Integrand size = 30, antiderivative size = 501 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^{3/2}} \, dx=-\frac {\left (5 a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )-2 b e \left (24 d^2 e^2-44 c d e f+19 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 e f^3 \sqrt {e+f x^2}}-\frac {(b e-a f) x \left (c+d x^2\right )^{5/2}}{e f \sqrt {e+f x^2}}-\frac {d (b e (24 d e-23 c f)-5 a f (4 d e-3 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 e f^3}+\frac {d (6 b e-5 a f) x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 e f^2}+\frac {\left (5 a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )-2 b e \left (24 d^2 e^2-44 c d e f+19 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 \sqrt {e} f^{7/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\sqrt {e} \left (10 a d f (2 d e-3 c f)-b \left (24 d^2 e^2-41 c d e f+15 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 f^{7/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
-(-a*f+b*e)*x*(d*x^2+c)^(5/2)/e/f/(f*x^2+e)^(1/2)-1/15*(5*a*f*(3*c^2*f^2-1 3*c*d*e*f+8*d^2*e^2)-2*b*e*(19*c^2*f^2-44*c*d*e*f+24*d^2*e^2))*x*(d*x^2+c) ^(1/2)/e/f^3/(f*x^2+e)^(1/2)+1/15*(5*a*f*(3*c^2*f^2-13*c*d*e*f+8*d^2*e^2)- 2*b*e*(19*c^2*f^2-44*c*d*e*f+24*d^2*e^2))*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e )^(1/2)*EllipticE(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*( d*x^2+c)^(1/2)/f^(7/2)/e^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^( 1/2)-1/15*(10*a*d*f*(-3*c*f+2*d*e)-b*(15*c^2*f^2-41*c*d*e*f+24*d^2*e^2))*( 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))*e^(1/2)*(d*x^2+c)^(1/2)/f^(7/2)/(e*(d*x^2+c) /c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/5*d*(-5*a*f+6*b*e)*x*(d*x^2+c)^(3/2) *(f*x^2+e)^(1/2)/e/f^2-1/15*d*(b*e*(-23*c*f+24*d*e)-5*a*f*(-3*c*f+4*d*e))* x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/e/f^3
Result contains complex when optimal does not.
Time = 5.81 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (5 a f \left (-6 c d e f+3 c^2 f^2+d^2 e \left (4 e+f x^2\right )\right )+b e \left (-15 c^2 f^2+c d f \left (41 e+11 f x^2\right )-3 d^2 \left (8 e^2+2 e f x^2-f^2 x^4\right )\right )\right )-i d e \left (-5 a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )+2 b e \left (24 d^2 e^2-44 c d e f+19 c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i e (-d e+c f) \left (5 a d f (-8 d e+9 c f)+b \left (48 d^2 e^2-64 c d e f+15 c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{15 \sqrt {\frac {d}{c}} e f^4 \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
(Sqrt[d/c]*f*x*(c + d*x^2)*(5*a*f*(-6*c*d*e*f + 3*c^2*f^2 + d^2*e*(4*e + f *x^2)) + b*e*(-15*c^2*f^2 + c*d*f*(41*e + 11*f*x^2) - 3*d^2*(8*e^2 + 2*e*f *x^2 - f^2*x^4))) - I*d*e*(-5*a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + 2 *b*e*(24*d^2*e^2 - 44*c*d*e*f + 19*c^2*f^2))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*e*(-(d*e) + c*f)*(5*a*d*f*(-8*d*e + 9*c*f) + b*(48*d^2*e^2 - 64*c*d*e*f + 15*c^2*f^2)) *Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(15*Sqrt[d/c]*e*f^4*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])
Time = 0.68 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.92, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {401, 25, 403, 25, 403, 406, 320, 388, 313}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle -\frac {\int -\frac {\left (d x^2+c\right )^{3/2} \left (d (6 b e-5 a f) x^2+b c e\right )}{\sqrt {f x^2+e}}dx}{e f}-\frac {x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt {e+f x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (d x^2+c\right )^{3/2} \left (d (6 b e-5 a f) x^2+b c e\right )}{\sqrt {f x^2+e}}dx}{e f}-\frac {x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt {e+f x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int -\frac {\sqrt {d x^2+c} \left (d (b e (24 d e-23 c f)-5 a f (4 d e-3 c f)) x^2+c e (6 b d e-5 b c f-5 a d f)\right )}{\sqrt {f x^2+e}}dx}{5 f}+\frac {d x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (6 b e-5 a f)}{5 f}}{e f}-\frac {x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt {e+f x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {d x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (6 b e-5 a f)}{5 f}-\frac {\int \frac {\sqrt {d x^2+c} \left (d (b e (24 d e-23 c f)-5 a f (4 d e-3 c f)) x^2+c e (6 b d e-5 b c f-5 a d f)\right )}{\sqrt {f x^2+e}}dx}{5 f}}{e f}-\frac {x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt {e+f x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {d x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (6 b e-5 a f)}{5 f}-\frac {\frac {\int \frac {d \left (5 a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )-2 b e \left (24 d^2 e^2-44 c d f e+19 c^2 f^2\right )\right ) x^2+c e \left (10 a d f (2 d e-3 c f)-b \left (24 d^2 e^2-41 c d f e+15 c^2 f^2\right )\right )}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{3 f}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{3 f}}{5 f}}{e f}-\frac {x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt {e+f x^2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {d x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (6 b e-5 a f)}{5 f}-\frac {\frac {c e \left (10 a d f (2 d e-3 c f)-b \left (15 c^2 f^2-41 c d e f+24 d^2 e^2\right )\right ) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx+d \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{3 f}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{3 f}}{5 f}}{e f}-\frac {x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt {e+f x^2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {d x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (6 b e-5 a f)}{5 f}-\frac {\frac {d \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx+\frac {e^{3/2} \sqrt {c+d x^2} \left (10 a d f (2 d e-3 c f)-b \left (15 c^2 f^2-41 c d e f+24 d^2 e^2\right )\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}}{3 f}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{3 f}}{5 f}}{e f}-\frac {x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt {e+f x^2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {d x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (6 b e-5 a f)}{5 f}-\frac {\frac {d \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right ) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {e \int \frac {\sqrt {d x^2+c}}{\left (f x^2+e\right )^{3/2}}dx}{d}\right )+\frac {e^{3/2} \sqrt {c+d x^2} \left (10 a d f (2 d e-3 c f)-b \left (15 c^2 f^2-41 c d e f+24 d^2 e^2\right )\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}}{3 f}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{3 f}}{5 f}}{e f}-\frac {x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt {e+f x^2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {d x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2} (6 b e-5 a f)}{5 f}-\frac {\frac {d \left (5 a f \left (3 c^2 f^2-13 c d e f+8 d^2 e^2\right )-2 b e \left (19 c^2 f^2-44 c d e f+24 d^2 e^2\right )\right ) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )+\frac {e^{3/2} \sqrt {c+d x^2} \left (10 a d f (2 d e-3 c f)-b \left (15 c^2 f^2-41 c d e f+24 d^2 e^2\right )\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}}{3 f}+\frac {d x \sqrt {c+d x^2} \sqrt {e+f x^2} (b e (24 d e-23 c f)-5 a f (4 d e-3 c f))}{3 f}}{5 f}}{e f}-\frac {x \left (c+d x^2\right )^{5/2} (b e-a f)}{e f \sqrt {e+f x^2}}\) |
-(((b*e - a*f)*x*(c + d*x^2)^(5/2))/(e*f*Sqrt[e + f*x^2])) + ((d*(6*b*e - 5*a*f)*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*f) - ((d*(b*e*(24*d*e - 23* c*f) - 5*a*f*(4*d*e - 3*c*f))*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(3*f) + ( d*(5*a*f*(8*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) - 2*b*e*(24*d^2*e^2 - 44*c*d *e*f + 19*c^2*f^2))*((x*Sqrt[c + d*x^2])/(d*Sqrt[e + f*x^2]) - (Sqrt[e]*Sq rt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(d* Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])) + (e^(3/2) *(10*a*d*f*(2*d*e - 3*c*f) - b*(24*d^2*e^2 - 41*c*d*e*f + 15*c^2*f^2))*Sqr t[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqr t[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]))/(3*f))/(5*f)) /(e*f)
3.1.42.3.1 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[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, 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[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Time = 9.68 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.58
| method | result | size |
| risch | \(\frac {x d \left (3 b d f \,x^{2}+5 a d f +11 b c f -9 b d e \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{15 f^{3}}+\frac {\left (-\frac {d \left (35 a c d \,f^{2}-25 a \,d^{2} e f +23 b \,c^{2} f^{2}-58 b c d e f +33 b \,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 )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}+\frac {\left (45 a \,c^{2} d \,f^{3}-50 a c \,d^{2} e \,f^{2}+15 a \,d^{3} e^{2} f +15 b \,c^{3} f^{3}-56 b \,c^{2} d e \,f^{2}+54 b c \,d^{2} e^{2} f -15 b \,d^{3} e^{3}\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 )}{f \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {\left (15 a \,c^{3} f^{4}-45 a \,c^{2} d e \,f^{3}+45 a c \,d^{2} e^{2} f^{2}-15 a \,d^{3} e^{3} f -15 b \,c^{3} e \,f^{3}+45 b \,c^{2} d \,e^{2} f^{2}-45 b c \,d^{2} e^{3} f +15 b \,d^{3} e^{4}\right ) \left (\frac {\left (d f \,x^{2}+c f \right ) x}{e \left (c f -d e \right ) \sqrt {\left (x^{2}+\frac {e}{f}\right ) \left (d f \,x^{2}+c f \right )}}+\frac {\left (\frac {1}{e}-\frac {c f}{e \left (c f -d e \right )}\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 {d \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 )}{\left (c f -d e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{f}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{15 f^{3} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(794\) |
| elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {\left (d f \,x^{2}+c f \right ) \left (c^{2} a \,f^{3}-2 a c d e \,f^{2}+a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -b \,d^{2} e^{3}\right ) x}{e \,f^{4} \sqrt {\left (x^{2}+\frac {e}{f}\right ) \left (d f \,x^{2}+c f \right )}}+\frac {b \,d^{2} x^{3} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 f^{2}}+\frac {\left (\frac {d^{2} \left (a d f +3 b c f -b d e \right )}{f^{2}}-\frac {b \,d^{2} \left (4 c f +4 d e \right )}{5 f^{2}}\right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 d f}+\frac {\left (\frac {3 a \,c^{2} d \,f^{3}-3 a c \,d^{2} e \,f^{2}+a \,d^{3} e^{2} f +b \,c^{3} f^{3}-3 b \,c^{2} d e \,f^{2}+3 b c \,d^{2} e^{2} f -b \,d^{3} e^{3}}{f^{4}}+\frac {\left (c^{2} a \,f^{3}-2 a c d e \,f^{2}+a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -b \,d^{2} e^{3}\right ) \left (c f -d e \right )}{f^{4} e}-\frac {c \left (c^{2} a \,f^{3}-2 a c d e \,f^{2}+a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -b \,d^{2} e^{3}\right )}{f^{3} e}-\frac {\left (\frac {d^{2} \left (a d f +3 b c f -b d e \right )}{f^{2}}-\frac {b \,d^{2} \left (4 c f +4 d e \right )}{5 f^{2}}\right ) c e}{3 d f}\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 (\frac {d \left (3 a c d \,f^{2}-a \,d^{2} e f +3 b \,c^{2} f^{2}-3 b c d e f +b \,d^{2} e^{2}\right )}{f^{3}}-\frac {\left (c^{2} a \,f^{3}-2 a c d e \,f^{2}+a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -b \,d^{2} e^{3}\right ) d}{f^{3} e}-\frac {3 c \,d^{2} e b}{5 f^{2}}-\frac {\left (\frac {d^{2} \left (a d f +3 b c f -b d e \right )}{f^{2}}-\frac {b \,d^{2} \left (4 c f +4 d e \right )}{5 f^{2}}\right ) \left (2 c f +2 d e \right )}{3 d f}\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 )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(892\) |
| default | \(\text {Expression too large to display}\) | \(1169\) |
1/15*x*d*(3*b*d*f*x^2+5*a*d*f+11*b*c*f-9*b*d*e)*(d*x^2+c)^(1/2)*(f*x^2+e)^ (1/2)/f^3+1/15/f^3*(-d*(35*a*c*d*f^2-25*a*d^2*e*f+23*b*c^2*f^2-58*b*c*d*e* f+33*b*d^2*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)/f*(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)))+(45*a*c^2*d*f ^3-50*a*c*d^2*e*f^2+15*a*d^3*e^2*f+15*b*c^3*f^3-56*b*c^2*d*e*f^2+54*b*c*d^ 2*e^2*f-15*b*d^3*e^3)/f/(-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))+(15*a*c^3*f^4-45*a*c^2*d*e*f^3+45*a*c*d^2*e^2*f^2-15*a*d^3*e^3 *f-15*b*c^3*e*f^3+45*b*c^2*d*e^2*f^2-45*b*c*d^2*e^3*f+15*b*d^3*e^4)/f*((d* f*x^2+c*f)/e/(c*f-d*e)*x/((x^2+e/f)*(d*f*x^2+c*f))^(1/2)+(1/e-c*f/e/(c*f-d *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))+d/(c*f- d*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))-Ellipt icE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2)))))*((d*x^2+c)*(f*x^2+e))^(1/2 )/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)
Time = 0.10 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (48 \, b d^{3} e^{4} f - 15 \, a c^{2} d e f^{4} - 8 \, {\left (11 \, b c d^{2} + 5 \, a d^{3}\right )} e^{3} f^{2} + {\left (38 \, b c^{2} d + 65 \, a c d^{2}\right )} e^{2} f^{3}\right )} x^{3} + {\left (48 \, b d^{3} e^{5} - 15 \, a c^{2} d e^{2} f^{3} - 8 \, {\left (11 \, b c d^{2} + 5 \, a d^{3}\right )} e^{4} f + {\left (38 \, b c^{2} d + 65 \, a c d^{2}\right )} e^{3} f^{2}\right )} x\right )} \sqrt {d f} \sqrt {-\frac {e}{f}} E(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left ({\left (48 \, b d^{3} e^{4} f - 8 \, {\left (11 \, b c d^{2} + 5 \, a d^{3}\right )} e^{3} f^{2} + {\left (38 \, b c^{2} d + {\left (65 \, a + 24 \, b\right )} c d^{2}\right )} e^{2} f^{3} - {\left ({\left (15 \, a + 41 \, b\right )} c^{2} d + 20 \, a c d^{2}\right )} e f^{4} + 15 \, {\left (b c^{3} + 2 \, a c^{2} d\right )} f^{5}\right )} x^{3} + {\left (48 \, b d^{3} e^{5} - 8 \, {\left (11 \, b c d^{2} + 5 \, a d^{3}\right )} e^{4} f + {\left (38 \, b c^{2} d + {\left (65 \, a + 24 \, b\right )} c d^{2}\right )} e^{3} f^{2} - {\left ({\left (15 \, a + 41 \, b\right )} c^{2} d + 20 \, a c d^{2}\right )} e^{2} f^{3} + 15 \, {\left (b c^{3} + 2 \, a c^{2} d\right )} e f^{4}\right )} x\right )} \sqrt {d f} \sqrt {-\frac {e}{f}} F(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left (3 \, b d^{3} e f^{4} x^{6} + 48 \, b d^{3} e^{4} f - 15 \, a c^{2} d e f^{4} - 8 \, {\left (11 \, b c d^{2} + 5 \, a d^{3}\right )} e^{3} f^{2} + {\left (38 \, b c^{2} d + 65 \, a c d^{2}\right )} e^{2} f^{3} - {\left (6 \, b d^{3} e^{2} f^{3} - {\left (11 \, b c d^{2} + 5 \, a d^{3}\right )} e f^{4}\right )} x^{4} + {\left (24 \, b d^{3} e^{3} f^{2} - {\left (47 \, b c d^{2} + 20 \, a d^{3}\right )} e^{2} f^{3} + {\left (23 \, b c^{2} d + 35 \, a c d^{2}\right )} e f^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{15 \, {\left (d e f^{6} x^{3} + d e^{2} f^{5} x\right )}} \]
-1/15*(((48*b*d^3*e^4*f - 15*a*c^2*d*e*f^4 - 8*(11*b*c*d^2 + 5*a*d^3)*e^3* f^2 + (38*b*c^2*d + 65*a*c*d^2)*e^2*f^3)*x^3 + (48*b*d^3*e^5 - 15*a*c^2*d* e^2*f^3 - 8*(11*b*c*d^2 + 5*a*d^3)*e^4*f + (38*b*c^2*d + 65*a*c*d^2)*e^3*f ^2)*x)*sqrt(d*f)*sqrt(-e/f)*elliptic_e(arcsin(sqrt(-e/f)/x), c*f/(d*e)) - ((48*b*d^3*e^4*f - 8*(11*b*c*d^2 + 5*a*d^3)*e^3*f^2 + (38*b*c^2*d + (65*a + 24*b)*c*d^2)*e^2*f^3 - ((15*a + 41*b)*c^2*d + 20*a*c*d^2)*e*f^4 + 15*(b* c^3 + 2*a*c^2*d)*f^5)*x^3 + (48*b*d^3*e^5 - 8*(11*b*c*d^2 + 5*a*d^3)*e^4*f + (38*b*c^2*d + (65*a + 24*b)*c*d^2)*e^3*f^2 - ((15*a + 41*b)*c^2*d + 20* a*c*d^2)*e^2*f^3 + 15*(b*c^3 + 2*a*c^2*d)*e*f^4)*x)*sqrt(d*f)*sqrt(-e/f)*e lliptic_f(arcsin(sqrt(-e/f)/x), c*f/(d*e)) - (3*b*d^3*e*f^4*x^6 + 48*b*d^3 *e^4*f - 15*a*c^2*d*e*f^4 - 8*(11*b*c*d^2 + 5*a*d^3)*e^3*f^2 + (38*b*c^2*d + 65*a*c*d^2)*e^2*f^3 - (6*b*d^3*e^2*f^3 - (11*b*c*d^2 + 5*a*d^3)*e*f^4)* x^4 + (24*b*d^3*e^3*f^2 - (47*b*c*d^2 + 20*a*d^3)*e^2*f^3 + (23*b*c^2*d + 35*a*c*d^2)*e*f^4)*x^2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e))/(d*e*f^6*x^3 + d* e^2*f^5*x)
\[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}}}{\left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}}{\left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}}{{\left (f\,x^2+e\right )}^{3/2}} \,d x \]