Integrand size = 30, antiderivative size = 385 \[ \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=-\frac {(b c-a d) x \sqrt {e+f x^2}}{5 c d \left (c+d x^2\right )^{5/2}}+\frac {(a d (4 d e-3 c f)+b c (d e-2 c f)) x \sqrt {e+f x^2}}{15 c^2 d (d e-c f) \left (c+d x^2\right )^{3/2}}+\frac {\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \sqrt {e+f x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{15 c^{5/2} d^{3/2} (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {e^{3/2} \sqrt {f} (2 a d (2 d e-3 c f)+b c (d e+c f)) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 c^3 d (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
-1/15*e^(3/2)*(2*a*d*(-3*c*f+2*d*e)+b*c*(c*f+d*e))*(1/(1+f*x^2/e))^(1/2)*( 1+f*x^2/e)^(1/2)*EllipticF(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f) ^(1/2))*f^(1/2)*(d*x^2+c)^(1/2)/c^3/d/(-c*f+d*e)^2/(e*(d*x^2+c)/c/(f*x^2+e ))^(1/2)/(f*x^2+e)^(1/2)-1/5*(-a*d+b*c)*x*(f*x^2+e)^(1/2)/c/d/(d*x^2+c)^(5 /2)+1/15*(a*d*(-3*c*f+4*d*e)+b*c*(-2*c*f+d*e))*x*(f*x^2+e)^(1/2)/c^2/d/(-c *f+d*e)/(d*x^2+c)^(3/2)+1/15*(2*b*c*(c^2*f^2-c*d*e*f+d^2*e^2)+a*d*(3*c^2*f ^2-13*c*d*e*f+8*d^2*e^2))*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Elliptic E(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-c*f/d/e)^(1/2))*(f*x^2+e)^(1/2)/c ^(5/2)/d^(3/2)/(-c*f+d*e)^2/(d*x^2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2 )
Result contains complex when optimal does not.
Time = 3.86 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {-\sqrt {\frac {d}{c}} x \left (e+f x^2\right ) \left (3 c^2 (b c-a d) (d e-c f)^2-c (d e-c f) (a d (4 d e-3 c f)+b c (d e-2 c f)) \left (c+d x^2\right )-\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \left (c+d x^2\right )^2\right )+i e \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \left (\left (2 b c \left (d^2 e^2-c d e f+c^2 f^2\right )+a d \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-(-d e+c f) (b c (-2 d e+c f)+a d (-8 d e+9 c f)) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )}{15 c^4 \left (\frac {d}{c}\right )^{3/2} (d e-c f)^2 \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \]
(-(Sqrt[d/c]*x*(e + f*x^2)*(3*c^2*(b*c - a*d)*(d*e - c*f)^2 - c*(d*e - c*f )*(a*d*(4*d*e - 3*c*f) + b*c*(d*e - 2*c*f))*(c + d*x^2) - (2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*(c + d*x^ 2)^2)) + I*e*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*((2*b*c *(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2)) *EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (-(d*e) + c*f)*(b*c*(-2* d*e + c*f) + a*d*(-8*d*e + 9*c*f))*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f) /(d*e)]))/(15*c^4*(d/c)^(3/2)*(d*e - c*f)^2*(c + d*x^2)^(5/2)*Sqrt[e + f*x ^2])
Time = 0.56 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {401, 25, 402, 25, 400, 313, 320}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle -\frac {\int -\frac {(2 b c+3 a d) f x^2+(b c+4 a d) e}{\left (d x^2+c\right )^{5/2} \sqrt {f x^2+e}}dx}{5 c d}-\frac {x \sqrt {e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(2 b c+3 a d) f x^2+(b c+4 a d) e}{\left (d x^2+c\right )^{5/2} \sqrt {f x^2+e}}dx}{5 c d}-\frac {x \sqrt {e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {x \sqrt {e+f x^2} (d e (4 a d+b c)-c f (3 a d+2 b c))}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}-\frac {\int -\frac {f (a d (4 d e-3 c f)+b c (d e-2 c f)) x^2+e (a d (8 d e-9 c f)+b c (2 d e-c f))}{\left (d x^2+c\right )^{3/2} \sqrt {f x^2+e}}dx}{3 c (d e-c f)}}{5 c d}-\frac {x \sqrt {e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {f (a d (4 d e-3 c f)+b c (d e-2 c f)) x^2+e (a d (8 d e-9 c f)+b c (2 d e-c f))}{\left (d x^2+c\right )^{3/2} \sqrt {f x^2+e}}dx}{3 c (d e-c f)}+\frac {x \sqrt {e+f x^2} (d e (4 a d+b c)-c f (3 a d+2 b c))}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}}{5 c d}-\frac {x \sqrt {e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\frac {\frac {\left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) \int \frac {\sqrt {f x^2+e}}{\left (d x^2+c\right )^{3/2}}dx}{d e-c f}-\frac {e f (2 a d (2 d e-3 c f)+b c (c f+d e)) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{d e-c f}}{3 c (d e-c f)}+\frac {x \sqrt {e+f x^2} (d e (4 a d+b c)-c f (3 a d+2 b c))}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}}{5 c d}-\frac {x \sqrt {e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {e+f x^2} \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {e f (2 a d (2 d e-3 c f)+b c (c f+d e)) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{d e-c f}}{3 c (d e-c f)}+\frac {x \sqrt {e+f x^2} (d e (4 a d+b c)-c f (3 a d+2 b c))}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}}{5 c d}-\frac {x \sqrt {e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {e+f x^2} \left (a d \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )+2 b c \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {e^{3/2} \sqrt {f} \sqrt {c+d x^2} (2 a d (2 d e-3 c f)+b c (c f+d e)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}}{3 c (d e-c f)}+\frac {x \sqrt {e+f x^2} (d e (4 a d+b c)-c f (3 a d+2 b c))}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}}{5 c d}-\frac {x \sqrt {e+f x^2} (b c-a d)}{5 c d \left (c+d x^2\right )^{5/2}}\) |
-1/5*((b*c - a*d)*x*Sqrt[e + f*x^2])/(c*d*(c + d*x^2)^(5/2)) + (((d*(b*c + 4*a*d)*e - c*(2*b*c + 3*a*d)*f)*x*Sqrt[e + f*x^2])/(3*c*(d*e - c*f)*(c + d*x^2)^(3/2)) + (((2*b*c*(d^2*e^2 - c*d*e*f + c^2*f^2) + a*d*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt [c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*Sqrt[d]*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[ (c*(e + f*x^2))/(e*(c + d*x^2))]) - (e^(3/2)*Sqrt[f]*(2*a*d*(2*d*e - 3*c*f ) + b*c*(d*e + c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]] , 1 - (d*e)/(c*f)])/(c*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*S qrt[e + f*x^2]))/(3*c*(d*e - c*f)))/(5*c*d)
3.1.28.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Time = 3.50 (sec) , antiderivative size = 756, normalized size of antiderivative = 1.96
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {\left (a d -b c \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 c \,d^{4} \left (x^{2}+\frac {c}{d}\right )^{3}}+\frac {\left (3 a c d f -4 a e \,d^{2}+2 c^{2} b f -b c d e \right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{15 d^{3} c^{2} \left (c f -d e \right ) \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (d f \,x^{2}+d e \right ) x \left (3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{15 d^{2} c^{3} \left (c f -d e \right )^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (d f \,x^{2}+d e \right )}}+\frac {\left (\frac {f \left (3 a c d f -4 a e \,d^{2}+2 c^{2} b f -b c d e \right )}{15 c^{2} \left (c f -d e \right ) d^{2}}-\frac {3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}}{15 d^{2} \left (c f -d e \right ) c^{3}}-\frac {e \left (3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{15 d \,c^{3} \left (c f -d e \right )^{2}}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\left (3 a \,c^{2} d \,f^{2}-13 a c \,d^{2} e f +8 a \,d^{3} e^{2}+2 b \,c^{3} f^{2}-2 b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{15 d \,c^{3} \left (c f -d e \right )^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(756\) |
| default | \(\text {Expression too large to display}\) | \(2856\) |
((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(1/5*(a*d-b*c) /c/d^4*x*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/(x^2+c/d)^3+1/15*(3*a*c*d*f-4 *a*d^2*e+2*b*c^2*f-b*c*d*e)/d^3/c^2/(c*f-d*e)*x*(d*f*x^4+c*f*x^2+d*e*x^2+c *e)^(1/2)/(x^2+c/d)^2+1/15*(d*f*x^2+d*e)/d^2/c^3/(c*f-d*e)^2*x*(3*a*c^2*d* f^2-13*a*c*d^2*e*f+8*a*d^3*e^2+2*b*c^3*f^2-2*b*c^2*d*e*f+2*b*c*d^2*e^2)/(( x^2+c/d)*(d*f*x^2+d*e))^(1/2)+(1/15*f*(3*a*c*d*f-4*a*d^2*e+2*b*c^2*f-b*c*d *e)/c^2/(c*f-d*e)/d^2-1/15/d^2/(c*f-d*e)*(3*a*c^2*d*f^2-13*a*c*d^2*e*f+8*a *d^3*e^2+2*b*c^3*f^2-2*b*c^2*d*e*f+2*b*c*d^2*e^2)/c^3-1/15/d*e/c^3/(c*f-d* e)^2*(3*a*c^2*d*f^2-13*a*c*d^2*e*f+8*a*d^3*e^2+2*b*c^3*f^2-2*b*c^2*d*e*f+2 *b*c*d^2*e^2))/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c *f*x^2+d*e*x^2+c*e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2 ))+1/15/d*(3*a*c^2*d*f^2-13*a*c*d^2*e*f+8*a*d^3*e^2+2*b*c^3*f^2-2*b*c^2*d* e*f+2*b*c*d^2*e^2)/c^3/(c*f-d*e)^2*e/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x ^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*(EllipticF(x*(-d/c)^(1/2), (-1+(c*f+d*e)/e/d)^(1/2))-EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2 ))))
Leaf count of result is larger than twice the leaf count of optimal. 1105 vs. \(2 (367) = 734\).
Time = 0.13 (sec) , antiderivative size = 1105, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
-1/15*(((2*(b*c*d^6 + 4*a*d^7)*e^2 - (2*b*c^2*d^5 + 13*a*c*d^6)*e*f + (2*b *c^3*d^4 + 3*a*c^2*d^5)*f^2)*x^6 + 3*(2*(b*c^2*d^5 + 4*a*c*d^6)*e^2 - (2*b *c^3*d^4 + 13*a*c^2*d^5)*e*f + (2*b*c^4*d^3 + 3*a*c^3*d^4)*f^2)*x^4 + 2*(b *c^4*d^3 + 4*a*c^3*d^4)*e^2 - (2*b*c^5*d^2 + 13*a*c^4*d^3)*e*f + (2*b*c^6* d + 3*a*c^5*d^2)*f^2 + 3*(2*(b*c^3*d^4 + 4*a*c^2*d^5)*e^2 - (2*b*c^4*d^3 + 13*a*c^3*d^4)*e*f + (2*b*c^5*d^2 + 3*a*c^4*d^3)*f^2)*x^2)*sqrt(c*e)*sqrt( -d/c)*elliptic_e(arcsin(x*sqrt(-d/c)), c*f/(d*e)) - ((2*(b*c*d^6 + 4*a*d^7 )*e^2 + (b*c^3*d^4 + 2*(2*a - b)*c^2*d^5 - 13*a*c*d^6)*e*f + (b*c^4*d^3 - 2*(3*a - b)*c^3*d^4 + 3*a*c^2*d^5)*f^2)*x^6 + 3*(2*(b*c^2*d^5 + 4*a*c*d^6) *e^2 + (b*c^4*d^3 + 2*(2*a - b)*c^3*d^4 - 13*a*c^2*d^5)*e*f + (b*c^5*d^2 - 2*(3*a - b)*c^4*d^3 + 3*a*c^3*d^4)*f^2)*x^4 + 2*(b*c^4*d^3 + 4*a*c^3*d^4) *e^2 + (b*c^6*d + 2*(2*a - b)*c^5*d^2 - 13*a*c^4*d^3)*e*f + (b*c^7 - 2*(3* a - b)*c^6*d + 3*a*c^5*d^2)*f^2 + 3*(2*(b*c^3*d^4 + 4*a*c^2*d^5)*e^2 + (b* c^5*d^2 + 2*(2*a - b)*c^4*d^3 - 13*a*c^3*d^4)*e*f + (b*c^6*d - 2*(3*a - b) *c^5*d^2 + 3*a*c^4*d^3)*f^2)*x^2)*sqrt(c*e)*sqrt(-d/c)*elliptic_f(arcsin(x *sqrt(-d/c)), c*f/(d*e)) - ((2*(b*c^2*d^5 + 4*a*c*d^6)*e^2 - (2*b*c^3*d^4 + 13*a*c^2*d^5)*e*f + (2*b*c^4*d^3 + 3*a*c^3*d^4)*f^2)*x^5 + (5*(b*c^3*d^4 + 4*a*c^2*d^5)*e^2 - (7*b*c^4*d^3 + 33*a*c^3*d^4)*e*f + 3*(2*b*c^5*d^2 + 3*a*c^4*d^3)*f^2)*x^3 + (15*a*c^3*d^4*e^2 + (b*c^5*d^2 - 26*a*c^4*d^3)*e*f + (b*c^6*d + 9*a*c^5*d^2)*f^2)*x)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e))/(c^...
\[ \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\left (a + b x^{2}\right ) \sqrt {e + f x^{2}}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \sqrt {f x^{2} + e}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right ) \sqrt {e+f x^2}}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\left (b\,x^2+a\right )\,\sqrt {f\,x^2+e}}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \]