Integrand size = 29, antiderivative size = 417 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=-\frac {d \sqrt {d+e x^2}}{2 a x^2}+\frac {\sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a}+\frac {\sqrt {d} (b d-2 a e) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2}-\frac {\sqrt {c} \left (b^2 d^2+b d \left (\sqrt {b^2-4 a c} d-2 a e\right )-2 a \left (c d^2+e \left (\sqrt {b^2-4 a c} d-a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {c} \left (b^2 d^2-b d \left (\sqrt {b^2-4 a c} d+2 a e\right )-2 a \left (c d^2-e \left (\sqrt {b^2-4 a c} d+a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
1/2*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))*d^(1/2)/a+(-2*a*e+b*d)*arctanh((e*x ^2+d)^(1/2)/d^(1/2))*d^(1/2)/a^2-1/2*d*(e*x^2+d)^(1/2)/a/x^2-1/2*arctanh(2 ^(1/2)*c^(1/2)*(e*x^2+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*c^( 1/2)*(b^2*d^2+b*d*(-2*a*e+d*(-4*a*c+b^2)^(1/2))-2*a*(c*d^2+e*(-a*e+d*(-4*a *c+b^2)^(1/2))))/a^2*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^( 1/2)))^(1/2)+1/2*arctanh(2^(1/2)*c^(1/2)*(e*x^2+d)^(1/2)/(2*c*d-e*(b+(-4*a *c+b^2)^(1/2)))^(1/2))*c^(1/2)*(b^2*d^2-b*d*(2*a*e+d*(-4*a*c+b^2)^(1/2))-2 *a*(c*d^2-e*(a*e+d*(-4*a*c+b^2)^(1/2))))/a^2*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2 *c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\frac {-\frac {a d \sqrt {d+e x^2}}{x^2}+\frac {\sqrt {2} \sqrt {c} \left (-i b^2 d^2+b d \left (\sqrt {-b^2+4 a c} d+2 i a e\right )-2 i a \left (-c d^2+e \left (-i \sqrt {-b^2+4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (i b^2 d^2+b d \left (\sqrt {-b^2+4 a c} d-2 i a e\right )+2 i a \left (-c d^2+e \left (i \sqrt {-b^2+4 a c} d+a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}+\sqrt {d} (2 b d-3 a e) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a^2} \]
(-((a*d*Sqrt[d + e*x^2])/x^2) + (Sqrt[2]*Sqrt[c]*((-I)*b^2*d^2 + b*d*(Sqrt [-b^2 + 4*a*c]*d + (2*I)*a*e) - (2*I)*a*(-(c*d^2) + e*((-I)*Sqrt[-b^2 + 4* a*c]*d + a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b - I*Sqrt [-b^2 + 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*(I*b^2*d^2 + b*d*(Sqrt[-b^2 + 4*a*c ]*d - (2*I)*a*e) + (2*I)*a*(-(c*d^2) + e*(I*Sqrt[-b^2 + 4*a*c]*d + a*e)))* ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])* e]) + Sqrt[d]*(2*b*d - 3*a*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*a^2)
Time = 1.80 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1578, 1199, 2009}
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (e x^2+d\right )^{3/2}}{x^4 \left (c x^4+b x^2+a\right )}dx^2\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle \frac {\int \left (\frac {d^2 e^2}{a \left (d-x^4\right )^2}+\frac {d (b d-2 a e) e}{a^2 \left (d-x^4\right )}-\frac {\left ((b d-a e) \left (c d^2-b e d+a e^2\right )-c d (b d-2 a e) x^4\right ) e}{a^2 \left (c x^8-(2 c d-b e) x^4+c d^2+a e^2-b d e\right )}\right )d\sqrt {e x^2+d}}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\sqrt {c} e \left (-2 a \left (e \left (d \sqrt {b^2-4 a c}-a e\right )+c d^2\right )+b d \left (d \sqrt {b^2-4 a c}-2 a e\right )+b^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {c} e \left (-b d \left (d \sqrt {b^2-4 a c}+2 a e\right )+2 a e \left (d \sqrt {b^2-4 a c}+a e\right )-2 a c d^2+b^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {\sqrt {d} e (b d-2 a e) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^2}+\frac {\sqrt {d} e^2 \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a}+\frac {d e^2 \sqrt {d+e x^2}}{2 a \left (d-x^4\right )}}{e}\) |
((d*e^2*Sqrt[d + e*x^2])/(2*a*(d - x^4)) + (Sqrt[d]*e^2*ArcTanh[Sqrt[d + e *x^2]/Sqrt[d]])/(2*a) + (Sqrt[d]*e*(b*d - 2*a*e)*ArcTanh[Sqrt[d + e*x^2]/S qrt[d]])/a^2 - (Sqrt[c]*e*(b^2*d^2 + b*d*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) - 2 *a*(c*d^2 + e*(Sqrt[b^2 - 4*a*c]*d - a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[ d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*a^2*Sqrt[b^ 2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (Sqrt[c]*e*(b^2*d^2 - 2*a*c*d^2 + 2*a*e*(Sqrt[b^2 - 4*a*c]*d + a*e) - b*d*(Sqrt[b^2 - 4*a*c]*d + 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqr t[b^2 - 4*a*c])*e]])/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt [b^2 - 4*a*c])*e]))/e
3.4.70.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e Subs t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer Q[n] && FractionQ[m]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Time = 0.85 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(-\frac {d \sqrt {e \,x^{2}+d}}{2 a \,x^{2}}-\frac {\frac {\sqrt {d}\, \left (3 a e -2 b d \right ) \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{a}+\frac {c \sqrt {2}\, \left (-\frac {\left (-2 e^{3} a^{2}+2 a b d \,e^{2}+2 a c \,d^{2} e -b^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (2 e^{3} a^{2}-2 a b d \,e^{2}-2 a c \,d^{2} e +b^{2} d^{2} e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}\right ) \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}}{2 a}\) | \(399\) |
| pseudoelliptic | \(-\frac {\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, c \left (\left (-d^{\frac {3}{2}} a e +\frac {b \,d^{\frac {5}{2}}}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (-a c +\frac {b^{2}}{2}\right ) d^{\frac {5}{2}}+a e \left (e a \sqrt {d}-d^{\frac {3}{2}} b \right )\right )\right ) x^{2} \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\sqrt {2}\, c \,x^{2} \left (\left (d^{\frac {3}{2}} a e -\frac {b \,d^{\frac {5}{2}}}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (-a c +\frac {b^{2}}{2}\right ) d^{\frac {5}{2}}+a e \left (e a \sqrt {d}-d^{\frac {3}{2}} b \right )\right )\right ) \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\frac {\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (3 e d a -2 b \,d^{2}\right ) x^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{\sqrt {d}}\right )+\sqrt {e \,x^{2}+d}\, d^{\frac {3}{2}} a \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{\sqrt {d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, a^{2} x^{2}}\) | \(468\) |
| default | \(\frac {-\frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}+\frac {3 e \left (\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )\right )}{2 d}}{a}-\frac {b \left (\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )\right )}{a^{2}}-\frac {\left (\left (-e d a +\frac {1}{2} b \,d^{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (e^{2} a^{2}-a b d e -d^{2} a c +\frac {1}{2} b^{2} d^{2}\right )\right ) \sqrt {2}\, c \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\sqrt {2}\, \left (d \left (a e -\frac {b d}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (e^{2} a^{2}-a b d e -d^{2} a c +\frac {1}{2} b^{2} d^{2}\right )\right ) c \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\left (-\frac {b \,x^{2}}{3}+a \right ) e -\frac {4 b d}{3}\right ) \sqrt {e \,x^{2}+d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{a^{2} \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}\) | \(563\) |
-1/2*d*(e*x^2+d)^(1/2)/a/x^2-1/2/a*(d^(1/2)*(3*a*e-2*b*d)/a*ln((2*d+2*d^(1 /2)*(e*x^2+d)^(1/2))/x)+1/a*c*2^(1/2)/(-e^2*(4*a*c-b^2))^(1/2)*(-(-2*e^3*a ^2+2*a*b*d*e^2+2*a*c*d^2*e-b^2*d^2*e+2*(-e^2*(4*a*c-b^2))^(1/2)*a*d*e-(-e^ 2*(4*a*c-b^2))^(1/2)*b*d^2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2 )*arctanh(c*(e*x^2+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2)) *c)^(1/2))+(2*e^3*a^2-2*a*b*d*e^2-2*a*c*d^2*e+b^2*d^2*e+2*(-e^2*(4*a*c-b^2 ))^(1/2)*a*d*e-(-e^2*(4*a*c-b^2))^(1/2)*b*d^2)/((b*e-2*c*d+(-e^2*(4*a*c-b^ 2))^(1/2))*c)^(1/2)*arctan(c*(e*x^2+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4* a*c-b^2))^(1/2))*c)^(1/2))))
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{3} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (c x^{4} + b x^{2} + a\right )} x^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 898 vs. \(2 (355) = 710\).
Time = 0.33 (sec) , antiderivative size = 898, normalized size of antiderivative = 2.15 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=-\frac {{\left (2 \, b d^{2} - 3 \, a d e\right )} \arctan \left (\frac {\sqrt {e x^{2} + d}}{\sqrt {-d}}\right )}{2 \, a^{2} \sqrt {-d}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{3} - 4 \, a b c\right )} d^{2} - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d e\right )} e^{2} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} b c d^{3} + 2 \, \sqrt {b^{2} - 4 \, a c} a b d e^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} e^{3} - {\left (b^{2} + a c\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | e \right |} + {\left (2 \, a^{2} b e^{4} - 2 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d^{3} e + {\left (b^{3} + 2 \, a b c\right )} d^{2} e^{2} - 2 \, {\left (a b^{2} + 2 \, a^{2} c\right )} d e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d}}{\sqrt {-\frac {2 \, a^{2} c d - a^{2} b e + \sqrt {-4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} a^{2} c + {\left (2 \, a^{2} c d - a^{2} b e\right )}^{2}}}{a^{2} c}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{2} c d^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} b d e + \sqrt {b^{2} - 4 \, a c} a^{3} e^{2}\right )} {\left | c \right |} {\left | e \right |}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{3} - 4 \, a b c\right )} d^{2} - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d e\right )} e^{2} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} b c d^{3} + 2 \, \sqrt {b^{2} - 4 \, a c} a b d e^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} e^{3} - {\left (b^{2} + a c\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | e \right |} + {\left (2 \, a^{2} b e^{4} - 2 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d^{3} e + {\left (b^{3} + 2 \, a b c\right )} d^{2} e^{2} - 2 \, {\left (a b^{2} + 2 \, a^{2} c\right )} d e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d}}{\sqrt {-\frac {2 \, a^{2} c d - a^{2} b e - \sqrt {-4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} a^{2} c + {\left (2 \, a^{2} c d - a^{2} b e\right )}^{2}}}{a^{2} c}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{2} c d^{2} - \sqrt {b^{2} - 4 \, a c} a^{2} b d e + \sqrt {b^{2} - 4 \, a c} a^{3} e^{2}\right )} {\left | c \right |} {\left | e \right |}} - \frac {\sqrt {e x^{2} + d} d}{2 \, a x^{2}} \]
-1/2*(2*b*d^2 - 3*a*d*e)*arctan(sqrt(e*x^2 + d)/sqrt(-d))/(a^2*sqrt(-d)) + 1/8*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^3 - 4*a*b*c)*d^ 2 - 2*(a*b^2 - 4*a^2*c)*d*e)*e^2 - 2*(sqrt(b^2 - 4*a*c)*b*c*d^3 + 2*sqrt(b ^2 - 4*a*c)*a*b*d*e^2 - sqrt(b^2 - 4*a*c)*a^2*e^3 - (b^2 + a*c)*sqrt(b^2 - 4*a*c)*d^2*e)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(e) + ( 2*a^2*b*e^4 - 2*(b^2*c - 2*a*c^2)*d^3*e + (b^3 + 2*a*b*c)*d^2*e^2 - 2*(a*b ^2 + 2*a^2*c)*d*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arc tan(2*sqrt(1/2)*sqrt(e*x^2 + d)/sqrt(-(2*a^2*c*d - a^2*b*e + sqrt(-4*(a^2* c*d^2 - a^2*b*d*e + a^3*e^2)*a^2*c + (2*a^2*c*d - a^2*b*e)^2))/(a^2*c)))/( (sqrt(b^2 - 4*a*c)*a^2*c*d^2 - sqrt(b^2 - 4*a*c)*a^2*b*d*e + sqrt(b^2 - 4* a*c)*a^3*e^2)*abs(c)*abs(e)) - 1/8*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4* a*c)*c)*e)*((b^3 - 4*a*b*c)*d^2 - 2*(a*b^2 - 4*a^2*c)*d*e)*e^2 + 2*(sqrt(b ^2 - 4*a*c)*b*c*d^3 + 2*sqrt(b^2 - 4*a*c)*a*b*d*e^2 - sqrt(b^2 - 4*a*c)*a^ 2*e^3 - (b^2 + a*c)*sqrt(b^2 - 4*a*c)*d^2*e)*sqrt(-4*c^2*d + 2*(b*c + sqrt (b^2 - 4*a*c)*c)*e)*abs(e) + (2*a^2*b*e^4 - 2*(b^2*c - 2*a*c^2)*d^3*e + (b ^3 + 2*a*b*c)*d^2*e^2 - 2*(a*b^2 + 2*a^2*c)*d*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x^2 + d)/sqrt(-(2*a^2 *c*d - a^2*b*e - sqrt(-4*(a^2*c*d^2 - a^2*b*d*e + a^3*e^2)*a^2*c + (2*a^2* c*d - a^2*b*e)^2))/(a^2*c)))/((sqrt(b^2 - 4*a*c)*a^2*c*d^2 - sqrt(b^2 - 4* a*c)*a^2*b*d*e + sqrt(b^2 - 4*a*c)*a^3*e^2)*abs(c)*abs(e)) - 1/2*sqrt(e...
Time = 11.06 (sec) , antiderivative size = 35855, normalized size of antiderivative = 85.98 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
(d^(1/2)*atan(((d^(1/2)*(3*a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(4*a^6*c^3*e^1 6 + 4*a^2*c^7*d^8*e^8 - 2*a^3*c^6*d^6*e^10 + 132*a^4*c^5*d^4*e^12 - 2*a^5* c^4*d^2*e^14 + 4*b^4*c^5*d^8*e^8 + 129*a^2*b^2*c^5*d^6*e^10 - 32*a^2*b^3*c ^4*d^5*e^11 + 8*a^2*b^4*c^3*d^4*e^12 + 88*a^3*b^2*c^4*d^4*e^12 - 28*a^3*b^ 3*c^3*d^3*e^13 + 33*a^4*b^2*c^3*d^2*e^14 - 16*a^5*b*c^3*d*e^15 - 8*a*b^2*c ^6*d^8*e^8 - 28*a*b^3*c^5*d^7*e^9 + 8*a^2*b*c^6*d^7*e^9 - 228*a^3*b*c^5*d^ 5*e^11 - 60*a^4*b*c^4*d^3*e^13))/(2*a^4) - (d^(1/2)*((56*a^4*c^6*d^6*e^9 - 44*a^5*c^5*d^4*e^11 - 100*a^6*c^4*d^2*e^13 + 40*a^2*b^3*c^5*d^7*e^8 - 39* a^2*b^5*c^3*d^5*e^10 - 11*a^2*b^6*c^2*d^4*e^11 - 108*a^3*b^2*c^5*d^6*e^9 + 96*a^3*b^3*c^4*d^5*e^10 + 111*a^3*b^4*c^3*d^4*e^11 + 22*a^3*b^5*c^2*d^3*e ^12 - 237*a^4*b^2*c^4*d^4*e^11 - 161*a^4*b^3*c^3*d^3*e^12 - 19*a^4*b^4*c^2 *d^2*e^13 + 111*a^5*b^2*c^3*d^2*e^13 - 28*a^6*b*c^3*d*e^14 - 8*a*b^5*c^4*d ^7*e^8 + 6*a*b^6*c^3*d^6*e^9 + 2*a*b^7*c^2*d^5*e^10 - 32*a^3*b*c^6*d^7*e^8 + 92*a^4*b*c^5*d^5*e^10 + 252*a^5*b*c^4*d^3*e^12 + 6*a^5*b^3*c^2*d*e^14)/ a^4 + (d^(1/2)*(3*a*e - 2*b*d)*(((d + e*x^2)^(1/2)*(64*a^7*b*c^3*e^13 + 35 2*a^7*c^4*d*e^12 - 16*a^6*b^3*c^2*e^13 - 160*a^5*c^6*d^5*e^8 + 736*a^6*c^5 *d^3*e^10 + 32*a^2*b^6*c^3*d^5*e^8 - 32*a^2*b^7*c^2*d^4*e^9 - 224*a^3*b^4* c^4*d^5*e^8 + 144*a^3*b^5*c^3*d^4*e^9 + 112*a^3*b^6*c^2*d^3*e^10 + 432*a^4 *b^2*c^5*d^5*e^8 + 144*a^4*b^3*c^4*d^4*e^9 - 716*a^4*b^4*c^3*d^3*e^10 - 13 2*a^4*b^5*c^2*d^2*e^11 + 936*a^5*b^2*c^4*d^3*e^10 + 860*a^5*b^3*c^3*d^2...