Integrand size = 29, antiderivative size = 361 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^3 \left (d+e x^2\right )} \, dx=-\frac {\sqrt {a+b x^2+c x^4}}{2 d x^2}-\frac {b \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {a} d}+\frac {\sqrt {a} e \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 d^2}+\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 d}-\frac {b e \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {c} d^2}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {c} d^2}+\frac {\sqrt {c d^2-b d e+a e^2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 d^2} \]
-1/4*b*arctanh(1/2*(b*x^2+2*a)/a^(1/2)/(c*x^4+b*x^2+a)^(1/2))/d/a^(1/2)+1/ 2*e*arctanh(1/2*(b*x^2+2*a)/a^(1/2)/(c*x^4+b*x^2+a)^(1/2))*a^(1/2)/d^2-1/4 *b*e*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/d^2/c^(1/2)-1/ 4*(-b*e+2*c*d)*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/d^2/ c^(1/2)+1/2*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))*c^(1/2) /d+1/2*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x^2)/(a*e^2-b*d*e+c*d^2)^(1/2)/ (c*x^4+b*x^2+a)^(1/2))*(a*e^2-b*d*e+c*d^2)^(1/2)/d^2-1/2*(c*x^4+b*x^2+a)^( 1/2)/d/x^2
Time = 0.64 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^3 \left (d+e x^2\right )} \, dx=\frac {-\frac {d \sqrt {a+b x^2+c x^4}}{x^2}+2 \sqrt {-c d^2+b d e-a e^2} \arctan \left (\frac {\sqrt {c} \left (d+e x^2\right )-e \sqrt {a+b x^2+c x^4}}{\sqrt {-c d^2+b d e-a e^2}}\right )+\frac {(b d-2 a e) \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 d^2} \]
(-((d*Sqrt[a + b*x^2 + c*x^4])/x^2) + 2*Sqrt[-(c*d^2) + b*d*e - a*e^2]*Arc Tan[(Sqrt[c]*(d + e*x^2) - e*Sqrt[a + b*x^2 + c*x^4])/Sqrt[-(c*d^2) + b*d* e - a*e^2]] + ((b*d - 2*a*e)*ArcTanh[(Sqrt[c]*x^2 - Sqrt[a + b*x^2 + c*x^4 ])/Sqrt[a]])/Sqrt[a])/(2*d^2)
Time = 0.64 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1578, 1289, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2+c x^4}}{x^3 \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt {c x^4+b x^2+a}}{x^4 \left (e x^2+d\right )}dx^2\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\sqrt {c x^4+b x^2+a} e^2}{d^2 \left (e x^2+d\right )}-\frac {\sqrt {c x^4+b x^2+a} e}{d^2 x^2}+\frac {\sqrt {c x^4+b x^2+a}}{d x^4}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {\sqrt {a e^2-b d e+c d^2} \text {arctanh}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{d^2}+\frac {\sqrt {a} e \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{d^2}-\frac {b e \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c} d^2}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c} d^2}-\frac {b \text {arctanh}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a} d}+\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{d}-\frac {\sqrt {a+b x^2+c x^4}}{d x^2}\right )\) |
(-(Sqrt[a + b*x^2 + c*x^4]/(d*x^2)) - (b*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]* Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[a]*d) + (Sqrt[a]*e*ArcTanh[(2*a + b*x^2 )/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/d^2 + (Sqrt[c]*ArcTanh[(b + 2*c*x^ 2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/d - (b*e*ArcTanh[(b + 2*c*x^2)/(2 *Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[c]*d^2) - ((2*c*d - b*e)*ArcTa nh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[c]*d^2) + ( Sqrt[c*d^2 - b*d*e + a*e^2]*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*S qrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x^2 + c*x^4])])/d^2)/2
3.4.15.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
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 = 0.45 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.62
| method | result | size |
| pseudoelliptic | \(-\frac {a \,x^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e +\left (b \,x^{2}+2 a \right ) e -d \left (2 c \,x^{2}+b \right )}{e \,x^{2}+d}\right )+e \left (\frac {x^{2} \left (b \sqrt {a}\, d -2 a^{\frac {3}{2}} e \right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2}+a d \sqrt {c \,x^{4}+b \,x^{2}+a}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e a \,x^{2} d^{2}}\) | \(225\) |
| risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 d \,x^{2}}-\frac {-\frac {\left (2 a e -b d \right ) \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2 d \sqrt {a}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{d e \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{2 d}\) | \(262\) |
| default | \(\frac {-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \sqrt {c \,x^{4}+b \,x^{2}+a}}{2 a}-\frac {b \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{4 \sqrt {a}}+\frac {c \sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{2 a}+\frac {\sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2}}{d}-\frac {e \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2}+\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 \sqrt {c}}-\frac {\sqrt {a}\, \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )}{2}\right )}{d^{2}}+\frac {e \left (\sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x^{2}+\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{2 d^{2}}\) | \(589\) |
| elliptic | \(\frac {-\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{a \,x^{2}}+\frac {b \left (\sqrt {c \,x^{4}+b \,x^{2}+a}+\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )\right )}{2 a}+\frac {2 c \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}\right )}{a}}{2 d}-\frac {e \left (\sqrt {c \,x^{4}+b \,x^{2}+a}+\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right )\right )}{2 d^{2}}+\frac {e \left (\sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x^{2}+\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{2 d^{2}}\) | \(644\) |
-1/2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(a*x^2*(a*e^2-b*d*e+c*d^2)*ln((2*(c*x ^4+b*x^2+a)^(1/2)*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*e+(b*x^2+2*a)*e-d*(2*c*x ^2+b))/(e*x^2+d))+e*(1/2*x^2*(b*a^(1/2)*d-2*a^(3/2)*e)*ln((2*a+b*x^2+2*a^( 1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)+a*d*(c*x^4+b*x^2+a)^(1/2))*((a*e^2-b*d*e+ c*d^2)/e^2)^(1/2))/e/a/x^2/d^2
Time = 0.48 (sec) , antiderivative size = 1094, normalized size of antiderivative = 3.03 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^3 \left (d+e x^2\right )} \, dx=\left [\frac {2 \, \sqrt {c d^{2} - b d e + a e^{2}} a x^{2} \log \left (-\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + {\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{4} - 8 \, a b d e + 8 \, a^{2} e^{2} + {\left (b^{2} + 4 \, a c\right )} d^{2} + 2 \, {\left (4 \, b c d^{2} + 4 \, a b e^{2} - {\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} \sqrt {c d^{2} - b d e + a e^{2}} {\left ({\left (2 \, c d - b e\right )} x^{2} + b d - 2 \, a e\right )}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) - {\left (b d - 2 \, a e\right )} \sqrt {a} x^{2} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, \sqrt {c x^{4} + b x^{2} + a} a d}{8 \, a d^{2} x^{2}}, \frac {4 \, \sqrt {-c d^{2} + b d e - a e^{2}} a x^{2} \arctan \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} \sqrt {-c d^{2} + b d e - a e^{2}} {\left ({\left (2 \, c d - b e\right )} x^{2} + b d - 2 \, a e\right )}}{2 \, {\left ({\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} x^{4} + a c d^{2} - a b d e + a^{2} e^{2} + {\left (b c d^{2} - b^{2} d e + a b e^{2}\right )} x^{2}\right )}}\right ) - {\left (b d - 2 \, a e\right )} \sqrt {a} x^{2} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, \sqrt {c x^{4} + b x^{2} + a} a d}{8 \, a d^{2} x^{2}}, \frac {{\left (b d - 2 \, a e\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) + \sqrt {c d^{2} - b d e + a e^{2}} a x^{2} \log \left (-\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + {\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{4} - 8 \, a b d e + 8 \, a^{2} e^{2} + {\left (b^{2} + 4 \, a c\right )} d^{2} + 2 \, {\left (4 \, b c d^{2} + 4 \, a b e^{2} - {\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} \sqrt {c d^{2} - b d e + a e^{2}} {\left ({\left (2 \, c d - b e\right )} x^{2} + b d - 2 \, a e\right )}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) - 2 \, \sqrt {c x^{4} + b x^{2} + a} a d}{4 \, a d^{2} x^{2}}, \frac {2 \, \sqrt {-c d^{2} + b d e - a e^{2}} a x^{2} \arctan \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} \sqrt {-c d^{2} + b d e - a e^{2}} {\left ({\left (2 \, c d - b e\right )} x^{2} + b d - 2 \, a e\right )}}{2 \, {\left ({\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} x^{4} + a c d^{2} - a b d e + a^{2} e^{2} + {\left (b c d^{2} - b^{2} d e + a b e^{2}\right )} x^{2}\right )}}\right ) + {\left (b d - 2 \, a e\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{4} + a b x^{2} + a^{2}\right )}}\right ) - 2 \, \sqrt {c x^{4} + b x^{2} + a} a d}{4 \, a d^{2} x^{2}}\right ] \]
[1/8*(2*sqrt(c*d^2 - b*d*e + a*e^2)*a*x^2*log(-((8*c^2*d^2 - 8*b*c*d*e + ( b^2 + 4*a*c)*e^2)*x^4 - 8*a*b*d*e + 8*a^2*e^2 + (b^2 + 4*a*c)*d^2 + 2*(4*b *c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x^2 + 4*sqrt(c*x^4 + b*x^2 + a)* sqrt(c*d^2 - b*d*e + a*e^2)*((2*c*d - b*e)*x^2 + b*d - 2*a*e))/(e^2*x^4 + 2*d*e*x^2 + d^2)) - (b*d - 2*a*e)*sqrt(a)*x^2*log(-((b^2 + 4*a*c)*x^4 + 8* a*b*x^2 + 4*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(a) + 8*a^2)/x^4) - 4*sqrt(c*x^4 + b*x^2 + a)*a*d)/(a*d^2*x^2), 1/8*(4*sqrt(-c*d^2 + b*d*e - a *e^2)*a*x^2*arctan(-1/2*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c*d^2 + b*d*e - a*e^ 2)*((2*c*d - b*e)*x^2 + b*d - 2*a*e)/((c^2*d^2 - b*c*d*e + a*c*e^2)*x^4 + a*c*d^2 - a*b*d*e + a^2*e^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x^2)) - (b*d - 2*a*e)*sqrt(a)*x^2*log(-((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 4*sqrt(c*x^4 + b *x^2 + a)*(b*x^2 + 2*a)*sqrt(a) + 8*a^2)/x^4) - 4*sqrt(c*x^4 + b*x^2 + a)* a*d)/(a*d^2*x^2), 1/4*((b*d - 2*a*e)*sqrt(-a)*x^2*arctan(1/2*sqrt(c*x^4 + b*x^2 + a)*(b*x^2 + 2*a)*sqrt(-a)/(a*c*x^4 + a*b*x^2 + a^2)) + sqrt(c*d^2 - b*d*e + a*e^2)*a*x^2*log(-((8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x ^4 - 8*a*b*d*e + 8*a^2*e^2 + (b^2 + 4*a*c)*d^2 + 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x^2 + 4*sqrt(c*x^4 + b*x^2 + a)*sqrt(c*d^2 - b*d*e + a*e^2)*((2*c*d - b*e)*x^2 + b*d - 2*a*e))/(e^2*x^4 + 2*d*e*x^2 + d^2)) - 2*sqrt(c*x^4 + b*x^2 + a)*a*d)/(a*d^2*x^2), 1/4*(2*sqrt(-c*d^2 + b*d*e - a*e^2)*a*x^2*arctan(-1/2*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c*d^2 + b*d*e - ...
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^3 \left (d+e x^2\right )} \, dx=\int \frac {\sqrt {a + b x^{2} + c x^{4}}}{x^{3} \left (d + e x^{2}\right )}\, dx \]
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^3 \left (d+e x^2\right )} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a}}{{\left (e x^{2} + d\right )} x^{3}} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^3 \left (d+e x^2\right )} \, dx=\frac {{\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{\sqrt {-c d^{2} + b d e - a e^{2}} d^{2}} + \frac {{\left (b d - 2 \, a e\right )} \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} d^{2}} + \frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} b + 2 \, a \sqrt {c}}{2 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )}^{2} - a\right )} d} \]
(c*d^2 - b*d*e + a*e^2)*arctan(-((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/(sqrt(-c*d^2 + b*d*e - a*e^2)* d^2) + 1/2*(b*d - 2*a*e)*arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))/s qrt(-a))/(sqrt(-a)*d^2) + 1/2*((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*b + 2*a*sqrt(c))/(((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))^2 - a)*d)
Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{x^3 \left (d+e x^2\right )} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2+a}}{x^3\,\left (e\,x^2+d\right )} \,d x \]