Integrand size = 20, antiderivative size = 190 \[ \int \frac {1}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\frac {2 e}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {4 c d e}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}} \]
2/3*e/(-a*e^2+c*d^2)/(e*x+d)^(3/2)-c^(3/4)*arctanh(c^(1/4)*(e*x+d)^(1/2)/( -e*a^(1/2)+d*c^(1/2))^(1/2))/a^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(5/2)+c^(3/4)* arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))/a^(1/2)/(e*a^(1 /2)+d*c^(1/2))^(5/2)+4*c*d*e/(-a*e^2+c*d^2)^2/(e*x+d)^(1/2)
Time = 0.96 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\frac {-2 a e^3+2 c d e (7 d+6 e x)}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {c \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {c \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \sqrt {-c d+\sqrt {a} \sqrt {c} e}} \]
(-2*a*e^3 + 2*c*d*e*(7*d + 6*e*x))/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + (c*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + S qrt[a]*e)])/(Sqrt[a]*(Sqrt[c]*d + Sqrt[a]*e)^2*Sqrt[-(c*d) - Sqrt[a]*Sqrt[ c]*e]) - (c*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[ c]*d - Sqrt[a]*e)])/(Sqrt[a]*(Sqrt[c]*d - Sqrt[a]*e)^2*Sqrt[-(c*d) + Sqrt[ a]*Sqrt[c]*e])
Time = 0.45 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.42, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {482, 655, 25, 654, 25, 27, 1480, 221}
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.
| \(\displaystyle \int \frac {1}{\left (a-c x^2\right ) (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 482 |
| \(\displaystyle \frac {c \int \frac {d-e x}{(d+e x)^{3/2} \left (a-c x^2\right )}dx}{c d^2-a e^2}+\frac {2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 655 |
| \(\displaystyle \frac {c \left (\frac {4 d e}{\sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {\int -\frac {c d^2-2 c e x d+a e^2}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c d^2-a e^2}\right )}{c d^2-a e^2}+\frac {2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {c d^2-2 c e x d+a e^2}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c d^2-a e^2}+\frac {4 d e}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{c d^2-a e^2}+\frac {2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {c \left (\frac {2 \int -\frac {e \left (3 c d^2-2 c (d+e x) d+a e^2\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c d^2-a e^2}+\frac {4 d e}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{c d^2-a e^2}+\frac {2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \left (\frac {4 d e}{\sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {2 \int \frac {e \left (3 c d^2-2 c (d+e x) d+a e^2\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c d^2-a e^2}\right )}{c d^2-a e^2}+\frac {2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {4 d e}{\sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {2 e \int \frac {3 c d^2-2 c (d+e x) d+a e^2}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c d^2-a e^2}\right )}{c d^2-a e^2}+\frac {2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {c \left (\frac {4 d e}{\sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {2 e \left (\frac {\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )^2 \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{c d^2-a e^2}\right )}{c d^2-a e^2}+\frac {2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c \left (\frac {4 d e}{\sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {2 e \left (\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^2 \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c} e \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c} e \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{c d^2-a e^2}\right )}{c d^2-a e^2}+\frac {2 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
(2*e)/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)) + (c*((4*d*e)/((c*d^2 - a*e^2)*S qrt[d + e*x]) - (2*e*(((Sqrt[c]*d + Sqrt[a]*e)^2*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)*e*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) - ((Sqrt[c]*d - Sqrt[a]*e)^2*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/ Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(1/4)*e*Sqrt[Sqrt[c]*d + Sqrt[a ]*e])))/(c*d^2 - a*e^2)))/(c*d^2 - a*e^2)
3.7.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[d*((c + d*x)^(n + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + Simp[b/(b*c^2 + a*d^2) I nt[(c + d*x)^(n + 1)*((c - d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[n, -1]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 + a*e^2)) ), x] + Simp[1/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d*f + a*e*g - c*(e*f - d*g)*x, x]/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 2.09 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.18
| method | result | size |
| derivativedivides | \(-2 e \left (\frac {1}{3 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 c d}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}+\frac {c^{2} \left (-\frac {\left (e^{2} a +c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-e^{2} a -c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2}}\right )\) | \(224\) |
| default | \(2 e \left (-\frac {1}{3 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c d}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {c^{2} \left (-\frac {\left (e^{2} a +c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-e^{2} a -c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2}}\right )\) | \(225\) |
| pseudoelliptic | \(-\frac {2 \left (-\frac {3 c^{2} \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (e^{2} a +c \,d^{2}+2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-\frac {3 c^{2} \left (e x +d \right )^{\frac {3}{2}} \left (e^{2} a +c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\left (-6 d e x -7 d^{2}\right ) c +e^{2} a \right )\right )\right ) e}{3 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \left (e x +d \right )^{\frac {3}{2}} \left (e^{2} a -c \,d^{2}\right )^{2}}\) | \(267\) |
-2*e*(1/3/(a*e^2-c*d^2)/(e*x+d)^(3/2)-2*c*d/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)+ c^2/(a*e^2-c*d^2)^2*(-1/2*(e^2*a+c*d^2-2*(a*c*e^2)^(1/2)*d)/(a*c*e^2)^(1/2 )/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^ (1/2))*c)^(1/2))+1/2*(-e^2*a-c*d^2-2*(a*c*e^2)^(1/2)*d)/(a*c*e^2)^(1/2)/(( -c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/ 2))*c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 5142 vs. \(2 (144) = 288\).
Time = 0.38 (sec) , antiderivative size = 5142, normalized size of antiderivative = 27.06 \[ \int \frac {1}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\int { -\frac {1}{{\left (c x^{2} - a\right )} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1165 vs. \(2 (144) = 288\).
Time = 0.35 (sec) , antiderivative size = 1165, normalized size of antiderivative = 6.13 \[ \int \frac {1}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\frac {{\left (2 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}^{2} \sqrt {-c^{2} d - \sqrt {a c} c e} \sqrt {a c} a d e {\left | c \right |} - {\left (3 \, a c^{3} d^{6} e - 5 \, a^{2} c^{2} d^{4} e^{3} + a^{3} c d^{2} e^{5} + a^{4} e^{7}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |} {\left | c \right |} + {\left (\sqrt {a c} c^{5} d^{11} e - 3 \, \sqrt {a c} a c^{4} d^{9} e^{3} + 2 \, \sqrt {a c} a^{2} c^{3} d^{7} e^{5} + 2 \, \sqrt {a c} a^{3} c^{2} d^{5} e^{7} - 3 \, \sqrt {a c} a^{4} c d^{3} e^{9} + \sqrt {a c} a^{5} d e^{11}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} + \sqrt {{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}^{2} - {\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 (c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}}\right )}{{\left (a c^{6} d^{10} - 5 \, a^{2} c^{5} d^{8} e^{2} + 10 \, a^{3} c^{4} d^{6} e^{4} - 10 \, a^{4} c^{3} d^{4} e^{6} + 5 \, a^{5} c^{2} d^{2} e^{8} - a^{6} c e^{10}\right )} {\left | c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |}} - \frac {{\left (2 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}^{2} \sqrt {-c^{2} d + \sqrt {a c} c e} \sqrt {a c} a d e {\left | c \right |} + {\left (3 \, a c^{3} d^{6} e - 5 \, a^{2} c^{2} d^{4} e^{3} + a^{3} c d^{2} e^{5} + a^{4} e^{7}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |} {\left | c \right |} + {\left (\sqrt {a c} c^{5} d^{11} e - 3 \, \sqrt {a c} a c^{4} d^{9} e^{3} + 2 \, \sqrt {a c} a^{2} c^{3} d^{7} e^{5} + 2 \, \sqrt {a c} a^{3} c^{2} d^{5} e^{7} - 3 \, \sqrt {a c} a^{4} c d^{3} e^{9} + \sqrt {a c} a^{5} d e^{11}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} - \sqrt {{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}^{2} - {\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 (c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} - 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}}\right )}{{\left (a c^{6} d^{10} - 5 \, a^{2} c^{5} d^{8} e^{2} + 10 \, a^{3} c^{4} d^{6} e^{4} - 10 \, a^{4} c^{3} d^{4} e^{6} + 5 \, a^{5} c^{2} d^{2} e^{8} - a^{6} c e^{10}\right )} {\left | c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5} \right |}} + \frac {2 \, {\left (6 \, {\left (e x + d\right )} c d e + c d^{2} e - a e^{3}\right )}}{3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \]
(2*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)^2*sqrt(-c^2*d - sqrt(a*c)*c*e)*sq rt(a*c)*a*d*e*abs(c) - (3*a*c^3*d^6*e - 5*a^2*c^2*d^4*e^3 + a^3*c*d^2*e^5 + a^4*e^7)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(c^2*d^4*e - 2*a*c*d^2*e^3 + a^ 2*e^5)*abs(c) + (sqrt(a*c)*c^5*d^11*e - 3*sqrt(a*c)*a*c^4*d^9*e^3 + 2*sqrt (a*c)*a^2*c^3*d^7*e^5 + 2*sqrt(a*c)*a^3*c^2*d^5*e^7 - 3*sqrt(a*c)*a^4*c*d^ 3*e^9 + sqrt(a*c)*a^5*d*e^11)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(c))*arctan( sqrt(e*x + d)/sqrt(-(c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4 + sqrt((c^3*d ^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)^2 - (c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2 *c*d^2*e^4 - a^3*e^6)*(c^3*d^4 - 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/(c^3*d^4 - 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/((a*c^6*d^10 - 5*a^2*c^5*d^8*e^2 + 10*a^3* c^4*d^6*e^4 - 10*a^4*c^3*d^4*e^6 + 5*a^5*c^2*d^2*e^8 - a^6*c*e^10)*abs(c^2 *d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)) - (2*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e ^5)^2*sqrt(-c^2*d + sqrt(a*c)*c*e)*sqrt(a*c)*a*d*e*abs(c) + (3*a*c^3*d^6*e - 5*a^2*c^2*d^4*e^3 + a^3*c*d^2*e^5 + a^4*e^7)*sqrt(-c^2*d + sqrt(a*c)*c* e)*abs(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*abs(c) + (sqrt(a*c)*c^5*d^11*e - 3*sqrt(a*c)*a*c^4*d^9*e^3 + 2*sqrt(a*c)*a^2*c^3*d^7*e^5 + 2*sqrt(a*c)*a ^3*c^2*d^5*e^7 - 3*sqrt(a*c)*a^4*c*d^3*e^9 + sqrt(a*c)*a^5*d*e^11)*sqrt(-c ^2*d + sqrt(a*c)*c*e)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c^3*d^5 - 2*a*c^ 2*d^3*e^2 + a^2*c*d*e^4 - sqrt((c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)^2 - (c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*(c^3*d^4 - 2...
Time = 11.71 (sec) , antiderivative size = 7831, normalized size of antiderivative = 41.22 \[ \int \frac {1}{(d+e x)^{5/2} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]
- ((2*e)/(3*(a*e^2 - c*d^2)) - (4*c*d*e*(d + e*x))/(a*e^2 - c*d^2)^2)/(d + e*x)^(3/2) - atan((((d + e*x)^(1/2)*(16*a^8*c^5*e^18 + 16*c^13*d^16*e^2 - 320*a^2*c^11*d^12*e^6 + 1024*a^3*c^10*d^10*e^8 - 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 - 320*a^6*c^7*d^4*e^14) + (-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3 )^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5* a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^ 6)))^(1/2)*(32*a^10*c^4*e^21 - (d + e*x)^(1/2)*(-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3) ^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - 5*a ^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 - 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6 )))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 - 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 - 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e ^10 - 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 - 7680*a^8*c^7*d^7* e^16 + 2880*a^9*c^6*d^5*e^18 - 640*a^10*c^5*d^3*e^20) + 96*a*c^13*d^18*e^3 - 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 - 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 - 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 + 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^5*(a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 + 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(a^3*c^3) ^(1/2) + 10*a*c*d^2*e^3*(a^3*c^3)^(1/2))/(4*(a^7*e^10 - a^2*c^5*d^10 - ...