Integrand size = 29, antiderivative size = 324 \[ \int \frac {x^5 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=-\frac {b \sqrt {d+e x^2}}{c^2}+\frac {\left (d+e x^2\right )^{3/2}}{3 c e}+\frac {\left (b c d-b^2 e+a c e-\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (b c d-b^2 e+a c e+\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
1/3*(e*x^2+d)^(3/2)/c/e-b*(e*x^2+d)^(1/2)/c^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))*(b*c*d-b^2*e+a*c*e +(-3*a*b*c*e+2*a*c^2*d+b^3*e-b^2*c*d)/(-4*a*c+b^2)^(1/2))/c^(5/2)*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))*(b*c*d-b^2*e+a*c*e+(3*a*b *c*e-2*a*c^2*d-b^3*e+b^2*c*d)/(-4*a*c+b^2)^(1/2))/c^(5/2)*2^(1/2)/(2*c*d-e *(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Time = 0.98 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.18 \[ \int \frac {x^5 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {d+e x^2} \left (-3 b e+c \left (d+e x^2\right )\right )}{e}+\frac {3 \sqrt {2} \left (-b^3 e+b c \left (-\sqrt {b^2-4 a c} d+3 a e\right )+b^2 \left (c d+\sqrt {b^2-4 a c} e\right )-a c \left (2 c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {3 \sqrt {2} \left (b^3 e-b c \left (\sqrt {b^2-4 a c} d+3 a e\right )+a c \left (2 c d-\sqrt {b^2-4 a c} e\right )+b^2 \left (-c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{6 c^{5/2}} \]
((2*Sqrt[c]*Sqrt[d + e*x^2]*(-3*b*e + c*(d + e*x^2)))/e + (3*Sqrt[2]*(-(b^ 3*e) + b*c*(-(Sqrt[b^2 - 4*a*c]*d) + 3*a*e) + b^2*(c*d + Sqrt[b^2 - 4*a*c] *e) - a*c*(2*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]) + (3*Sqrt[2]*(b^3*e - b*c*(Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a*c*(2*c*d - Sqrt[b^2 - 4*a*c]*e) + b^2*(-(c*d) + Sq rt[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]))/(6*c^(5/2))
Time = 2.10 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.99, 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.
| \(\displaystyle \int \frac {x^5 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4 \sqrt {e x^2+d}}{c x^4+b x^2+a}dx^2\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle \frac {\int \left (\frac {x^4}{c}-\frac {b e}{c^2}+\frac {b \left (c d^2-b e d+a e^2\right )-\left (-e b^2+c d b+a c e\right ) x^4}{c^2 e \left (\frac {c x^8}{e^2}-\frac {(2 c d-b e) x^4}{e^2}+a+\frac {d (c d-b e)}{e^2}\right )}\right )d\sqrt {e x^2+d}}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {e \left (-\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {e \left (\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {b e \sqrt {d+e x^2}}{c^2}+\frac {x^6}{3 c}}{e}\) |
(x^6/(3*c) - (b*e*Sqrt[d + e*x^2])/c^2 + (e*(b*c*d - b^2*e + a*c*e - (b^2* c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*S qrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2] *c^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (e*(b*c*d - b^2*e + a* c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh [(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e] ])/(Sqrt[2]*c^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/e
3.4.55.3.1 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 = 0.87 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.22
| method | result | size |
| risch | \(-\frac {\left (-c \,x^{2} e +3 b e -c d \right ) \sqrt {e \,x^{2}+d}}{3 e \,c^{2}}-\frac {\sqrt {2}\, \left (-\left (\left (\left (a e +b d \right ) c -b^{2} e \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+2 a \,c^{2} d e +\left (-3 a b \,e^{2}-b^{2} d e \right ) c +b^{3} e^{2}\right ) \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (\left (a e +b d \right ) c -b^{2} e \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-2 a \,c^{2} d e +\left (3 a b \,e^{2}+b^{2} d e \right ) c -b^{3} e^{2}\right )\right )}{2 c^{2} \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}\) | \(396\) |
| default | \(\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 c e}-\frac {b \sqrt {e \,x^{2}+d}+\frac {\left (3 a b \,e^{2} c -2 a \,c^{2} d e -b^{3} e^{2}+b^{2} c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-3 a b \,e^{2} c +2 a \,c^{2} d e +b^{3} e^{2}-b^{2} c d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c^{2}}\) | \(422\) |
| pseudoelliptic | \(-\frac {-\left (\left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-3 a b c +b^{3}\right ) e^{2}+d \left (2 a \,c^{2}-b^{2} c \right ) e \right ) e \sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (e \sqrt {2}\, \left (\left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (3 a b c -b^{3}\right ) e^{2}+d \left (-2 a \,c^{2}+b^{2} c \right ) e \right ) \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+2 \left (\left (-\frac {c \,x^{2}}{3}+b \right ) e -\frac {c d}{3}\right ) \sqrt {e \,x^{2}+d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{2 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, e \,c^{2}}\) | \(437\) |
-1/3*(-c*e*x^2+3*b*e-c*d)*(e*x^2+d)^(1/2)/e/c^2-1/2/c^2/((b*e-2*c*d+(-4*e^ 2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*2^(1/2)*(-(((a*e+b*d)*c-b^2*e)*(-4*e^2*(a *c-1/4*b^2))^(1/2)+2*a*c^2*d*e+(-3*a*b*e^2-b^2*d*e)*c+b^3*e^2)*((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)*(((a*e+b*d)*c-b^2*e)*(-4*e^2*(a* c-1/4*b^2))^(1/2)-2*a*c^2*d*e+(3*a*b*e^2+b^2*d*e)*c-b^3*e^2))/((-b*e+2*c*d +(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)/(-4*e^2*(a*c-1/4*b^2))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 4182 vs. \(2 (280) = 560\).
Time = 102.89 (sec) , antiderivative size = 4182, normalized size of antiderivative = 12.91 \[ \int \frac {x^5 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
-1/12*(3*sqrt(1/2)*c^2*e*sqrt(((b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e + (b^2*c^5 - 4*a*c^6)*sqrt(((b^6*c^2 - 4*a*b^ 4*c^3 + 4*a^2*b^2*c^4)*d^2 - 2*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^ 3*b*c^4)*d*e + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4 )*e^2)/(b^2*c^10 - 4*a*c^11)))/(b^2*c^5 - 4*a*c^6))*log(-(2*(a^2*b^4*c - 2 *a^3*b^2*c^2)*d^2 - 2*(a^2*b^5 - 2*a^3*b^3*c - a^4*b*c^2)*d*e + 2*(a^3*b^4 - 3*a^4*b^2*c + a^5*c^2)*e^2 + ((a^2*b^4*c - 2*a^3*b^2*c^2)*d*e - (a^2*b^ 5 - 3*a^3*b^3*c + a^4*b*c^2)*e^2)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*((b^7* c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 8*a^3*b*c^4)*d - (b^8 - 8*a*b^6*c + 20* a^2*b^4*c^2 - 17*a^3*b^2*c^3 + 4*a^4*c^4)*e - (b^5*c^5 - 7*a*b^3*c^6 + 12* a^2*b*c^7)*sqrt(((b^6*c^2 - 4*a*b^4*c^3 + 4*a^2*b^2*c^4)*d^2 - 2*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e + (b^8 - 6*a*b^6*c + 11*a^2 *b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^2)/(b^2*c^10 - 4*a*c^11)))*sqrt(((b^ 4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e + (b^ 2*c^5 - 4*a*c^6)*sqrt(((b^6*c^2 - 4*a*b^4*c^3 + 4*a^2*b^2*c^4)*d^2 - 2*(b^ 7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^2)/(b^2*c^10 - 4*a*c^11)))/(b^ 2*c^5 - 4*a*c^6)) - ((a^2*b^2*c^5 - 4*a^3*c^6)*e*x^2 + 2*(a^2*b^2*c^5 - 4* a^3*c^6)*d)*sqrt(((b^6*c^2 - 4*a*b^4*c^3 + 4*a^2*b^2*c^4)*d^2 - 2*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e + (b^8 - 6*a*b^6*c + 11...
\[ \int \frac {x^5 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\int \frac {x^{5} \sqrt {d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \]
\[ \int \frac {x^5 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\int { \frac {\sqrt {e x^{2} + d} x^{5}}{c x^{4} + b x^{2} + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (280) = 560\).
Time = 0.32 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.40 \[ \int \frac {x^5 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\frac {{\left ({\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} c^{2} e^{2} + 2 \, {\left (b^{2} c^{4} - 2 \, a c^{5}\right )} d^{2} e - {\left (3 \, b^{3} c^{3} - 8 \, a b c^{4}\right )} d e^{2} + {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{3} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} - \sqrt {b^{2} - 4 \, a c} b^{2} c^{2} d e + \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{2}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d}}{\sqrt {-\frac {2 \, c^{4} d e^{4} - b c^{3} e^{5} + \sqrt {-4 \, {\left (c^{4} d^{2} e^{4} - b c^{3} d e^{5} + a c^{3} e^{6}\right )} c^{4} e^{4} + {\left (2 \, c^{4} d e^{4} - b c^{3} e^{5}\right )}^{2}}}{c^{4} e^{4}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{3} d - {\left (b^{2} c^{2} - 4 \, a c^{3} + \sqrt {b^{2} - 4 \, a c} b c^{2}\right )} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} {\left | e \right |}} - \frac {{\left ({\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} c^{2} e^{2} + 2 \, {\left (b^{2} c^{4} - 2 \, a c^{5}\right )} d^{2} e - {\left (3 \, b^{3} c^{3} - 8 \, a b c^{4}\right )} d e^{2} + {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} e^{3} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} b c^{3} d^{2} - \sqrt {b^{2} - 4 \, a c} b^{2} c^{2} d e + \sqrt {b^{2} - 4 \, a c} a b c^{2} e^{2}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d}}{\sqrt {-\frac {2 \, c^{4} d e^{4} - b c^{3} e^{5} - \sqrt {-4 \, {\left (c^{4} d^{2} e^{4} - b c^{3} d e^{5} + a c^{3} e^{6}\right )} c^{4} e^{4} + {\left (2 \, c^{4} d e^{4} - b c^{3} e^{5}\right )}^{2}}}{c^{4} e^{4}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{3} d + {\left (b^{2} c^{2} - 4 \, a c^{3} - \sqrt {b^{2} - 4 \, a c} b c^{2}\right )} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2} {\left | e \right |}} + \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} c^{2} e^{2} - 3 \, \sqrt {e x^{2} + d} b c e^{3}}{3 \, c^{3} e^{3}} \]
(((b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*c^2*e^2 + 2*(b^ 2*c^4 - 2*a*c^5)*d^2*e - (3*b^3*c^3 - 8*a*b*c^4)*d*e^2 + (b^4*c^2 - 3*a*b^ 2*c^3)*e^3 - 2*(sqrt(b^2 - 4*a*c)*b*c^3*d^2 - sqrt(b^2 - 4*a*c)*b^2*c^2*d* e + sqrt(b^2 - 4*a*c)*a*b*c^2*e^2)*abs(c)*abs(e))*arctan(2*sqrt(1/2)*sqrt( e*x^2 + d)/sqrt(-(2*c^4*d*e^4 - b*c^3*e^5 + sqrt(-4*(c^4*d^2*e^4 - b*c^3*d *e^5 + a*c^3*e^6)*c^4*e^4 + (2*c^4*d*e^4 - b*c^3*e^5)^2))/(c^4*e^4)))/((2* sqrt(b^2 - 4*a*c)*c^3*d - (b^2*c^2 - 4*a*c^3 + sqrt(b^2 - 4*a*c)*b*c^2)*e) *sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^2*abs(e)) - (((b^3*c - 4*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*c^2*e^2 + 2*(b^2*c^4 - 2* a*c^5)*d^2*e - (3*b^3*c^3 - 8*a*b*c^4)*d*e^2 + (b^4*c^2 - 3*a*b^2*c^3)*e^3 + 2*(sqrt(b^2 - 4*a*c)*b*c^3*d^2 - sqrt(b^2 - 4*a*c)*b^2*c^2*d*e + sqrt(b ^2 - 4*a*c)*a*b*c^2*e^2)*abs(c)*abs(e))*arctan(2*sqrt(1/2)*sqrt(e*x^2 + d) /sqrt(-(2*c^4*d*e^4 - b*c^3*e^5 - sqrt(-4*(c^4*d^2*e^4 - b*c^3*d*e^5 + a*c ^3*e^6)*c^4*e^4 + (2*c^4*d*e^4 - b*c^3*e^5)^2))/(c^4*e^4)))/((2*sqrt(b^2 - 4*a*c)*c^3*d + (b^2*c^2 - 4*a*c^3 - sqrt(b^2 - 4*a*c)*b*c^2)*e)*sqrt(-4*c ^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c^2*abs(e)) + 1/3*((e*x^2 + d)^(3/ 2)*c^2*e^2 - 3*sqrt(e*x^2 + d)*b*c*e^3)/(c^3*e^3)
Time = 8.68 (sec) , antiderivative size = 8222, normalized size of antiderivative = 25.38 \[ \int \frac {x^5 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
(d + e*x^2)^(3/2)/(3*c*e) - atan(((((4*a*b^3*c^3*e^4 - 16*a^2*b*c^4*e^4 - 4*b^4*c^3*d*e^3 + 4*b^3*c^4*d^2*e^2 - 16*a*b*c^5*d^2*e^2 + 16*a*b^2*c^4*d* e^3)/c^3 - (2*(d + e*x^2)^(1/2)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e* (-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c ^6)))^(1/2)*(4*b^3*c^5*e^3 - 8*b^2*c^6*d*e^2 - 16*a*b*c^6*e^3 + 32*a*c^7*d *e^2))/c^3)*(-(b^7*e + 8*a^3*c^4*d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6* c*d - 18*a^2*b^2*c^3*d + 25*a^2*b^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^( 1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2*d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b ^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^7 + b^4*c^5 - 8*a*b^2*c^6)))^(1/2) - (2*(d + e*x^2)^(1/2)*(b^6*e^4 - 2*a^3*c^3*e^4 + 9*a^2*b^2*c^2*e^4 + 2*a^2*c^4*d^ 2*e^2 + b^4*c^2*d^2*e^2 - 6*a*b^4*c*e^4 - 2*b^5*c*d*e^3 + 10*a*b^3*c^2*d*e ^3 - 10*a^2*b*c^3*d*e^3 - 4*a*b^2*c^3*d^2*e^2))/c^3)*(-(b^7*e + 8*a^3*c^4* d + b^4*e*(-(4*a*c - b^2)^3)^(1/2) - b^6*c*d - 18*a^2*b^2*c^3*d + 25*a^2*b ^3*c^2*e + a^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e + 8*a*b^4*c^2* d - 20*a^3*b*c^3*e - b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) + 2*a*b*c^2*d*(-(4*a *c - b^2)^3)^(1/2) - 3*a*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c...