Integrand size = 39, antiderivative size = 331 \[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2}{24 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {35 c^3 d^3 \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {35 c^3 d^3 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 \left (c d^2-a e^2\right )^{9/2}} \]
-35/8*c^3*d^3*arctan(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^(1/2)/(-a*e^2+c*d^2)^(9/2)+1/3/(-a*e^2+c* d^2)/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+7/12*c*d/(-a*e^ 2+c*d^2)^2/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/24*c^2 *d^2/(-a*e^2+c*d^2)^3/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2 )-35/8*c^3*d^3*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e *x^2)^(1/2)
Time = 0.75 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {c d^2-a e^2} \left (8 a^3 e^6-2 a^2 c d e^4 (19 d+7 e x)+a c^2 d^2 e^2 \left (87 d^2+98 d e x+35 e^2 x^2\right )+c^3 d^3 \left (48 d^3+231 d^2 e x+280 d e^2 x^2+105 e^3 x^3\right )\right )-105 c^3 d^3 \sqrt {e} \sqrt {a e+c d x} (d+e x)^3 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{24 \left (c d^2-a e^2\right )^{9/2} (d+e x)^{5/2} \sqrt {(a e+c d x) (d+e x)}} \]
(-(Sqrt[c*d^2 - a*e^2]*(8*a^3*e^6 - 2*a^2*c*d*e^4*(19*d + 7*e*x) + a*c^2*d ^2*e^2*(87*d^2 + 98*d*e*x + 35*e^2*x^2) + c^3*d^3*(48*d^3 + 231*d^2*e*x + 280*d*e^2*x^2 + 105*e^3*x^3))) - 105*c^3*d^3*Sqrt[e]*Sqrt[a*e + c*d*x]*(d + e*x)^3*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(24*(c*d ^2 - a*e^2)^(9/2)*(d + e*x)^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
Time = 0.53 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1135, 1135, 1135, 1132, 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.
| \(\displaystyle \int \frac {1}{(d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {7 c d \int \frac {1}{(d+e x)^{3/2} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{6 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {7 c d \left (\frac {5 c d \int \frac {1}{\sqrt {d+e x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{6 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {7 c d \left (\frac {5 c d \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{6 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {7 c d \left (\frac {5 c d \left (\frac {3 c d \left (-\frac {e \int \frac {1}{\sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d^2-a e^2}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{6 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {7 c d \left (\frac {5 c d \left (\frac {3 c d \left (-\frac {2 e^2 \int \frac {1}{\frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}+\left (c d^2-a e^2\right ) e}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d^2-a e^2}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{6 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {7 c d \left (\frac {5 c d \left (\frac {3 c d \left (-\frac {2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{6 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\) |
1/(3*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d* e*x^2]) + (7*c*d*(1/(2*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (5*c*d*(1/((c*d^2 - a*e^2)*Sqrt[d + e*x]*Sqrt[ a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3*c*d*((-2*Sqrt[d + e*x])/((c*d ^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (2*Sqrt[e]*ArcT an[(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])])/(c*d^2 - a*e^2)^(3/2)))/(2*(c*d^2 - a*e^2))))/(4*(c*d ^2 - a*e^2))))/(6*(c*d^2 - a*e^2))
3.21.71.3.1 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[(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Free Q[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ [0, m, 1] && 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]
Time = 2.86 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.66
| method | result | size |
| default | \(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (105 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{3} e^{4} x^{3}+315 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{4} e^{3} x^{2}+315 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{5} e^{2} x -105 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{3} e^{3} x^{3}+105 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c^{3} d^{6} e -35 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{2} e^{4} x^{2}-280 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{4} e^{2} x^{2}+14 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c d \,e^{5} x -98 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{3} e^{3} x -231 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{5} e x -8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} e^{6}+38 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c \,d^{2} e^{4}-87 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{2} d^{4} e^{2}-48 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{3} d^{6}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\) | \(549\) |
1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(105*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2- c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c^3*d^3*e^4*x^3+315*arctanh(e*(c*d*x+a* e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c^3*d^4*e^3*x^2+315*ar ctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c^3*d ^5*e^2*x-105*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^3*e^3*x^3+105*arctanh(e*(c*d*x+ a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c^3*d^6*e-35*((a*e^2 -c*d^2)*e)^(1/2)*a*c^2*d^2*e^4*x^2-280*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^4*e^2 *x^2+14*((a*e^2-c*d^2)*e)^(1/2)*a^2*c*d*e^5*x-98*((a*e^2-c*d^2)*e)^(1/2)*a *c^2*d^3*e^3*x-231*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^5*e*x-8*((a*e^2-c*d^2)*e) ^(1/2)*a^3*e^6+38*((a*e^2-c*d^2)*e)^(1/2)*a^2*c*d^2*e^4-87*((a*e^2-c*d^2)* e)^(1/2)*a*c^2*d^4*e^2-48*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^6)/(e*x+d)^(7/2)/( c*d*x+a*e)/(a*e^2-c*d^2)^4/((a*e^2-c*d^2)*e)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 781 vs. \(2 (293) = 586\).
Time = 0.67 (sec) , antiderivative size = 1584, normalized size of antiderivative = 4.79 \[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/48*(105*(c^4*d^4*e^4*x^5 + a*c^3*d^7*e + (4*c^4*d^5*e^3 + a*c^3*d^3*e^5 )*x^4 + 2*(3*c^4*d^6*e^2 + 2*a*c^3*d^4*e^4)*x^3 + 2*(2*c^4*d^7*e + 3*a*c^3 *d^5*e^3)*x^2 + (c^4*d^8 + 4*a*c^3*d^6*e^2)*x)*sqrt(-e/(c*d^2 - a*e^2))*lo g(-(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)*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2 )))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(105*c^3*d^3*e^3*x^3 + 48*c^3*d^6 + 87* a*c^2*d^4*e^2 - 38*a^2*c*d^2*e^4 + 8*a^3*e^6 + 35*(8*c^3*d^4*e^2 + a*c^2*d ^2*e^4)*x^2 + 7*(33*c^3*d^5*e + 14*a*c^2*d^3*e^3 - 2*a^2*c*d*e^5)*x)*sqrt( c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a*c^4*d^12*e - 4*a^ 2*c^3*d^10*e^3 + 6*a^3*c^2*d^8*e^5 - 4*a^4*c*d^6*e^7 + a^5*d^4*e^9 + (c^5* d^9*e^4 - 4*a*c^4*d^7*e^6 + 6*a^2*c^3*d^5*e^8 - 4*a^3*c^2*d^3*e^10 + a^4*c *d*e^12)*x^5 + (4*c^5*d^10*e^3 - 15*a*c^4*d^8*e^5 + 20*a^2*c^3*d^6*e^7 - 1 0*a^3*c^2*d^4*e^9 + a^5*e^13)*x^4 + 2*(3*c^5*d^11*e^2 - 10*a*c^4*d^9*e^4 + 10*a^2*c^3*d^7*e^6 - 5*a^4*c*d^3*e^10 + 2*a^5*d*e^12)*x^3 + 2*(2*c^5*d^12 *e - 5*a*c^4*d^10*e^3 + 10*a^3*c^2*d^6*e^7 - 10*a^4*c*d^4*e^9 + 3*a^5*d^2* e^11)*x^2 + (c^5*d^13 - 10*a^2*c^3*d^9*e^4 + 20*a^3*c^2*d^7*e^6 - 15*a^4*c *d^5*e^8 + 4*a^5*d^3*e^10)*x), -1/24*(105*(c^4*d^4*e^4*x^5 + a*c^3*d^7*e + (4*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^4 + 2*(3*c^4*d^6*e^2 + 2*a*c^3*d^4*e^4) *x^3 + 2*(2*c^4*d^7*e + 3*a*c^3*d^5*e^3)*x^2 + (c^4*d^8 + 4*a*c^3*d^6*e^2) *x)*sqrt(e/(c*d^2 - a*e^2))*arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a...
\[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Time = 0.49 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.60 \[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {1}{24} \, {\left (\frac {105 \, c^{3} d^{3} e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{4} d^{8} {\left | e \right |} - 4 \, a c^{3} d^{6} e^{2} {\left | e \right |} + 6 \, a^{2} c^{2} d^{4} e^{4} {\left | e \right |} - 4 \, a^{3} c d^{2} e^{6} {\left | e \right |} + a^{4} e^{8} {\left | e \right |}\right )} \sqrt {c d^{2} e - a e^{3}}} + \frac {48 \, c^{3} d^{3} e}{{\left (c^{4} d^{8} {\left | e \right |} - 4 \, a c^{3} d^{6} e^{2} {\left | e \right |} + 6 \, a^{2} c^{2} d^{4} e^{4} {\left | e \right |} - 4 \, a^{3} c d^{2} e^{6} {\left | e \right |} + a^{4} e^{8} {\left | e \right |}\right )} \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}} + \frac {87 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{5} d^{7} e^{3} - 174 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{4} d^{5} e^{5} + 87 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{3} d^{3} e^{7} + 136 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{4} d^{5} e^{2} - 136 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{3} d^{3} e^{4} + 57 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{3} d^{3} e}{{\left (c^{4} d^{8} {\left | e \right |} - 4 \, a c^{3} d^{6} e^{2} {\left | e \right |} + 6 \, a^{2} c^{2} d^{4} e^{4} {\left | e \right |} - 4 \, a^{3} c d^{2} e^{6} {\left | e \right |} + a^{4} e^{8} {\left | e \right |}\right )} {\left (e x + d\right )}^{3} c^{3} d^{3} e^{3}}\right )} e \]
-1/24*(105*c^3*d^3*e*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c *d^2*e - a*e^3))/((c^4*d^8*abs(e) - 4*a*c^3*d^6*e^2*abs(e) + 6*a^2*c^2*d^4 *e^4*abs(e) - 4*a^3*c*d^2*e^6*abs(e) + a^4*e^8*abs(e))*sqrt(c*d^2*e - a*e^ 3)) + 48*c^3*d^3*e/((c^4*d^8*abs(e) - 4*a*c^3*d^6*e^2*abs(e) + 6*a^2*c^2*d ^4*e^4*abs(e) - 4*a^3*c*d^2*e^6*abs(e) + a^4*e^8*abs(e))*sqrt((e*x + d)*c* d*e - c*d^2*e + a*e^3)) + (87*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^5* d^7*e^3 - 174*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^4*d^5*e^5 + 87*s qrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^3*d^3*e^7 + 136*((e*x + d)*c* d*e - c*d^2*e + a*e^3)^(3/2)*c^4*d^5*e^2 - 136*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^3*d^3*e^4 + 57*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2 )*c^3*d^3*e)/((c^4*d^8*abs(e) - 4*a*c^3*d^6*e^2*abs(e) + 6*a^2*c^2*d^4*e^4 *abs(e) - 4*a^3*c*d^2*e^6*abs(e) + a^4*e^8*abs(e))*(e*x + d)^3*c^3*d^3*e^3 ))*e
Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \]