Integrand size = 39, antiderivative size = 457 \[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {1}{4 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {11 c d}{24 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {33 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {77 c^3 d^3 \sqrt {d+e x}}{32 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {385 c^3 d^3 e}{64 \left (c d^2-a e^2\right )^5 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {1155 c^4 d^4 e \sqrt {d+e x}}{64 \left (c d^2-a e^2\right )^6 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {1155 c^4 d^4 e^{3/2} \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{64 \left (c d^2-a e^2\right )^{13/2}} \]
1/4/(-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)^(3/2)+1 1/24*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)^ (3/2)+1155/64*c^4*d^4*e^(3/2)*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))/(-a*e^2+c*d^2)^(13/2)+33/32 *c^2*d^2/(-a*e^2+c*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^ (1/2)-77/32*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)^(3/2)-385/64*c^3*d^3*e/(-a*e^2+c*d^2)^5/(e*x+d)^(1/2)/(a*d*e+(a *e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1155/64*c^4*d^4*e*(e*x+d)^(1/2)/(-a*e^2+c*d ^2)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
Time = 1.46 (sec) , antiderivative size = 361, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {c^4 d^4 (d+e x)^{5/2} \left (-\frac {(a e+c d x) \left (48 a^5 e^{10}-8 a^4 c d e^8 (41 d+11 e x)+2 a^3 c^2 d^2 e^6 \left (515 d^2+374 d e x+99 e^2 x^2\right )-3 a^2 c^3 d^3 e^4 \left (765 d^3+1265 d^2 e x+891 d e^2 x^2+231 e^3 x^3\right )-2 a c^4 d^4 e^2 \left (1024 d^4+6391 d^3 e x+11484 d^2 e^2 x^2+8547 d e^3 x^3+2310 e^4 x^4\right )+c^5 d^5 \left (128 d^5-1408 d^4 e x-9207 d^3 e^2 x^2-16863 d^2 e^3 x^3-12705 d e^4 x^4-3465 e^5 x^5\right )\right )}{c^4 d^4 \left (c d^2-a e^2\right )^6 (d+e x)^4}+\frac {3465 e^{3/2} (a e+c d x)^{5/2} \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{13/2}}\right )}{192 ((a e+c d x) (d+e x))^{5/2}} \]
(c^4*d^4*(d + e*x)^(5/2)*(-(((a*e + c*d*x)*(48*a^5*e^10 - 8*a^4*c*d*e^8*(4 1*d + 11*e*x) + 2*a^3*c^2*d^2*e^6*(515*d^2 + 374*d*e*x + 99*e^2*x^2) - 3*a ^2*c^3*d^3*e^4*(765*d^3 + 1265*d^2*e*x + 891*d*e^2*x^2 + 231*e^3*x^3) - 2* a*c^4*d^4*e^2*(1024*d^4 + 6391*d^3*e*x + 11484*d^2*e^2*x^2 + 8547*d*e^3*x^ 3 + 2310*e^4*x^4) + c^5*d^5*(128*d^5 - 1408*d^4*e*x - 9207*d^3*e^2*x^2 - 1 6863*d^2*e^3*x^3 - 12705*d*e^4*x^4 - 3465*e^5*x^5)))/(c^4*d^4*(c*d^2 - a*e ^2)^6*(d + e*x)^4)) + (3465*e^(3/2)*(a*e + c*d*x)^(5/2)*ArcTan[(Sqrt[e]*Sq rt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(13/2)))/(192*((a*e + c*d*x)*(d + e*x))^(5/2))
Time = 0.71 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {1135, 1135, 1135, 1132, 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.
| \(\displaystyle \int \frac {1}{(d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {11 c d \int \frac {1}{(d+e x)^{3/2} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{8 \left (c d^2-a e^2\right )}+\frac {1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {11 c d \left (\frac {3 c d \int \frac {1}{\sqrt {d+e x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{8 \left (c d^2-a e^2\right )}+\frac {1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {11 c d \left (\frac {3 c d \left (\frac {7 c d \int \frac {\sqrt {d+e x}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}dx}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{8 \left (c d^2-a e^2\right )}+\frac {1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {11 c d \left (\frac {3 c d \left (\frac {7 c d \left (-\frac {5 e \int \frac {1}{\sqrt {d+e x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{8 \left (c d^2-a e^2\right )}+\frac {1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {11 c d \left (\frac {3 c d \left (\frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{8 \left (c d^2-a e^2\right )}+\frac {1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {11 c d \left (\frac {3 c d \left (\frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \left (-\frac {e \int \frac {1}{\sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d^2-a e^2}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{8 \left (c d^2-a e^2\right )}+\frac {1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {11 c d \left (\frac {3 c d \left (\frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \left (-\frac {2 e^2 \int \frac {1}{\frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}+\left (c d^2-a e^2\right ) e}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d^2-a e^2}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{8 \left (c d^2-a e^2\right )}+\frac {1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {11 c d \left (\frac {3 c d \left (\frac {7 c d \left (-\frac {5 e \left (\frac {3 c d \left (-\frac {2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}}-\frac {2 \sqrt {d+e x}}{\left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{3 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {d+e x}}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{4 \left (c d^2-a e^2\right )}+\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{2 \left (c d^2-a e^2\right )}+\frac {1}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\right )}{8 \left (c d^2-a e^2\right )}+\frac {1}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
1/(4*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 2)^(3/2)) + (11*c*d*(1/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (3*c*d*(1/(2*(c*d^2 - a*e^2)*Sqrt[d + e*x ]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (7*c*d*((-2*Sqrt[d + e* x])/(3*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (5 *e*(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]*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])])/(c*d^2 - a* e^2)^(3/2)))/(2*(c*d^2 - a*e^2))))/(3*(c*d^2 - a*e^2))))/(4*(c*d^2 - a*e^2 ))))/(2*(c*d^2 - a*e^2))))/(8*(c*d^2 - a*e^2))
3.21.79.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1214\) vs. \(2(407)=814\).
Time = 2.98 (sec) , antiderivative size = 1215, normalized size of antiderivative = 2.66
-1/192*((c*d*x+a*e)*(e*x+d))^(1/2)*(48*((a*e^2-c*d^2)*e)^(1/2)*a^5*e^10+34 65*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^4*d^4*e^7*x^4* (c*d*x+a*e)^(1/2)+13860*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2 ))*a*c^4*d^5*e^6*x^3*(c*d*x+a*e)^(1/2)+20790*arctanh(e*(c*d*x+a*e)^(1/2)/( (a*e^2-c*d^2)*e)^(1/2))*a*c^4*d^6*e^5*x^2*(c*d*x+a*e)^(1/2)+13860*arctanh( e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^4*d^7*e^4*x*(c*d*x+a*e)^( 1/2)-4620*((a*e^2-c*d^2)*e)^(1/2)*a*c^4*d^4*e^6*x^4+128*((a*e^2-c*d^2)*e)^ (1/2)*c^5*d^10+1030*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^2*d^4*e^6-2295*((a*e^2-c *d^2)*e)^(1/2)*a^2*c^3*d^6*e^4-328*((a*e^2-c*d^2)*e)^(1/2)*a^4*c*d^2*e^8-2 2968*((a*e^2-c*d^2)*e)^(1/2)*a*c^4*d^6*e^4*x^2-12782*((a*e^2-c*d^2)*e)^(1/ 2)*a*c^4*d^7*e^3*x+13860*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/ 2))*c^5*d^8*e^3*x^2*(c*d*x+a*e)^(1/2)+3465*arctanh(e*(c*d*x+a*e)^(1/2)/((a *e^2-c*d^2)*e)^(1/2))*c^5*d^9*e^2*x*(c*d*x+a*e)^(1/2)+3465*arctanh(e*(c*d* x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*a*c^4*d^8*e^3*(c*d*x+a*e)^(1/2)-3465 *((a*e^2-c*d^2)*e)^(1/2)*c^5*d^5*e^5*x^5+198*((a*e^2-c*d^2)*e)^(1/2)*a^3*c ^2*d^2*e^8*x^2-2673*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^3*d^4*e^6*x^2-88*((a*e^2 -c*d^2)*e)^(1/2)*a^4*c*d*e^9*x+748*((a*e^2-c*d^2)*e)^(1/2)*a^3*c^2*d^3*e^7 *x-3795*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^3*d^5*e^5*x+3465*arctanh(e*(c*d*x+a* e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^5*d^5*e^6*x^5*(c*d*x+a*e)^(1/2)+13860* arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^5*d^6*e^5*x^4*(c...
Leaf count of result is larger than twice the leaf count of optimal. 1528 vs. \(2 (407) = 814\).
Time = 3.13 (sec) , antiderivative size = 3078, normalized size of antiderivative = 6.74 \[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/384*(3465*(c^6*d^6*e^6*x^7 + a^2*c^4*d^9*e^3 + (5*c^6*d^7*e^5 + 2*a*c^5 *d^5*e^7)*x^6 + (10*c^6*d^8*e^4 + 10*a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^5 + 5*(2*c^6*d^9*e^3 + 4*a*c^5*d^7*e^5 + a^2*c^4*d^5*e^7)*x^4 + 5*(c^6*d^10* e^2 + 4*a*c^5*d^8*e^4 + 2*a^2*c^4*d^6*e^6)*x^3 + (c^6*d^11*e + 10*a*c^5*d^ 9*e^3 + 10*a^2*c^4*d^7*e^5)*x^2 + (2*a*c^5*d^10*e^2 + 5*a^2*c^4*d^8*e^4)*x )*sqrt(-e/(c*d^2 - a*e^2))*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)*(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*(3465*c^5* d^5*e^5*x^5 - 128*c^5*d^10 + 2048*a*c^4*d^8*e^2 + 2295*a^2*c^3*d^6*e^4 - 1 030*a^3*c^2*d^4*e^6 + 328*a^4*c*d^2*e^8 - 48*a^5*e^10 + 1155*(11*c^5*d^6*e ^4 + 4*a*c^4*d^4*e^6)*x^4 + 231*(73*c^5*d^7*e^3 + 74*a*c^4*d^5*e^5 + 3*a^2 *c^3*d^3*e^7)*x^3 + 99*(93*c^5*d^8*e^2 + 232*a*c^4*d^6*e^4 + 27*a^2*c^3*d^ 4*e^6 - 2*a^3*c^2*d^2*e^8)*x^2 + 11*(128*c^5*d^9*e + 1162*a*c^4*d^7*e^3 + 345*a^2*c^3*d^5*e^5 - 68*a^3*c^2*d^3*e^7 + 8*a^4*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))/(a^2*c^6*d^17*e^2 - 6*a^3*c^ 5*d^15*e^4 + 15*a^4*c^4*d^13*e^6 - 20*a^5*c^3*d^11*e^8 + 15*a^6*c^2*d^9*e^ 10 - 6*a^7*c*d^7*e^12 + a^8*d^5*e^14 + (c^8*d^14*e^5 - 6*a*c^7*d^12*e^7 + 15*a^2*c^6*d^10*e^9 - 20*a^3*c^5*d^8*e^11 + 15*a^4*c^4*d^6*e^13 - 6*a^5*c^ 3*d^4*e^15 + a^6*c^2*d^2*e^17)*x^7 + (5*c^8*d^15*e^4 - 28*a*c^7*d^13*e^6 + 63*a^2*c^6*d^11*e^8 - 70*a^3*c^5*d^9*e^10 + 35*a^4*c^4*d^7*e^12 - 7*a^...
\[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 820 vs. \(2 (407) = 814\).
Time = 0.76 (sec) , antiderivative size = 820, normalized size of antiderivative = 1.79 \[ \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 )^{5/2}} \, dx=\frac {1}{192} \, {\left (\frac {3465 \, c^{4} d^{4} 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^{6} d^{12} {\left | e \right |} - 6 \, a c^{5} d^{10} e^{2} {\left | e \right |} + 15 \, a^{2} c^{4} d^{8} e^{4} {\left | e \right |} - 20 \, a^{3} c^{3} d^{6} e^{6} {\left | e \right |} + 15 \, a^{4} c^{2} d^{4} e^{8} {\left | e \right |} - 6 \, a^{5} c d^{2} e^{10} {\left | e \right |} + a^{6} e^{12} {\left | e \right |}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {128 \, {\left (c^{5} d^{6} e^{2} - a c^{4} d^{4} e^{4} - 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c^{4} d^{4} e\right )}}{{\left (c^{6} d^{12} {\left | e \right |} - 6 \, a c^{5} d^{10} e^{2} {\left | e \right |} + 15 \, a^{2} c^{4} d^{8} e^{4} {\left | e \right |} - 20 \, a^{3} c^{3} d^{6} e^{6} {\left | e \right |} + 15 \, a^{4} c^{2} d^{4} e^{8} {\left | e \right |} - 6 \, a^{5} c d^{2} e^{10} {\left | e \right |} + a^{6} e^{12} {\left | e \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}} + \frac {2295 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{7} d^{10} e^{4} - 6885 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{6} d^{8} e^{6} + 6885 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{5} d^{6} e^{8} - 2295 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{4} d^{4} e^{10} + 5855 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{6} d^{8} e^{3} - 11710 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{5} d^{6} e^{5} + 5855 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{4} d^{4} e^{7} + 5153 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{5} d^{6} e^{2} - 5153 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{4} d^{4} e^{4} + 1545 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{4} d^{4} e}{{\left (c^{6} d^{12} {\left | e \right |} - 6 \, a c^{5} d^{10} e^{2} {\left | e \right |} + 15 \, a^{2} c^{4} d^{8} e^{4} {\left | e \right |} - 20 \, a^{3} c^{3} d^{6} e^{6} {\left | e \right |} + 15 \, a^{4} c^{2} d^{4} e^{8} {\left | e \right |} - 6 \, a^{5} c d^{2} e^{10} {\left | e \right |} + a^{6} e^{12} {\left | e \right |}\right )} {\left (e x + d\right )}^{4} c^{4} d^{4} e^{4}}\right )} e^{2} \]
1/192*(3465*c^4*d^4*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^6*d^12*abs(e) - 6*a*c^5*d^10*e^2*abs(e) + 15*a^2*c^4 *d^8*e^4*abs(e) - 20*a^3*c^3*d^6*e^6*abs(e) + 15*a^4*c^2*d^4*e^8*abs(e) - 6*a^5*c*d^2*e^10*abs(e) + a^6*e^12*abs(e))*sqrt(c*d^2*e - a*e^3)) - 128*(c ^5*d^6*e^2 - a*c^4*d^4*e^4 - 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^4*d^ 4*e)/((c^6*d^12*abs(e) - 6*a*c^5*d^10*e^2*abs(e) + 15*a^2*c^4*d^8*e^4*abs( e) - 20*a^3*c^3*d^6*e^6*abs(e) + 15*a^4*c^2*d^4*e^8*abs(e) - 6*a^5*c*d^2*e ^10*abs(e) + a^6*e^12*abs(e))*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)) + (2295*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^7*d^10*e^4 - 6885*sqrt((e *x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^6*d^8*e^6 + 6885*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^5*d^6*e^8 - 2295*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^4*d^4*e^10 + 5855*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2) *c^6*d^8*e^3 - 11710*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^5*d^6*e ^5 + 5855*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^4*d^4*e^7 + 5153 *((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^5*d^6*e^2 - 5153*((e*x + d)*c *d*e - c*d^2*e + a*e^3)^(5/2)*a*c^4*d^4*e^4 + 1545*((e*x + d)*c*d*e - c*d^ 2*e + a*e^3)^(7/2)*c^4*d^4*e)/((c^6*d^12*abs(e) - 6*a*c^5*d^10*e^2*abs(e) + 15*a^2*c^4*d^8*e^4*abs(e) - 20*a^3*c^3*d^6*e^6*abs(e) + 15*a^4*c^2*d^4*e ^8*abs(e) - 6*a^5*c*d^2*e^10*abs(e) + a^6*e^12*abs(e))*(e*x + d)^4*c^4*d^4 *e^4))*e^2
Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \]