Integrand size = 29, antiderivative size = 406 \[ \int \frac {x^7 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{c^3}-\frac {(c d+b e) \left (d+e x^2\right )^{3/2}}{3 c^2 e^2}+\frac {\left (d+e x^2\right )^{5/2}}{5 c e^2}-\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e-\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e+\frac {b^3 c d-3 a b c^2 d-b^4 e+4 a b^2 c e-2 a^2 c^2 e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-1/3*(b*e+c*d)*(e*x^2+d)^(3/2)/c^2/e^2+1/5*(e*x^2+d)^(5/2)/c/e^2+(-a*c+b^2 )*(e*x^2+d)^(1/2)/c^3-1/2*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^2*c*d-a*c^2*d-b^3*e+2*a*b*c*e+(2*a^2*c^ 2*e-4*a*b^2*c*e+3*a*b*c^2*d+b^4*e-b^3*c*d)/(-4*a*c+b^2)^(1/2))/c^(7/2)*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^2*c*d-a*c^2*d-b^3 *e+2*a*b*c*e+(-2*a^2*c^2*e+4*a*b^2*c*e-3*a*b*c^2*d-b^4*e+b^3*c*d)/(-4*a*c+ b^2)^(1/2))/c^(7/2)*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Time = 1.30 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.17 \[ \int \frac {x^7 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\frac {\frac {2 \sqrt {c} \sqrt {d+e x^2} \left (15 b^2 e^2+c^2 \left (-2 d^2+d e x^2+3 e^2 x^4\right )-5 c e \left (3 a e+b \left (d+e x^2\right )\right )\right )}{e^2}-\frac {15 \sqrt {2} \left (-b^4 e+a c^2 \left (\sqrt {b^2-4 a c} d-2 a e\right )+b^2 c \left (-\sqrt {b^2-4 a c} d+4 a e\right )+b^3 \left (c d+\sqrt {b^2-4 a c} e\right )-a b c \left (3 c d+2 \sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {15 \sqrt {2} \left (b^4 e+a c^2 \left (\sqrt {b^2-4 a c} d+2 a e\right )-b^2 c \left (\sqrt {b^2-4 a c} d+4 a e\right )+a b c \left (3 c d-2 \sqrt {b^2-4 a c} e\right )+b^3 \left (-c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}}{30 c^{7/2}} \]
((2*Sqrt[c]*Sqrt[d + e*x^2]*(15*b^2*e^2 + c^2*(-2*d^2 + d*e*x^2 + 3*e^2*x^ 4) - 5*c*e*(3*a*e + b*(d + e*x^2))))/e^2 - (15*Sqrt[2]*(-(b^4*e) + a*c^2*( Sqrt[b^2 - 4*a*c]*d - 2*a*e) + b^2*c*(-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e) + b^ 3*(c*d + Sqrt[b^2 - 4*a*c]*e) - a*b*c*(3*c*d + 2*Sqrt[b^2 - 4*a*c]*e))*Arc Tan[(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]) - (15* Sqrt[2]*(b^4*e + a*c^2*(Sqrt[b^2 - 4*a*c]*d + 2*a*e) - b^2*c*(Sqrt[b^2 - 4 *a*c]*d + 4*a*e) + a*b*c*(3*c*d - 2*Sqrt[b^2 - 4*a*c]*e) + b^3*(-(c*d) + S qrt[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]))/(30*c^(7/2))
Time = 4.30 (sec) , antiderivative size = 397, 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, 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.
| \(\displaystyle \int \frac {x^7 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {x^6 \sqrt {e x^2+d}}{c x^4+b x^2+a}dx^2\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle \frac {\int \left (\frac {x^8}{c e}-\frac {(c d+b e) x^4}{c^2 e}+\frac {\left (b^2-a c\right ) e}{c^3}-\frac {\left (b^2-a c\right ) \left (c d^2-b e d+a e^2\right )-\left (-e b^3+c d b^2+2 a c e b-a c^2 d\right ) x^4}{c^3 e \left (\frac {c x^8}{e^2}-\frac {(2 c d-b e) x^4}{e^2}+a+\frac {d (c d-b e)}{e^2}\right )}\right )d\sqrt {e x^2+d}}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt {b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {e \left (\frac {-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d}{\sqrt {b^2-4 a c}}+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{7/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {e \left (b^2-a c\right ) \sqrt {d+e x^2}}{c^3}-\frac {x^6 (b e+c d)}{3 c^2 e}+\frac {x^{10}}{5 c e}}{e}\) |
(-1/3*((c*d + b*e)*x^6)/(c^2*e) + x^10/(5*c*e) + ((b^2 - a*c)*e*Sqrt[d + e *x^2])/c^3 - (e*(b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e - (b^3*c*d - 3*a*b* c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqr t[2]*Sqrt[c]*Sqrt[d + e*x^2])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(S qrt[2]*c^(7/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (e*(b^2*c*d - a* c^2*d - b^3*e + 2*a*b*c*e + (b^3*c*d - 3*a*b*c^2*d - b^4*e + 4*a*b^2*c*e - 2*a^2*c^2*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^(7/2)*Sqrt[2*c*d - ( b + Sqrt[b^2 - 4*a*c])*e]))/e
3.4.54.3.1 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 = 2.39 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.17
| method | result | size |
| risch | \(-\frac {\left (-3 e^{2} c^{2} x^{4}+5 b c \,e^{2} x^{2}-c^{2} d e \,x^{2}+15 e^{2} a c -15 b^{2} e^{2}+5 b c d e +2 c^{2} d^{2}\right ) \sqrt {e \,x^{2}+d}}{15 e^{2} c^{3}}-\frac {\sqrt {2}\, \left (\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (-\frac {a \,c^{2} d}{2}+b \left (a e +\frac {b d}{2}\right ) c -\frac {b^{3} e}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (a \left (a e +\frac {3 b d}{2}\right ) c^{2}+\left (-2 a \,b^{2} e -\frac {1}{2} b^{3} d \right ) c +\frac {b^{4} e}{2}\right )\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (\frac {a \,c^{2} d}{2}+\left (-a b e -\frac {1}{2} b^{2} d \right ) c +\frac {b^{3} e}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (a \left (a e +\frac {3 b d}{2}\right ) c^{2}+\left (-2 a \,b^{2} e -\frac {1}{2} b^{3} d \right ) c +\frac {b^{4} e}{2}\right )\right ) \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )\right )}{c^{3} \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}\) | \(473\) |
| pseudoelliptic | \(-\frac {\left (\left (\left (a b c -\frac {1}{2} b^{3}\right ) e -\frac {d c \left (a c -b^{2}\right )}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (-2 a \,b^{2} c +\frac {1}{2} b^{4}+a^{2} c^{2}\right ) e +\frac {3 c d b \left (a c -\frac {b^{2}}{3}\right )}{2}\right )\right ) e^{2} \sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (e^{2} \sqrt {2}\, \left (\left (\left (-a b c +\frac {1}{2} b^{3}\right ) e +\frac {d c \left (a c -b^{2}\right )}{2}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (\left (-2 a \,b^{2} c +\frac {1}{2} b^{4}+a^{2} c^{2}\right ) e +\frac {3 c d b \left (a c -\frac {b^{2}}{3}\right )}{2}\right )\right ) \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {e \,x^{2}+d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (\left (-\frac {c^{2} x^{4}}{5}+\left (\frac {b \,x^{2}}{3}+a \right ) c -b^{2}\right ) e^{2}+\frac {c d \left (-\frac {c \,x^{2}}{5}+b \right ) e}{3}+\frac {2 c^{2} d^{2}}{15}\right )\right )}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, e^{2} c^{3}}\) | \(514\) |
| default | \(\frac {\frac {x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{5 e}-\frac {2 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{15 e^{2}}}{c}-\frac {b \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} e}+\frac {-\left (a c -b^{2}\right ) \sqrt {e \,x^{2}+d}+\frac {\left (-2 a^{2} c^{2} e^{2}+4 a \,b^{2} c \,e^{2}-3 a b \,c^{2} d e -b^{4} e^{2}+b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (2 a^{2} c^{2} e^{2}-4 a \,b^{2} c \,e^{2}+3 a b \,c^{2} d e +b^{4} e^{2}-b^{3} c d e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{c^{3}}\) | \(552\) |
-1/15*(-3*c^2*e^2*x^4+5*b*c*e^2*x^2-c^2*d*e*x^2+15*a*c*e^2-15*b^2*e^2+5*b* c*d*e+2*c^2*d^2)*(e*x^2+d)^(1/2)/e^2/c^3-1/c^3/((b*e-2*c*d+(-4*e^2*(a*c-1/ 4*b^2))^(1/2))*c)^(1/2)*2^(1/2)/((-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)*(((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2)) ^(1/2))*c)^(1/2)*((-1/2*a*c^2*d+b*(a*e+1/2*b*d)*c-1/2*b^3*e)*(-4*e^2*(a*c- 1/4*b^2))^(1/2)+e*(a*(a*e+3/2*b*d)*c^2+(-2*a*b^2*e-1/2*b^3*d)*c+1/2*b^4*e) )*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))+((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*((1/2*a *c^2*d+(-a*b*e-1/2*b^2*d)*c+1/2*b^3*e)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+e*(a*( a*e+3/2*b*d)*c^2+(-2*a*b^2*e-1/2*b^3*d)*c+1/2*b^4*e))*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)))
Leaf count of result is larger than twice the leaf count of optimal. 5829 vs. \(2 (356) = 712\).
Time = 257.63 (sec) , antiderivative size = 5829, normalized size of antiderivative = 14.36 \[ \int \frac {x^7 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
\[ \int \frac {x^7 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\int \frac {x^{7} \sqrt {d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \]
\[ \int \frac {x^7 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\int { \frac {\sqrt {e x^{2} + d} x^{7}}{c x^{4} + b x^{2} + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 959 vs. \(2 (356) = 712\).
Time = 0.33 (sec) , antiderivative size = 959, normalized size of antiderivative = 2.36 \[ \int \frac {x^7 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=-\frac {{\left ({\left ({\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e\right )} c^{2} e^{2} + 2 \, {\left (b^{3} c^{4} - 3 \, a b c^{5}\right )} d^{2} e - {\left (3 \, b^{4} c^{3} - 11 \, a b^{2} c^{4} + 4 \, a^{2} c^{5}\right )} d e^{2} + {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3} + 2 \, a^{2} b c^{4}\right )} e^{3} - 2 \, {\left ({\left (b^{2} c^{3} - a c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} c^{2} - a b c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} c^{2} - a^{2} c^{3}\right )} \sqrt {b^{2} - 4 \, a c} 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^{6} d e^{12} - b c^{5} e^{13} + \sqrt {-4 \, {\left (c^{6} d^{2} e^{12} - b c^{5} d e^{13} + a c^{5} e^{14}\right )} c^{6} e^{12} + {\left (2 \, c^{6} d e^{12} - b c^{5} e^{13}\right )}^{2}}}{c^{6} e^{12}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{4} d - {\left (b^{2} c^{3} - 4 \, a c^{4} + \sqrt {b^{2} - 4 \, a c} b c^{3}\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^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} e\right )} c^{2} e^{2} + 2 \, {\left (b^{3} c^{4} - 3 \, a b c^{5}\right )} d^{2} e - {\left (3 \, b^{4} c^{3} - 11 \, a b^{2} c^{4} + 4 \, a^{2} c^{5}\right )} d e^{2} + {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3} + 2 \, a^{2} b c^{4}\right )} e^{3} + 2 \, {\left ({\left (b^{2} c^{3} - a c^{4}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} c^{2} - a b c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} c^{2} - a^{2} c^{3}\right )} \sqrt {b^{2} - 4 \, a c} 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^{6} d e^{12} - b c^{5} e^{13} - \sqrt {-4 \, {\left (c^{6} d^{2} e^{12} - b c^{5} d e^{13} + a c^{5} e^{14}\right )} c^{6} e^{12} + {\left (2 \, c^{6} d e^{12} - b c^{5} e^{13}\right )}^{2}}}{c^{6} e^{12}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{4} d + {\left (b^{2} c^{3} - 4 \, a c^{4} - \sqrt {b^{2} - 4 \, a c} b c^{3}\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 {3 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} c^{4} e^{8} - 5 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} c^{4} d e^{8} - 5 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} b c^{3} e^{9} + 15 \, \sqrt {e x^{2} + d} b^{2} c^{2} e^{10} - 15 \, \sqrt {e x^{2} + d} a c^{3} e^{10}}{15 \, c^{5} e^{10}} \]
-(((b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d - (b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*e )*c^2*e^2 + 2*(b^3*c^4 - 3*a*b*c^5)*d^2*e - (3*b^4*c^3 - 11*a*b^2*c^4 + 4* a^2*c^5)*d*e^2 + (b^5*c^2 - 4*a*b^3*c^3 + 2*a^2*b*c^4)*e^3 - 2*((b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c^2 - a*b*c^3)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2*c^2 - a^2*c^3)*sqrt(b^2 - 4*a*c)*e^2)*abs(c)*abs(e))*arctan(2*sqr t(1/2)*sqrt(e*x^2 + d)/sqrt(-(2*c^6*d*e^12 - b*c^5*e^13 + sqrt(-4*(c^6*d^2 *e^12 - b*c^5*d*e^13 + a*c^5*e^14)*c^6*e^12 + (2*c^6*d*e^12 - b*c^5*e^13)^ 2))/(c^6*e^12)))/((2*sqrt(b^2 - 4*a*c)*c^4*d - (b^2*c^3 - 4*a*c^4 + sqrt(b ^2 - 4*a*c)*b*c^3)*e)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^2 *abs(e)) + (((b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d - (b^5 - 6*a*b^3*c + 8*a^ 2*b*c^2)*e)*c^2*e^2 + 2*(b^3*c^4 - 3*a*b*c^5)*d^2*e - (3*b^4*c^3 - 11*a*b^ 2*c^4 + 4*a^2*c^5)*d*e^2 + (b^5*c^2 - 4*a*b^3*c^3 + 2*a^2*b*c^4)*e^3 + 2*( (b^2*c^3 - a*c^4)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c^2 - a*b*c^3)*sqrt(b^2 - 4 *a*c)*d*e + (a*b^2*c^2 - a^2*c^3)*sqrt(b^2 - 4*a*c)*e^2)*abs(c)*abs(e))*ar ctan(2*sqrt(1/2)*sqrt(e*x^2 + d)/sqrt(-(2*c^6*d*e^12 - b*c^5*e^13 - sqrt(- 4*(c^6*d^2*e^12 - b*c^5*d*e^13 + a*c^5*e^14)*c^6*e^12 + (2*c^6*d*e^12 - b* c^5*e^13)^2))/(c^6*e^12)))/((2*sqrt(b^2 - 4*a*c)*c^4*d + (b^2*c^3 - 4*a*c^ 4 - sqrt(b^2 - 4*a*c)*b*c^3)*e)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c) *c)*e)*c^2*abs(e)) + 1/15*(3*(e*x^2 + d)^(5/2)*c^4*e^8 - 5*(e*x^2 + d)^(3/ 2)*c^4*d*e^8 - 5*(e*x^2 + d)^(3/2)*b*c^3*e^9 + 15*sqrt(e*x^2 + d)*b^2*c...
Time = 8.90 (sec) , antiderivative size = 11195, normalized size of antiderivative = 27.57 \[ \int \frac {x^7 \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
(d + e*x^2)^(1/2)*((3*d^2)/(c*e^2) - (a*e^4 + c*d^2*e^2 - b*d*e^3)/(c^2*e^ 4) + (((3*d)/(c*e^2) + (b*e^3 - 2*c*d*e^2)/(c^2*e^4))*(b*e^3 - 2*c*d*e^2)) /(c*e^2)) - (d + e*x^2)^(3/2)*(d/(c*e^2) + (b*e^3 - 2*c*d*e^2)/(3*c^2*e^4) ) + atan(((((16*a^3*c^6*e^4 + 4*a*b^4*c^4*e^4 - 4*b^5*c^4*d*e^3 - 20*a^2*b ^2*c^5*e^4 + 16*a^2*c^7*d^2*e^2 + 4*b^4*c^5*d^2*e^2 + 20*a*b^3*c^5*d*e^3 - 16*a^2*b*c^6*d*e^3 - 20*a*b^2*c^6*d^2*e^2)/c^5 - (2*(d + e*x^2)^(1/2)*(-( b^9*e - 8*a^4*c^5*d - b^6*e*(-(4*a*c - b^2)^3)^(1/2) - b^8*c*d - 33*a^2*b^ 4*c^3*d + 38*a^3*b^2*c^4*d + 42*a^2*b^5*c^2*e - 63*a^3*b^3*c^3*e + a^3*c^3 *e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*c^2*d + 28*a^4*b*c^4 *e + b^5*c*d*(-(4*a*c - b^2)^3)^(1/2) + 5*a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/ 2) - 4*a*b^3*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b*c^3*d*(-(4*a*c - b^2 )^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^9 + b^ 4*c^7 - 8*a*b^2*c^8)))^(1/2)*(4*b^3*c^7*e^3 - 8*b^2*c^8*d*e^2 - 16*a*b*c^8 *e^3 + 32*a*c^9*d*e^2))/c^5)*(-(b^9*e - 8*a^4*c^5*d - b^6*e*(-(4*a*c - b^2 )^3)^(1/2) - b^8*c*d - 33*a^2*b^4*c^3*d + 38*a^3*b^2*c^4*d + 42*a^2*b^5*c^ 2*e - 63*a^3*b^3*c^3*e + a^3*c^3*e*(-(4*a*c - b^2)^3)^(1/2) - 11*a*b^7*c*e + 10*a*b^6*c^2*d + 28*a^4*b*c^4*e + b^5*c*d*(-(4*a*c - b^2)^3)^(1/2) + 5* a*b^4*c*e*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b^3*c^2*d*(-(4*a*c - b^2)^3)^(1/2 ) + 3*a^2*b*c^3*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a^2*b^2*c^2*e*(-(4*a*c - b^ 2)^3)^(1/2))/(8*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2) - (2*(d + ...