Integrand size = 39, antiderivative size = 250 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=-\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^2 (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}+\frac {c^3 d^3 \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 e^{5/2} \left (c d^2-a e^2\right )^{3/2}} \]
-1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e/(e*x+d)^(9/2)+1/8*c^3*d^3*a rctan(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^(1/2) /(e*x+d)^(1/2))/e^(5/2)/(-a*e^2+c*d^2)^(3/2)-1/4*c*d*(a*d*e+(a*e^2+c*d^2)* x+c*d*e*x^2)^(1/2)/e^2/(e*x+d)^(5/2)+1/8*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c* d*e*x^2)^(1/2)/e^2/(-a*e^2+c*d^2)/(e*x+d)^(3/2)
Time = 0.61 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {e} \sqrt {c d^2-a e^2} \sqrt {a e+c d x} \left (8 a^2 e^4-2 a c d e^2 (d-7 e x)+c^2 d^2 \left (-3 d^2-8 d e x+3 e^2 x^2\right )\right )+3 c^3 d^3 (d+e x)^3 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{24 e^{5/2} \left (c d^2-a e^2\right )^{3/2} \sqrt {a e+c d x} (d+e x)^{7/2}} \]
(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e]*Sqrt[c*d^2 - a*e^2]*Sqrt[a*e + c*d *x]*(8*a^2*e^4 - 2*a*c*d*e^2*(d - 7*e*x) + c^2*d^2*(-3*d^2 - 8*d*e*x + 3*e ^2*x^2)) + 3*c^3*d^3*(d + e*x)^3*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c *d^2 - a*e^2]]))/(24*e^(5/2)*(c*d^2 - a*e^2)^(3/2)*Sqrt[a*e + c*d*x]*(d + e*x)^(7/2))
Time = 0.39 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1130, 1130, 1135, 1136, 218}
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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1130 |
| \(\displaystyle \frac {c d \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{(d+e x)^{7/2}}dx}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1130 |
| \(\displaystyle \frac {c d \left (\frac {c d \int \frac {1}{(d+e x)^{3/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {c d \left (\frac {c d \left (\frac {c d \int \frac {1}{\sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {c d \left (\frac {c d \left (\frac {c d e \int \frac {1}{\frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}+\left (c d^2-a e^2\right ) e}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d^2-a e^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {c d \left (\frac {c d \left (\frac {c d \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\) |
-1/3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(e*(d + e*x)^(9/2)) + ( c*d*(-1/2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(e*(d + e*x)^(5/2)) + (c*d*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/((c*d^2 - a*e^2)*(d + e*x)^(3/2)) + (c*d*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e* x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(Sqrt[e]*(c*d^2 - a*e^2)^(3/2) )))/(4*e)))/(2*e)
3.21.44.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + p + 1))), x] - Simp[c*(p/(e^2*(m + p + 1))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] & & IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e))) Int [(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && I ntegerQ[2*p]
Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x _Symbol] :> Simp[2*e Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(446\) vs. \(2(218)=436\).
Time = 2.43 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.79
| method | result | size |
| default | \(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{3} e^{3} x^{3}+9 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{4} e^{2} x^{2}+9 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{5} e x +3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{3} d^{6}-3 c^{2} d^{2} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-14 a c d \,e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}+8 c^{2} d^{3} e x \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}-8 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} e^{4}+2 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}+3 \sqrt {c d x +a e}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \sqrt {c d x +a e}\, \left (e^{2} a -c \,d^{2}\right ) e^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\) | \(447\) |
1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c* d^2)*e)^(1/2))*c^3*d^3*e^3*x^3+9*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2 )*e)^(1/2))*c^3*d^4*e^2*x^2+9*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e )^(1/2))*c^3*d^5*e*x+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2) )*c^3*d^6-3*c^2*d^2*e^2*x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-14*a *c*d*e^3*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+8*c^2*d^3*e*x*(c*d*x+ a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-8*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^( 1/2)*a^2*e^4+2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a*c*d^2*e^2+3*(c* d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^4)/(e*x+d)^(7/2)/(c*d*x+a*e)^ (1/2)/(a*e^2-c*d^2)/e^2/((a*e^2-c*d^2)*e)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (218) = 436\).
Time = 0.39 (sec) , antiderivative size = 967, normalized size of antiderivative = 3.87 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\left [-\frac {3 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {-c d^{2} e + a e^{3}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d^{2} e + a e^{3}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (3 \, c^{3} d^{6} e - a c^{2} d^{4} e^{3} - 10 \, a^{2} c d^{2} e^{5} + 8 \, a^{3} e^{7} - 3 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + 2 \, {\left (4 \, c^{3} d^{5} e^{2} - 11 \, a c^{2} d^{3} e^{4} + 7 \, a^{2} c d e^{6}\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}}{48 \, {\left (c^{2} d^{8} e^{3} - 2 \, a c d^{6} e^{5} + a^{2} d^{4} e^{7} + {\left (c^{2} d^{4} e^{7} - 2 \, a c d^{2} e^{9} + a^{2} e^{11}\right )} x^{4} + 4 \, {\left (c^{2} d^{5} e^{6} - 2 \, a c d^{3} e^{8} + a^{2} d e^{10}\right )} x^{3} + 6 \, {\left (c^{2} d^{6} e^{5} - 2 \, a c d^{4} e^{7} + a^{2} d^{2} e^{9}\right )} x^{2} + 4 \, {\left (c^{2} d^{7} e^{4} - 2 \, a c d^{5} e^{6} + a^{2} d^{3} e^{8}\right )} x\right )}}, -\frac {3 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d^{2} e - a e^{3}} \sqrt {e x + d}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) + {\left (3 \, c^{3} d^{6} e - a c^{2} d^{4} e^{3} - 10 \, a^{2} c d^{2} e^{5} + 8 \, a^{3} e^{7} - 3 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} + 2 \, {\left (4 \, c^{3} d^{5} e^{2} - 11 \, a c^{2} d^{3} e^{4} + 7 \, a^{2} c d e^{6}\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}}{24 \, {\left (c^{2} d^{8} e^{3} - 2 \, a c d^{6} e^{5} + a^{2} d^{4} e^{7} + {\left (c^{2} d^{4} e^{7} - 2 \, a c d^{2} e^{9} + a^{2} e^{11}\right )} x^{4} + 4 \, {\left (c^{2} d^{5} e^{6} - 2 \, a c d^{3} e^{8} + a^{2} d e^{10}\right )} x^{3} + 6 \, {\left (c^{2} d^{6} e^{5} - 2 \, a c d^{4} e^{7} + a^{2} d^{2} e^{9}\right )} x^{2} + 4 \, {\left (c^{2} d^{7} e^{4} - 2 \, a c d^{5} e^{6} + a^{2} d^{3} e^{8}\right )} x\right )}}\right ] \]
[-1/48*(3*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c^3 *d^6*e*x + c^3*d^7)*sqrt(-c*d^2*e + a*e^3)*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c *d^2*e + a*e^3)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(3*c^3*d^6*e - a*c^2*d^4*e^3 - 10*a^2*c*d^2*e^5 + 8*a^3*e^7 - 3*(c^3*d^4*e^3 - a*c^2*d ^2*e^5)*x^2 + 2*(4*c^3*d^5*e^2 - 11*a*c^2*d^3*e^4 + 7*a^2*c*d*e^6)*x)*sqrt (c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^2*d^8*e^3 - 2*a* c*d^6*e^5 + a^2*d^4*e^7 + (c^2*d^4*e^7 - 2*a*c*d^2*e^9 + a^2*e^11)*x^4 + 4 *(c^2*d^5*e^6 - 2*a*c*d^3*e^8 + a^2*d*e^10)*x^3 + 6*(c^2*d^6*e^5 - 2*a*c*d ^4*e^7 + a^2*d^2*e^9)*x^2 + 4*(c^2*d^7*e^4 - 2*a*c*d^5*e^6 + a^2*d^3*e^8)* x), -1/24*(3*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4* c^3*d^6*e*x + c^3*d^7)*sqrt(c*d^2*e - a*e^3)*arctan(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)*sqrt(e*x + d)/(c*d*e^2*x^2 + a *d*e^2 + (c*d^2*e + a*e^3)*x)) + (3*c^3*d^6*e - a*c^2*d^4*e^3 - 10*a^2*c*d ^2*e^5 + 8*a^3*e^7 - 3*(c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 + 2*(4*c^3*d^5*e^ 2 - 11*a*c^2*d^3*e^4 + 7*a^2*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^2*d^8*e^3 - 2*a*c*d^6*e^5 + a^2*d^4*e^7 + (c^ 2*d^4*e^7 - 2*a*c*d^2*e^9 + a^2*e^11)*x^4 + 4*(c^2*d^5*e^6 - 2*a*c*d^3*e^8 + a^2*d*e^10)*x^3 + 6*(c^2*d^6*e^5 - 2*a*c*d^4*e^7 + a^2*d^2*e^9)*x^2 + 4 *(c^2*d^7*e^4 - 2*a*c*d^5*e^6 + a^2*d^3*e^8)*x)]
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]
Time = 0.44 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\frac {\frac {3 \, c^{4} d^{4} e {\left | e \right |} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{\sqrt {c d^{2} e - a e^{3}} {\left (c d^{2} - a e^{2}\right )}} - \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{6} d^{8} e^{3} {\left | e \right |} - 6 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{5} d^{6} e^{5} {\left | e \right |} + 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{4} d^{4} e^{7} {\left | e \right |} + 8 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{5} d^{6} e^{2} {\left | e \right |} - 8 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{4} d^{4} e^{4} {\left | e \right |} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{4} d^{4} e {\left | e \right |}}{{\left (c d^{2} - a e^{2}\right )} {\left (e x + d\right )}^{3} c^{3} d^{3} e^{3}}}{24 \, c d e^{4}} \]
1/24*(3*c^4*d^4*e*abs(e)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/sq rt(c*d^2*e - a*e^3))/(sqrt(c*d^2*e - a*e^3)*(c*d^2 - a*e^2)) - (3*sqrt((e* x + d)*c*d*e - c*d^2*e + a*e^3)*c^6*d^8*e^3*abs(e) - 6*sqrt((e*x + d)*c*d* e - c*d^2*e + a*e^3)*a*c^5*d^6*e^5*abs(e) + 3*sqrt((e*x + d)*c*d*e - c*d^2 *e + a*e^3)*a^2*c^4*d^4*e^7*abs(e) + 8*((e*x + d)*c*d*e - c*d^2*e + a*e^3) ^(3/2)*c^5*d^6*e^2*abs(e) - 8*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a* c^4*d^4*e^4*abs(e) - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^4*d^4*e *abs(e))/((c*d^2 - a*e^2)*(e*x + d)^3*c^3*d^3*e^3))/(c*d*e^4)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]