \,c^{2} d^{5} e^{5} x^{3}+5 c^{3} d^{7} e^{3} x^{3}+4290 a^{3} d^{2} e^{8} x^{2}+4680 a^{2} c \,d^{4} e^{6} x^{2}+45 a \,c^{2} d^{6} e^{4} x^{2}-6 c^{3} d^{8} e^{2} x^{2}+2860 a^{3} d^{3} e^{7} x +195 a^{2} c \,d^{5} e^{5} x -60 a \,c^{2} d^{7} e^{3} x +8 c^{3} d^{9} e x +715 a^{3} d^{4} e^{6}-390 a^{2} c \,d^{6} e^{4}+120 a \,c^{2} d^{8} e^{2}-16 c^{3} d^{10}\right ) \sqrt {e x +d}}{6435 e^{4}}\) | \(359\) |
| risch | \(\frac {2 \left (429 c^{3} d^{3} e^{7} x^{7}+1485 a \,c^{2} d^{2} e^{8} x^{6}+1518 c^{3} d^{4} e^{6} x^{6}+1755 a^{2} c d \,e^{9} x^{5}+5400 a \,c^{2} d^{3} e^{7} x^{5}+1854 c^{3} d^{5} e^{5} x^{5}+715 a^{3} e^{10} x^{4}+6630 a^{2} c \,d^{2} e^{8} x^{4}+6870 d^{4} a \,c^{2} e^{6} x^{4}+800 c^{3} d^{6} e^{4} x^{4}+2860 a^{3} d \,e^{9} x^{3}+8970 a^{2} c \,d^{3} e^{7} x^{3}+3180 a \,c^{2} d^{5} e^{5} x^{3}+5 c^{3} d^{7} e^{3} x^{3}+4290 a^{3} d^{2} e^{8} x^{2}+4680 a^{2} c \,d^{4} e^{6} x^{2}+45 a \,c^{2} d^{6} e^{4} x^{2}-6 c^{3} d^{8} e^{2} x^{2}+2860 a^{3} d^{3} e^{7} x +195 a^{2} c \,d^{5} e^{5} x -60 a \,c^{2} d^{7} e^{3} x +8 c^{3} d^{9} e x +715 a^{3} d^{4} e^{6}-390 a^{2} c \,d^{6} e^{4}+120 a \,c^{2} d^{8} e^{2}-16 c^{3} d^{10}\right ) \sqrt {e x +d}}{6435 e^{4}}\) | \(359\) |
Input:
int((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x,method=_RETURNVERB OSE)
Output:
2/e^4*(1/15*c^3*d^3*(e*x+d)^(15/2)+3/13*(a*e^2-c*d^2)*c^2*d^2*(e*x+d)^(13/ 2)+3/11*(a*e^2-c*d^2)^2*c*d*(e*x+d)^(11/2)+1/9*(a*e^2-c*d^2)^3*(e*x+d)^(9/ 2))
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (103) = 206\).
Time = 0.08 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.82 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {2 \, {\left (429 \, c^{3} d^{3} e^{7} x^{7} - 16 \, c^{3} d^{10} + 120 \, a c^{2} d^{8} e^{2} - 390 \, a^{2} c d^{6} e^{4} + 715 \, a^{3} d^{4} e^{6} + 33 \, {\left (46 \, c^{3} d^{4} e^{6} + 45 \, a c^{2} d^{2} e^{8}\right )} x^{6} + 9 \, {\left (206 \, c^{3} d^{5} e^{5} + 600 \, a c^{2} d^{3} e^{7} + 195 \, a^{2} c d e^{9}\right )} x^{5} + 5 \, {\left (160 \, c^{3} d^{6} e^{4} + 1374 \, a c^{2} d^{4} e^{6} + 1326 \, a^{2} c d^{2} e^{8} + 143 \, a^{3} e^{10}\right )} x^{4} + 5 \, {\left (c^{3} d^{7} e^{3} + 636 \, a c^{2} d^{5} e^{5} + 1794 \, a^{2} c d^{3} e^{7} + 572 \, a^{3} d e^{9}\right )} x^{3} - 3 \, {\left (2 \, c^{3} d^{8} e^{2} - 15 \, a c^{2} d^{6} e^{4} - 1560 \, a^{2} c d^{4} e^{6} - 1430 \, a^{3} d^{2} e^{8}\right )} x^{2} + {\left (8 \, c^{3} d^{9} e - 60 \, a c^{2} d^{7} e^{3} + 195 \, a^{2} c d^{5} e^{5} + 2860 \, a^{3} d^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{4}} \] Input:
integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm=" fricas")
Output:
2/6435*(429*c^3*d^3*e^7*x^7 - 16*c^3*d^10 + 120*a*c^2*d^8*e^2 - 390*a^2*c* d^6*e^4 + 715*a^3*d^4*e^6 + 33*(46*c^3*d^4*e^6 + 45*a*c^2*d^2*e^8)*x^6 + 9 *(206*c^3*d^5*e^5 + 600*a*c^2*d^3*e^7 + 195*a^2*c*d*e^9)*x^5 + 5*(160*c^3* d^6*e^4 + 1374*a*c^2*d^4*e^6 + 1326*a^2*c*d^2*e^8 + 143*a^3*e^10)*x^4 + 5* (c^3*d^7*e^3 + 636*a*c^2*d^5*e^5 + 1794*a^2*c*d^3*e^7 + 572*a^3*d*e^9)*x^3 - 3*(2*c^3*d^8*e^2 - 15*a*c^2*d^6*e^4 - 1560*a^2*c*d^4*e^6 - 1430*a^3*d^2 *e^8)*x^2 + (8*c^3*d^9*e - 60*a*c^2*d^7*e^3 + 195*a^2*c*d^5*e^5 + 2860*a^3 *d^3*e^7)*x)*sqrt(e*x + d)/e^4
Time = 0.98 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.50 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\begin {cases} \frac {2 \left (\frac {c^{3} d^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{3}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (3 a c^{2} d^{2} e^{2} - 3 c^{3} d^{4}\right )}{13 e^{3}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (3 a^{2} c d e^{4} - 6 a c^{2} d^{3} e^{2} + 3 c^{3} d^{5}\right )}{11 e^{3}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}\right )}{9 e^{3}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {c^{3} d^{\frac {13}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**(1/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
Output:
Piecewise((2*(c**3*d**3*(d + e*x)**(15/2)/(15*e**3) + (d + e*x)**(13/2)*(3 *a*c**2*d**2*e**2 - 3*c**3*d**4)/(13*e**3) + (d + e*x)**(11/2)*(3*a**2*c*d *e**4 - 6*a*c**2*d**3*e**2 + 3*c**3*d**5)/(11*e**3) + (d + e*x)**(9/2)*(a* *3*e**6 - 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 - c**3*d**6)/(9*e**3))/e , Ne(e, 0)), (c**3*d**(13/2)*x**4/4, True))
Time = 0.03 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.15 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} d^{3} - 1485 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 1755 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 715 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{6435 \, e^{4}} \] Input:
integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm=" maxima")
Output:
2/6435*(429*(e*x + d)^(15/2)*c^3*d^3 - 1485*(c^3*d^4 - a*c^2*d^2*e^2)*(e*x + d)^(13/2) + 1755*(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(1 1/2) - 715*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*(e*x + d)^(9/2))/e^4
Leaf count of result is larger than twice the leaf count of optimal. 1214 vs. \(2 (103) = 206\).
Time = 0.16 (sec) , antiderivative size = 1214, normalized size of antiderivative = 10.20 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm=" giac")
Output:
2/45045*(45045*sqrt(e*x + d)*a^3*d^4*e^3 + 45045*((e*x + d)^(3/2) - 3*sqrt (e*x + d)*d)*a^2*c*d^5*e + 60060*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3 *d^3*e^3 + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*c^2*d^6/e + 36036*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15 *sqrt(e*x + d)*d^2)*a^2*c*d^4*e + 18018*(3*(e*x + d)^(5/2) - 10*(e*x + d)^ (3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*d^2*e^3 + 1287*(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)*c^3*d^ 7/e^3 + 15444*(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^2*d^5/e + 23166*(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^2*c* d^3*e + 5148*(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^3*d*e^3 + 572*(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^3*d^6/e^3 + 2574*(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*d^4/e + 1716*(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^2*c*d^2*e + 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^3*e^ 3 + 390*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7...
Time = 5.16 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4} \] Input:
int((d + e*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)
Output:
(2*(a*e^2 - c*d^2)^3*(d + e*x)^(9/2))/(9*e^4) - ((6*c^3*d^4 - 6*a*c^2*d^2* e^2)*(d + e*x)^(13/2))/(13*e^4) + (2*c^3*d^3*(d + e*x)^(15/2))/(15*e^4) + (6*c*d*(a*e^2 - c*d^2)^2*(d + e*x)^(11/2))/(11*e^4)
Time = 0.20 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.00 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {2 \sqrt {e x +d}\, \left (429 c^{3} d^{3} e^{7} x^{7}+1485 a \,c^{2} d^{2} e^{8} x^{6}+1518 c^{3} d^{4} e^{6} x^{6}+1755 a^{2} c d \,e^{9} x^{5}+5400 a \,c^{2} d^{3} e^{7} x^{5}+1854 c^{3} d^{5} e^{5} x^{5}+715 a^{3} e^{10} x^{4}+6630 a^{2} c \,d^{2} e^{8} x^{4}+6870 a \,c^{2} d^{4} e^{6} x^{4}+800 c^{3} d^{6} e^{4} x^{4}+2860 a^{3} d \,e^{9} x^{3}+8970 a^{2} c \,d^{3} e^{7} x^{3}+3180 a \,c^{2} d^{5} e^{5} x^{3}+5 c^{3} d^{7} e^{3} x^{3}+4290 a^{3} d^{2} e^{8} x^{2}+4680 a^{2} c \,d^{4} e^{6} x^{2}+45 a \,c^{2} d^{6} e^{4} x^{2}-6 c^{3} d^{8} e^{2} x^{2}+2860 a^{3} d^{3} e^{7} x +195 a^{2} c \,d^{5} e^{5} x -60 a \,c^{2} d^{7} e^{3} x +8 c^{3} d^{9} e x +715 a^{3} d^{4} e^{6}-390 a^{2} c \,d^{6} e^{4}+120 a \,c^{2} d^{8} e^{2}-16 c^{3} d^{10}\right )}{6435 e^{4}} \] Input:
int((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
Output:
(2*sqrt(d + e*x)*(715*a**3*d**4*e**6 + 2860*a**3*d**3*e**7*x + 4290*a**3*d **2*e**8*x**2 + 2860*a**3*d*e**9*x**3 + 715*a**3*e**10*x**4 - 390*a**2*c*d **6*e**4 + 195*a**2*c*d**5*e**5*x + 4680*a**2*c*d**4*e**6*x**2 + 8970*a**2 *c*d**3*e**7*x**3 + 6630*a**2*c*d**2*e**8*x**4 + 1755*a**2*c*d*e**9*x**5 + 120*a*c**2*d**8*e**2 - 60*a*c**2*d**7*e**3*x + 45*a*c**2*d**6*e**4*x**2 + 3180*a*c**2*d**5*e**5*x**3 + 6870*a*c**2*d**4*e**6*x**4 + 5400*a*c**2*d** 3*e**7*x**5 + 1485*a*c**2*d**2*e**8*x**6 - 16*c**3*d**10 + 8*c**3*d**9*e*x - 6*c**3*d**8*e**2*x**2 + 5*c**3*d**7*e**3*x**3 + 800*c**3*d**6*e**4*x**4 + 1854*c**3*d**5*e**5*x**5 + 1518*c**3*d**4*e**6*x**6 + 429*c**3*d**3*e** 7*x**7))/(6435*e**4)