Integrand size = 29, antiderivative size = 552 \[ \int \frac {\sqrt {d+e x^2}}{x^5 \left (a+b x^2+c x^4\right )} \, dx=-\frac {\sqrt {d+e x^2}}{4 a x^4}+\frac {3 e \sqrt {d+e x^2}}{8 a d x^2}+\frac {(b d-a e) \sqrt {d+e x^2}}{2 a^2 d x^2}-\frac {3 e^2 \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{8 a d^{3/2}}-\frac {e (b d-a e) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a^2 d^{3/2}}-\frac {\left (b^2 d-a c d-a b e\right ) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a^3 \sqrt {d}}+\frac {\sqrt {c} \left (b^3 d-a c \left (\sqrt {b^2-4 a c} d-2 a e\right )+b^2 \left (\sqrt {b^2-4 a c} d-a e\right )-a b \left (3 c d+\sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {c} \left (b^3 d-b^2 \left (\sqrt {b^2-4 a c} d+a e\right )+a c \left (\sqrt {b^2-4 a c} d+2 a e\right )-a b \left (3 c d-\sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-3/8*e^2*arctanh((e*x^2+d)^(1/2)/d^(1/2))/a/d^(3/2)-1/2*e*(-a*e+b*d)*arcta nh((e*x^2+d)^(1/2)/d^(1/2))/a^2/d^(3/2)-(-a*b*e-a*c*d+b^2*d)*arctanh((e*x^ 2+d)^(1/2)/d^(1/2))/a^3/d^(1/2)-1/4*(e*x^2+d)^(1/2)/a/x^4+3/8*e*(e*x^2+d)^ (1/2)/a/d/x^2+1/2*(-a*e+b*d)*(e*x^2+d)^(1/2)/a^2/d/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^3*d-a*c*(-2*a*e+d*(-4*a*c+b^2)^(1/2))+b^2*(-a*e+d*(-4*a*c+b^2)^(1/2))-a* b*(3*c*d+e*(-4*a*c+b^2)^(1/2)))/a^3*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^3*d-b^2*(a*e+d*(-4*a*c+b^ 2)^(1/2))+a*c*(2*a*e+d*(-4*a*c+b^2)^(1/2))-a*b*(3*c*d-e*(-4*a*c+b^2)^(1/2) ))/a^3*2^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Time = 1.70 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {d+e x^2}}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\frac {\frac {a \sqrt {d+e x^2} \left (4 b d x^2-a \left (2 d+e x^2\right )\right )}{d x^4}+\frac {4 \sqrt {2} \sqrt {c} \left (-b^3 d+a c \left (\sqrt {b^2-4 a c} d-2 a e\right )+b^2 \left (-\sqrt {b^2-4 a c} d+a e\right )+a b \left (3 c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {4 \sqrt {2} \sqrt {c} \left (b^3 d-b^2 \left (\sqrt {b^2-4 a c} d+a e\right )+a c \left (\sqrt {b^2-4 a c} d+2 a e\right )+a b \left (-3 c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (-8 b^2 d^2+4 a b d e+a \left (8 c d^2+a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}}{8 a^3} \]
((a*Sqrt[d + e*x^2]*(4*b*d*x^2 - a*(2*d + e*x^2)))/(d*x^4) + (4*Sqrt[2]*Sq rt[c]*(-(b^3*d) + a*c*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*(-(Sqrt[b^2 - 4* a*c]*d) + a*e) + a*b*(3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c ]*Sqrt[d + e*x^2])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c])*e]) + (4*Sqrt[2]*Sqrt[c]*(b^3 *d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) + a*c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) + a*b*(-3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^ 2])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-2* c*d + (b + Sqrt[b^2 - 4*a*c])*e]) + ((-8*b^2*d^2 + 4*a*b*d*e + a*(8*c*d^2 + a*e^2))*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(3/2))/(8*a^3)
Time = 2.15 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.06, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x^2}}{x^5 \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt {e x^2+d}}{x^6 \left (c x^4+b x^2+a\right )}dx^2\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle \frac {\int \left (-\frac {d e^3}{a \left (d-x^4\right )^3}-\frac {(b d-a e) e^2}{a^2 \left (d-x^4\right )^2}-\frac {\left (d b^2-a e b-a c d\right ) e}{a^3 \left (d-x^4\right )}+\frac {\left (\left (b^2-a c\right ) \left (c d^2-b e d+a e^2\right )-c \left (d b^2-a e b-a c d\right ) x^4\right ) e}{a^3 \left (c x^8-(2 c d-b e) x^4+c d^2+a e^2-b d e\right )}\right )d\sqrt {e x^2+d}}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {e \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \left (-a b e-a c d+b^2 d\right )}{a^3 \sqrt {d}}+\frac {\sqrt {c} e \left (b^2 \left (d \sqrt {b^2-4 a c}-a e\right )-a b \left (e \sqrt {b^2-4 a c}+3 c d\right )-a c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b^3 d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {c} e \left (-b^2 \left (d \sqrt {b^2-4 a c}+a e\right )-a b \left (3 c d-e \sqrt {b^2-4 a c}\right )+a c \left (d \sqrt {b^2-4 a c}+2 a e\right )+b^3 d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {e^2 (b d-a e) \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a^2 d^{3/2}}-\frac {e^2 \sqrt {d+e x^2} (b d-a e)}{2 a^2 d \left (d-x^4\right )}-\frac {3 e^3 \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{8 a d^{3/2}}-\frac {3 e^3 \sqrt {d+e x^2}}{8 a d \left (d-x^4\right )}-\frac {e^3 \sqrt {d+e x^2}}{4 a \left (d-x^4\right )^2}}{e}\) |
(-1/4*(e^3*Sqrt[d + e*x^2])/(a*(d - x^4)^2) - (3*e^3*Sqrt[d + e*x^2])/(8*a *d*(d - x^4)) - (e^2*(b*d - a*e)*Sqrt[d + e*x^2])/(2*a^2*d*(d - x^4)) - (3 *e^3*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(8*a*d^(3/2)) - (e^2*(b*d - a*e)*Ar cTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*a^2*d^(3/2)) - (e*(b^2*d - a*c*d - a*b* e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(a^3*Sqrt[d]) + (Sqrt[c]*e*(b^3*d - a *c*(Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*(Sqrt[b^2 - 4*a*c]*d - a*e) - a*b*( 3*c*d + Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sq rt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*a^3*Sqrt[b^2 - 4*a*c]*Sqr t[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (Sqrt[c]*e*(b^3*d - b^2*(Sqrt[b^2 - 4*a*c]*d + a*e) + a*c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - a*b*(3*c*d - Sqrt[ b^2 - 4*a*c]*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^3*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/e
3.4.60.3.1 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 = 1.15 (sec) , antiderivative size = 463, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {\sqrt {e \,x^{2}+d}\, \left (a e \,x^{2}-4 b d \,x^{2}+2 d a \right )}{8 a^{2} x^{4} d}-\frac {-\frac {\left (e^{2} a^{2}+4 a b d e +8 d^{2} a c -8 b^{2} d^{2}\right ) \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{a \sqrt {d}}-\frac {8 d \sqrt {2}\, c \left (\left (\frac {\left (-a c d -b \left (a e -b d \right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (a \left (a e -\frac {3 b d}{2}\right ) c -\frac {b^{2} \left (a e -b d \right )}{2}\right )\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\frac {\left (a c d +b \left (a e -b d \right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (a \left (a e -\frac {3 b d}{2}\right ) c -\frac {b^{2} \left (a e -b d \right )}{2}\right )\right )\right )}{a \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}}{8 a^{2} d}\) | \(463\) |
| pseudoelliptic | \(-\frac {-8 \sqrt {2}\, c \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, x^{4} \left (\frac {\left (-a \,d^{\frac {3}{2}} b e -d^{\frac {5}{2}} \left (a c -b^{2}\right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (a e \left (a c -\frac {b^{2}}{2}\right ) d^{\frac {3}{2}}-\frac {3 d^{\frac {5}{2}} b \left (a c -\frac {b^{2}}{3}\right )}{2}\right )\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (-8 \sqrt {2}\, \left (\frac {\left (a \,d^{\frac {3}{2}} b e +d^{\frac {5}{2}} \left (a c -b^{2}\right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (a e \left (a c -\frac {b^{2}}{2}\right ) d^{\frac {3}{2}}-\frac {3 d^{\frac {5}{2}} b \left (a c -\frac {b^{2}}{3}\right )}{2}\right )\right ) c \,x^{4} \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (-\left (e^{2} a^{2}+4 \left (b d e +2 c \,d^{2}\right ) a -8 b^{2} d^{2}\right ) x^{4} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{\sqrt {d}}\right )+a \left (2 \left (-2 b \,x^{2}+a \right ) d^{\frac {3}{2}}+a \sqrt {d}\, e \,x^{2}\right ) \sqrt {e \,x^{2}+d}\right )\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{8 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, d^{\frac {3}{2}} \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, x^{4} a^{3}}\) | \(527\) |
| default | \(\frac {-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 d \,x^{4}}-\frac {e \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {e \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )}{2 d}\right )}{4 d}}{a}-\frac {b \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {e \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )}{2 d}\right )}{a^{2}}+\frac {\left (-a c +b^{2}\right ) \left (\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )\right )}{a^{3}}+\frac {\sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, c \left (\frac {\left (-a c d -b \left (a e -b d \right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (a \left (a e -\frac {3 b d}{2}\right ) c -\frac {b^{2} \left (a e -b d \right )}{2}\right )\right ) \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 \left (\frac {\left (a c d +b \left (a e -b d \right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (a \left (a e -\frac {3 b d}{2}\right ) c -\frac {b^{2} \left (a e -b d \right )}{2}\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 )+\sqrt {e \,x^{2}+d}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (a c -b^{2}\right )\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{a^{3} \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 )}}\) | \(631\) |
-1/8*(e*x^2+d)^(1/2)*(a*e*x^2-4*b*d*x^2+2*a*d)/a^2/x^4/d-1/8/a^2/d*(-(a^2* e^2+4*a*b*d*e+8*a*c*d^2-8*b^2*d^2)/a/d^(1/2)*ln((2*d+2*d^(1/2)*(e*x^2+d)^( 1/2))/x)-8*d/a/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*2^(1/2)* c/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*((1/2*(-a*c*d-b*(a*e -b*d))*(-4*e^2*(a*c-1/4*b^2))^(1/2)+e*(a*(a*e-3/2*b*d)*c-1/2*b^2*(a*e-b*d) ))*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x^2+d)^ (1/2)*2^(1/2)/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+arctan( c*(e*x^2+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/ 2))*((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(1/2*(a*c*d+b*(a*e -b*d))*(-4*e^2*(a*c-1/4*b^2))^(1/2)+e*(a*(a*e-3/2*b*d)*c-1/2*b^2*(a*e-b*d) )))/(-4*e^2*(a*c-1/4*b^2))^(1/2))
Timed out. \[ \int \frac {\sqrt {d+e x^2}}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {d+e x^2}}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\sqrt {d + e x^{2}}}{x^{5} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {\sqrt {d+e x^2}}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )} x^{5}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1057 vs. \(2 (468) = 936\).
Time = 0.35 (sec) , antiderivative size = 1057, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {d+e x^2}}{x^5 \left (a+b x^2+c x^4\right )} \, dx=-\frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e\right )} a^{2} e^{2} - 2 \, {\left ({\left (a b^{2} c - a^{2} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (a b^{3} - a^{2} b c\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a^{2} b^{2} - a^{3} c\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | a \right |} {\left | e \right |} - {\left (2 \, {\left (a^{2} b^{3} c - 3 \, a^{3} b c^{2}\right )} d^{2} e - {\left (a^{2} b^{4} - a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} d e^{2} + {\left (a^{3} b^{3} - 2 \, a^{4} b c\right )} 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^{3} c d - a^{3} b e + \sqrt {-4 \, {\left (a^{3} c d^{2} - a^{3} b d e + a^{4} e^{2}\right )} a^{3} c + {\left (2 \, a^{3} c d - a^{3} b e\right )}^{2}}}{a^{3} c}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{4} c d^{2} - \sqrt {b^{2} - 4 \, a c} a^{4} b d e + \sqrt {b^{2} - 4 \, a c} a^{5} e^{2}\right )} {\left | a \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^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e\right )} a^{2} e^{2} + 2 \, {\left ({\left (a b^{2} c - a^{2} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (a b^{3} - a^{2} b c\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a^{2} b^{2} - a^{3} c\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | a \right |} {\left | e \right |} - {\left (2 \, {\left (a^{2} b^{3} c - 3 \, a^{3} b c^{2}\right )} d^{2} e - {\left (a^{2} b^{4} - a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} d e^{2} + {\left (a^{3} b^{3} - 2 \, a^{4} b c\right )} 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^{3} c d - a^{3} b e - \sqrt {-4 \, {\left (a^{3} c d^{2} - a^{3} b d e + a^{4} e^{2}\right )} a^{3} c + {\left (2 \, a^{3} c d - a^{3} b e\right )}^{2}}}{a^{3} c}}}\right )}{8 \, {\left (\sqrt {b^{2} - 4 \, a c} a^{4} c d^{2} - \sqrt {b^{2} - 4 \, a c} a^{4} b d e + \sqrt {b^{2} - 4 \, a c} a^{5} e^{2}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}} + \frac {{\left (8 \, b^{2} d^{2} - 8 \, a c d^{2} - 4 \, a b d e - a^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {e x^{2} + d}}{\sqrt {-d}}\right )}{8 \, a^{3} \sqrt {-d} d} + \frac {4 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} b d e - 4 \, \sqrt {e x^{2} + d} b d^{2} e - {\left (e x^{2} + d\right )}^{\frac {3}{2}} a e^{2} - \sqrt {e x^{2} + d} a d e^{2}}{8 \, a^{2} d e^{2} x^{4}} \]
-1/8*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^4 - 5*a*b^2*c + 4*a^2*c^2)*d - (a*b^3 - 4*a^2*b*c)*e)*a^2*e^2 - 2*((a*b^2*c - a^2*c^2)*sq rt(b^2 - 4*a*c)*d^2 - (a*b^3 - a^2*b*c)*sqrt(b^2 - 4*a*c)*d*e + (a^2*b^2 - a^3*c)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)* c)*e)*abs(a)*abs(e) - (2*(a^2*b^3*c - 3*a^3*b*c^2)*d^2*e - (a^2*b^4 - a^3* b^2*c - 4*a^4*c^2)*d*e^2 + (a^3*b^3 - 2*a^4*b*c)*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^3*c*d - a^3*b*e + sqrt(-4*(a^3*c*d^2 - a^3*b*d*e + a^4*e^2)*a^3*c + (2*a ^3*c*d - a^3*b*e)^2))/(a^3*c)))/((sqrt(b^2 - 4*a*c)*a^4*c*d^2 - sqrt(b^2 - 4*a*c)*a^4*b*d*e + sqrt(b^2 - 4*a*c)*a^5*e^2)*abs(a)*abs(c)*abs(e)) + 1/8 *(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*((b^4 - 5*a*b^2*c + 4*a ^2*c^2)*d - (a*b^3 - 4*a^2*b*c)*e)*a^2*e^2 + 2*((a*b^2*c - a^2*c^2)*sqrt(b ^2 - 4*a*c)*d^2 - (a*b^3 - a^2*b*c)*sqrt(b^2 - 4*a*c)*d*e + (a^2*b^2 - a^3 *c)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e )*abs(a)*abs(e) - (2*(a^2*b^3*c - 3*a^3*b*c^2)*d^2*e - (a^2*b^4 - a^3*b^2* c - 4*a^4*c^2)*d*e^2 + (a^3*b^3 - 2*a^4*b*c)*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^3* c*d - a^3*b*e - sqrt(-4*(a^3*c*d^2 - a^3*b*d*e + a^4*e^2)*a^3*c + (2*a^3*c *d - a^3*b*e)^2))/(a^3*c)))/((sqrt(b^2 - 4*a*c)*a^4*c*d^2 - sqrt(b^2 - 4*a *c)*a^4*b*d*e + sqrt(b^2 - 4*a*c)*a^5*e^2)*abs(a)*abs(c)*abs(e)) + 1/8*...
Time = 11.97 (sec) , antiderivative size = 33925, normalized size of antiderivative = 61.46 \[ \int \frac {\sqrt {d+e x^2}}{x^5 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
atan(((((((2048*a^12*c^4*d*e^12 + 12288*a^10*c^6*d^5*e^8 + 14336*a^11*c^5* d^3*e^10 + 2048*a^8*b^4*c^4*d^5*e^8 - 1536*a^8*b^5*c^3*d^4*e^9 - 512*a^8*b ^6*c^2*d^3*e^10 - 11264*a^9*b^2*c^5*d^5*e^8 + 7168*a^9*b^3*c^4*d^4*e^9 + 6 272*a^9*b^4*c^3*d^3*e^10 + 384*a^9*b^5*c^2*d^2*e^11 - 20480*a^10*b^2*c^4*d ^3*e^10 - 3072*a^10*b^3*c^3*d^2*e^11 - 4096*a^10*b*c^5*d^4*e^9 + 128*a^10* b^4*c^2*d*e^12 + 6144*a^11*b*c^4*d^2*e^11 - 1024*a^11*b^2*c^3*d*e^12)/(64* a^8*d^2) - ((d + e*x^2)^(1/2)*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2 )^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^ 2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(- (4*a*c - b^2)^3)^(1/2))/(8*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)*(2 4576*a^12*c^5*d^4*e^8 + 16384*a^13*c^4*d^2*e^10 + 2048*a^10*b^4*c^3*d^4*e^ 8 - 2048*a^10*b^5*c^2*d^3*e^9 - 14336*a^11*b^2*c^4*d^4*e^8 + 15360*a^11*b^ 3*c^3*d^3*e^9 + 1024*a^11*b^4*c^2*d^2*e^10 - 8192*a^12*b^2*c^3*d^2*e^10 - 28672*a^12*b*c^4*d^3*e^9))/(32*a^8*d^2))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-( 4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25 *a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4 *e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d *(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*...