Integrand size = 39, antiderivative size = 240 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {2 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^4 d^4 (d+e x)^{7/2}}+\frac {2 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{3 c^4 d^4 (d+e x)^{9/2}}+\frac {6 e^2 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{11/2}}{11 c^4 d^4 (d+e x)^{11/2}}+\frac {2 e^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{13/2}}{13 c^4 d^4 (d+e x)^{13/2}} \] Output:
2/7*(-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)^(7/2)+2/3*e*(-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^ 4/d^4/(e*x+d)^(9/2)+6/11*e^2*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x ^2)^(11/2)/c^4/d^4/(e*x+d)^(11/2)+2/13*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^ 2)^(13/2)/c^4/d^4/(e*x+d)^(13/2)
Time = 0.13 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.59 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (-16 a^3 e^6+8 a^2 c d e^4 (13 d+7 e x)-2 a c^2 d^2 e^2 \left (143 d^2+182 d e x+63 e^2 x^2\right )+c^3 d^3 \left (429 d^3+1001 d^2 e x+819 d e^2 x^2+231 e^3 x^3\right )\right )}{3003 c^4 d^4 \sqrt {d+e x}} \] Input:
Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]
Output:
(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-16*a^3*e^6 + 8*a^2*c*d* e^4*(13*d + 7*e*x) - 2*a*c^2*d^2*e^2*(143*d^2 + 182*d*e*x + 63*e^2*x^2) + c^3*d^3*(429*d^3 + 1001*d^2*e*x + 819*d*e^2*x^2 + 231*e^3*x^3)))/(3003*c^4 *d^4*Sqrt[d + e*x])
Time = 0.66 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {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.
| \(\displaystyle \int \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {6 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{\sqrt {d+e x}}dx}{13 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^{3/2}}dx}{11 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}}\right )}{13 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^{5/2}}dx}{9 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}}\right )}{11 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}}\right )}{13 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}+\frac {6 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 c d (d+e x)^{3/2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 c d^2 (d+e x)^{7/2}}\right )}{11 d}\right )}{13 d}\) |
Input:
Int[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]
Output:
(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)
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 = 160, normalized size of antiderivative = 0.67
| method | result | size |
| default | \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (c d x +a e \right )^{3} \left (-231 c^{3} d^{3} e^{3} x^{3}+126 x^{2} a \,c^{2} d^{2} e^{4}-819 c^{3} d^{4} e^{2} x^{2}-56 x \,a^{2} c d \,e^{5}+364 x a \,c^{2} d^{3} e^{3}-1001 c^{3} d^{5} e x +16 e^{6} a^{3}-104 d^{2} e^{4} a^{2} c +286 d^{4} e^{2} a \,c^{2}-429 d^{6} c^{3}\right )}{3003 \sqrt {e x +d}\, d^{4} c^{4}}\) | \(160\) |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-231 c^{3} d^{3} e^{3} x^{3}+126 x^{2} a \,c^{2} d^{2} e^{4}-819 c^{3} d^{4} e^{2} x^{2}-56 x \,a^{2} c d \,e^{5}+364 x a \,c^{2} d^{3} e^{3}-1001 c^{3} d^{5} e x +16 e^{6} a^{3}-104 d^{2} e^{4} a^{2} c +286 d^{4} e^{2} a \,c^{2}-429 d^{6} c^{3}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 d^{4} c^{4} \left (e x +d \right )^{\frac {5}{2}}}\) | \(168\) |
| orering | \(-\frac {2 \left (-231 c^{3} d^{3} e^{3} x^{3}+126 x^{2} a \,c^{2} d^{2} e^{4}-819 c^{3} d^{4} e^{2} x^{2}-56 x \,a^{2} c d \,e^{5}+364 x a \,c^{2} d^{3} e^{3}-1001 c^{3} d^{5} e x +16 e^{6} a^{3}-104 d^{2} e^{4} a^{2} c +286 d^{4} e^{2} a \,c^{2}-429 d^{6} c^{3}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}{3003 d^{4} c^{4} \left (e x +d \right )^{\frac {5}{2}}}\) | \(169\) |
Input:
int((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RETURN VERBOSE)
Output:
-2/3003/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(c*d*x+a*e)^3*(-231*c^3* d^3*e^3*x^3+126*a*c^2*d^2*e^4*x^2-819*c^3*d^4*e^2*x^2-56*a^2*c*d*e^5*x+364 *a*c^2*d^3*e^3*x-1001*c^3*d^5*e*x+16*a^3*e^6-104*a^2*c*d^2*e^4+286*a*c^2*d ^4*e^2-429*c^3*d^6)/d^4/c^4
Time = 0.09 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.48 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {2 \, {\left (231 \, c^{6} d^{6} e^{3} x^{6} + 429 \, a^{3} c^{3} d^{6} e^{3} - 286 \, a^{4} c^{2} d^{4} e^{5} + 104 \, a^{5} c d^{2} e^{7} - 16 \, a^{6} e^{9} + 63 \, {\left (13 \, c^{6} d^{7} e^{2} + 9 \, a c^{5} d^{5} e^{4}\right )} x^{5} + 7 \, {\left (143 \, c^{6} d^{8} e + 299 \, a c^{5} d^{6} e^{3} + 53 \, a^{2} c^{4} d^{4} e^{5}\right )} x^{4} + {\left (429 \, c^{6} d^{9} + 2717 \, a c^{5} d^{7} e^{2} + 1469 \, a^{2} c^{4} d^{5} e^{4} + 5 \, a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (429 \, a c^{5} d^{8} e + 715 \, a^{2} c^{4} d^{6} e^{3} + 13 \, a^{3} c^{3} d^{4} e^{5} - 2 \, a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1287 \, a^{2} c^{4} d^{7} e^{2} + 143 \, a^{3} c^{3} d^{5} e^{4} - 52 \, a^{4} c^{2} d^{3} e^{6} + 8 \, a^{5} c d e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3003 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \] Input:
integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorit hm="fricas")
Output:
2/3003*(231*c^6*d^6*e^3*x^6 + 429*a^3*c^3*d^6*e^3 - 286*a^4*c^2*d^4*e^5 + 104*a^5*c*d^2*e^7 - 16*a^6*e^9 + 63*(13*c^6*d^7*e^2 + 9*a*c^5*d^5*e^4)*x^5 + 7*(143*c^6*d^8*e + 299*a*c^5*d^6*e^3 + 53*a^2*c^4*d^4*e^5)*x^4 + (429*c ^6*d^9 + 2717*a*c^5*d^7*e^2 + 1469*a^2*c^4*d^5*e^4 + 5*a^3*c^3*d^3*e^6)*x^ 3 + 3*(429*a*c^5*d^8*e + 715*a^2*c^4*d^6*e^3 + 13*a^3*c^3*d^4*e^5 - 2*a^4* c^2*d^2*e^7)*x^2 + (1287*a^2*c^4*d^7*e^2 + 143*a^3*c^3*d^5*e^4 - 52*a^4*c^ 2*d^3*e^6 + 8*a^5*c*d*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)* sqrt(e*x + d)/(c^4*d^4*e*x + c^4*d^5)
Timed out. \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(1/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
Output:
Timed out
Time = 0.07 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.40 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {2 \, {\left (231 \, c^{6} d^{6} e^{3} x^{6} + 429 \, a^{3} c^{3} d^{6} e^{3} - 286 \, a^{4} c^{2} d^{4} e^{5} + 104 \, a^{5} c d^{2} e^{7} - 16 \, a^{6} e^{9} + 63 \, {\left (13 \, c^{6} d^{7} e^{2} + 9 \, a c^{5} d^{5} e^{4}\right )} x^{5} + 7 \, {\left (143 \, c^{6} d^{8} e + 299 \, a c^{5} d^{6} e^{3} + 53 \, a^{2} c^{4} d^{4} e^{5}\right )} x^{4} + {\left (429 \, c^{6} d^{9} + 2717 \, a c^{5} d^{7} e^{2} + 1469 \, a^{2} c^{4} d^{5} e^{4} + 5 \, a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (429 \, a c^{5} d^{8} e + 715 \, a^{2} c^{4} d^{6} e^{3} + 13 \, a^{3} c^{3} d^{4} e^{5} - 2 \, a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1287 \, a^{2} c^{4} d^{7} e^{2} + 143 \, a^{3} c^{3} d^{5} e^{4} - 52 \, a^{4} c^{2} d^{3} e^{6} + 8 \, a^{5} c d e^{8}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{3003 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \] Input:
integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorit hm="maxima")
Output:
2/3003*(231*c^6*d^6*e^3*x^6 + 429*a^3*c^3*d^6*e^3 - 286*a^4*c^2*d^4*e^5 + 104*a^5*c*d^2*e^7 - 16*a^6*e^9 + 63*(13*c^6*d^7*e^2 + 9*a*c^5*d^5*e^4)*x^5 + 7*(143*c^6*d^8*e + 299*a*c^5*d^6*e^3 + 53*a^2*c^4*d^4*e^5)*x^4 + (429*c ^6*d^9 + 2717*a*c^5*d^7*e^2 + 1469*a^2*c^4*d^5*e^4 + 5*a^3*c^3*d^3*e^6)*x^ 3 + 3*(429*a*c^5*d^8*e + 715*a^2*c^4*d^6*e^3 + 13*a^3*c^3*d^4*e^5 - 2*a^4* c^2*d^2*e^7)*x^2 + (1287*a^2*c^4*d^7*e^2 + 143*a^3*c^3*d^5*e^4 - 52*a^4*c^ 2*d^3*e^6 + 8*a^5*c*d*e^8)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^4*d^4*e*x + c ^4*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 1195 vs. \(2 (216) = 432\).
Time = 0.16 (sec) , antiderivative size = 1195, normalized size of antiderivative = 4.98 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorit hm="giac")
Output:
2/15015*(15015*sqrt(c*d*x + a*e)*a^3*d^3*e^3 - 15015*(3*sqrt(c*d*x + a*e)* a*e - (c*d*x + a*e)^(3/2))*a^2*d^3*e^2 - 15015*(3*sqrt(c*d*x + a*e)*a*e - (c*d*x + a*e)^(3/2))*a^3*d*e^4/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))*a*d^3*e + 9009*(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*d*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))*a^3*e^5/(c^2*d) - 429*(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^3 - 3861*(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*d*e^2/c - 3861*(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^4/(c^2* d) - 429*(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^3*e^6/(c^3*d^3) + 14 3*(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))*d*e/c + 429*(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^3/(c^2*d) + 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^...
Time = 5.64 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.60 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (x^4\,\sqrt {d+e\,x}\,\left (\frac {106\,a^2\,e^4}{429}+\frac {46\,a\,c\,d^2\,e^2}{33}+\frac {2\,c^2\,d^4}{3}\right )-\frac {\sqrt {d+e\,x}\,\left (32\,a^6\,e^9-208\,a^5\,c\,d^2\,e^7+572\,a^4\,c^2\,d^4\,e^5-858\,a^3\,c^3\,d^6\,e^3\right )}{3003\,c^4\,d^4\,e}+\frac {2\,c^2\,d^2\,e^2\,x^6\,\sqrt {d+e\,x}}{13}+\frac {x^3\,\sqrt {d+e\,x}\,\left (10\,a^3\,c^3\,d^3\,e^6+2938\,a^2\,c^4\,d^5\,e^4+5434\,a\,c^5\,d^7\,e^2+858\,c^6\,d^9\right )}{3003\,c^4\,d^4\,e}+\frac {6\,c\,d\,e\,x^5\,\left (13\,c\,d^2+9\,a\,e^2\right )\,\sqrt {d+e\,x}}{143}+\frac {2\,a\,x^2\,\sqrt {d+e\,x}\,\left (-2\,a^3\,e^6+13\,a^2\,c\,d^2\,e^4+715\,a\,c^2\,d^4\,e^2+429\,c^3\,d^6\right )}{1001\,c^2\,d^2}+\frac {2\,a^2\,e\,x\,\sqrt {d+e\,x}\,\left (8\,a^3\,e^6-52\,a^2\,c\,d^2\,e^4+143\,a\,c^2\,d^4\,e^2+1287\,c^3\,d^6\right )}{3003\,c^3\,d^3}\right )}{x+\frac {d}{e}} \] Input:
int((d + e*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)
Output:
((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(x^4*(d + e*x)^(1/2)*((106* a^2*e^4)/429 + (2*c^2*d^4)/3 + (46*a*c*d^2*e^2)/33) - ((d + e*x)^(1/2)*(32 *a^6*e^9 - 208*a^5*c*d^2*e^7 - 858*a^3*c^3*d^6*e^3 + 572*a^4*c^2*d^4*e^5)) /(3003*c^4*d^4*e) + (2*c^2*d^2*e^2*x^6*(d + e*x)^(1/2))/13 + (x^3*(d + e*x )^(1/2)*(858*c^6*d^9 + 5434*a*c^5*d^7*e^2 + 2938*a^2*c^4*d^5*e^4 + 10*a^3* c^3*d^3*e^6))/(3003*c^4*d^4*e) + (6*c*d*e*x^5*(9*a*e^2 + 13*c*d^2)*(d + e* x)^(1/2))/143 + (2*a*x^2*(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))/(1001*c^2*d^2) + (2*a^2*e*x*(d + e*x)^(1/ 2)*(8*a^3*e^6 + 1287*c^3*d^6 + 143*a*c^2*d^4*e^2 - 52*a^2*c*d^2*e^4))/(300 3*c^3*d^3)))/(x + d/e)
Time = 0.21 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.39 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (231 c^{6} d^{6} e^{3} x^{6}+567 a \,c^{5} d^{5} e^{4} x^{5}+819 c^{6} d^{7} e^{2} x^{5}+371 a^{2} c^{4} d^{4} e^{5} x^{4}+2093 a \,c^{5} d^{6} e^{3} x^{4}+1001 c^{6} d^{8} e \,x^{4}+5 a^{3} c^{3} d^{3} e^{6} x^{3}+1469 a^{2} c^{4} d^{5} e^{4} x^{3}+2717 a \,c^{5} d^{7} e^{2} x^{3}+429 c^{6} d^{9} x^{3}-6 a^{4} c^{2} d^{2} e^{7} x^{2}+39 a^{3} c^{3} d^{4} e^{5} x^{2}+2145 a^{2} c^{4} d^{6} e^{3} x^{2}+1287 a \,c^{5} d^{8} e \,x^{2}+8 a^{5} c d \,e^{8} x -52 a^{4} c^{2} d^{3} e^{6} x +143 a^{3} c^{3} d^{5} e^{4} x +1287 a^{2} c^{4} d^{7} e^{2} x -16 a^{6} e^{9}+104 a^{5} c \,d^{2} e^{7}-286 a^{4} c^{2} d^{4} e^{5}+429 a^{3} c^{3} d^{6} e^{3}\right )}{3003 c^{4} d^{4}} \] Input:
int((e*x+d)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
Output:
(2*sqrt(a*e + c*d*x)*( - 16*a**6*e**9 + 104*a**5*c*d**2*e**7 + 8*a**5*c*d* e**8*x - 286*a**4*c**2*d**4*e**5 - 52*a**4*c**2*d**3*e**6*x - 6*a**4*c**2* d**2*e**7*x**2 + 429*a**3*c**3*d**6*e**3 + 143*a**3*c**3*d**5*e**4*x + 39* a**3*c**3*d**4*e**5*x**2 + 5*a**3*c**3*d**3*e**6*x**3 + 1287*a**2*c**4*d** 7*e**2*x + 2145*a**2*c**4*d**6*e**3*x**2 + 1469*a**2*c**4*d**5*e**4*x**3 + 371*a**2*c**4*d**4*e**5*x**4 + 1287*a*c**5*d**8*e*x**2 + 2717*a*c**5*d**7 *e**2*x**3 + 2093*a*c**5*d**6*e**3*x**4 + 567*a*c**5*d**5*e**4*x**5 + 429* c**6*d**9*x**3 + 1001*c**6*d**8*e*x**4 + 819*c**6*d**7*e**2*x**5 + 231*c** 6*d**6*e**3*x**6))/(3003*c**4*d**4)