3.88 \(\int \frac {1}{(a+b x^2) (c+d x^2)^{5/2} (e+f x^2)^{3/2}} \, dx\)

Optimal. Leaf size=814 \[ \frac {e^{3/2} \sqrt {d x^2+c} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) b^4}{a c (b c-a d)^2 \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {f^{3/2} \sqrt {d x^2+c} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) b^2}{(b c-a d)^2 \sqrt {e} (b e-a f) (d e-c f) \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {\sqrt {e} \sqrt {f} (2 b d e-b c f-a d f) \sqrt {d x^2+c} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) b^2}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {d \sqrt {f} \left (b c \left (5 d^2 e^2-7 c d f e-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d f e-3 c^2 f^2\right )\right ) \sqrt {d x^2+c} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 \sqrt {e} (d e-c f)^3 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {d^2 \sqrt {e} \sqrt {f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt {d x^2+c} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {d x^2+c} \sqrt {f x^2+e}}-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (d x^2+c\right )^{3/2} \sqrt {f x^2+e}} \]

[Out]

-1/3*d^2*x/c/(-a*d+b*c)/(-c*f+d*e)/(d*x^2+c)^(3/2)/(f*x^2+e)^(1/2)-1/3*d^2*(b*c*(-9*c*f+5*d*e)-2*a*d*(-3*c*f+d
*e))*x/c^2/(-a*d+b*c)^2/(-c*f+d*e)^2/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)+b^2*f^(3/2)*(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)/(-a*d+b*c)^2/(-a*f
+b*e)/(-c*f+d*e)/e^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+b^4*e^(3/2)*(1/(1+f*x^2/e))^(1/2)*(1+
f*x^2/e)^(1/2)*EllipticPi(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),1-b*e/a/f,(1-d*e/c/f)^(1/2))*(d*x^2+c)^(1/2)/a/c
/(-a*d+b*c)^2/(-a*f+b*e)^2/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-1/3*d*(b*c*(-6*c^2*f^2-7*c*
d*e*f+5*d^2*e^2)-a*d*(-3*c^2*f^2-7*c*d*e*f+2*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))*f^(1/2)*(d*x^2+c)^(1/2)/c^2/(-a*d+b*c)^2/(-c*f+d*e)^3/e^(1/2
)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-b^2*(-a*d*f-b*c*f+2*b*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))*e^(1/2)*f^(1/2)*(d*x^2+c)^(1/2)/c/(-a*
d+b*c)^2/(-a*f+b*e)^2/(-c*f+d*e)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/3*d^2*(b*c*(-15*c*f+7*d*e)-
a*d*(-9*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))*e^(1/2)*f^(1/2)*(d*x^2+c)^(1/2)/c^2/(-a*d+b*c)^2/(-c*f+d*e)^3/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*
x^2+e)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.96, antiderivative size = 814, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {546, 539, 525, 418, 411, 527} \[ \frac {e^{3/2} \sqrt {d x^2+c} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) b^4}{a c (b c-a d)^2 \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {f^{3/2} \sqrt {d x^2+c} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) b^2}{(b c-a d)^2 \sqrt {e} (b e-a f) (d e-c f) \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {\sqrt {e} \sqrt {f} (2 b d e-b c f-a d f) \sqrt {d x^2+c} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) b^2}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {d \sqrt {f} \left (b c \left (5 d^2 e^2-7 c d f e-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d f e-3 c^2 f^2\right )\right ) \sqrt {d x^2+c} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 \sqrt {e} (d e-c f)^3 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {d^2 \sqrt {e} \sqrt {f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt {d x^2+c} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {d x^2+c} \sqrt {f x^2+e}}-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (d x^2+c\right )^{3/2} \sqrt {f x^2+e}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^2)*(c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]

[Out]

-(d^2*x)/(3*c*(b*c - a*d)*(d*e - c*f)*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2]) - (d^2*(b*c*(5*d*e - 9*c*f) - 2*a*d*(
d*e - 3*c*f))*x)/(3*c^2*(b*c - a*d)^2*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]) + (b^2*f^(3/2)*Sqrt[c + d
*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/((b*c - a*d)^2*Sqrt[e]*(b*e - a*f)*(d*e - c*f)*
Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[f]*(b*c*(5*d^2*e^2 - 7*c*d*e*f - 6*c^2*f^2) -
a*d*(2*d^2*e^2 - 7*c*d*e*f - 3*c^2*f^2))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*
f)])/(3*c^2*(b*c - a*d)^2*Sqrt[e]*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) - (b^2*
Sqrt[e]*Sqrt[f]*(2*b*d*e - b*c*f - a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*
f)])/(c*(b*c - a*d)^2*(b*e - a*f)^2*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d^2*
Sqrt[e]*Sqrt[f]*(b*c*(7*d*e - 15*c*f) - a*d*(d*e - 9*c*f))*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e
]], 1 - (d*e)/(c*f)])/(3*c^2*(b*c - a*d)^2*(d*e - c*f)^3*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]
) + (b^4*e^(3/2)*Sqrt[c + d*x^2]*EllipticPi[1 - (b*e)/(a*f), ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(a
*c*(b*c - a*d)^2*Sqrt[f]*(b*e - a*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 525

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(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]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 539

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(c*Sqrt[e +
f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[Rt[d/c, 2]*x], 1 - (c*f)/(d*e)])/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sq
rt[(c*(e + f*x^2))/(e*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rule 546

Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Dist[b^2/(b*c
- a*d)^2, Int[((c + d*x^2)^(q + 2)*(e + f*x^2)^r)/(a + b*x^2), x], x] - Dist[d/(b*c - a*d)^2, Int[(c + d*x^2)^
q*(e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r}, x] && LtQ[q, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx &=\frac {b^2 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx}{(b c-a d)^2}-\frac {d \int \frac {2 b c-a d+b d x^2}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx}{(b c-a d)^2}\\ &=-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}+\frac {b^4 \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{(b c-a d)^2 (b e-a f)^2}-\frac {\left (b^2 f\right ) \int \frac {2 b e-a f+b f x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx}{(b c-a d)^2 (b e-a f)^2}+\frac {d \int \frac {-b c (5 d e-6 c f)+a d (2 d e-3 c f)-3 d (b c-a d) f x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx}{3 c (b c-a d)^2 (d e-c f)}\\ &=-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}-\frac {d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {b^4 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^2 \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d \int \frac {-c f (2 b c (d e-3 c f)+a d (d e+3 c f))+d f (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx}{3 c^2 (b c-a d)^2 (d e-c f)^2}+\frac {\left (b^2 f^2\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{(b c-a d)^2 (b e-a f) (d e-c f)}-\frac {\left (b^2 f (2 b d e-b c f-a d f)\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{(b c-a d)^2 (b e-a f)^2 (d e-c f)}\\ &=-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}-\frac {d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {b^2 f^{3/2} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{(b c-a d)^2 \sqrt {e} (b e-a f) (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {b^2 \sqrt {e} \sqrt {f} (2 b d e-b c f-a d f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^4 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^2 \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\left (d^2 f (b c (7 d e-15 c f)-a d (d e-9 c f))\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 c (b c-a d)^2 (d e-c f)^3}-\frac {\left (d f \left (b c \left (5 d^2 e^2-7 c d e f-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 c^2 (b c-a d)^2 (d e-c f)^3}\\ &=-\frac {d^2 x}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}-\frac {d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {b^2 f^{3/2} \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{(b c-a d)^2 \sqrt {e} (b e-a f) (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {d \sqrt {f} \left (b c \left (5 d^2 e^2-7 c d e f-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 \sqrt {e} (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {b^2 \sqrt {e} \sqrt {f} (2 b d e-b c f-a d f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d^2 \sqrt {e} \sqrt {f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^4 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a c (b c-a d)^2 \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 7.27, size = 1645, normalized size = 2.02 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^2)*(c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]

[Out]

((-I)*a*d*e*(2*a*b*d*(d*e - 3*c*f)*(d*e + c*f)^2 + a^2*d^2*f*(-2*d^2*e^2 + 7*c*d*e*f + 3*c^2*f^2) + b^2*c*(-5*
d^3*e^3 + 10*c*d^2*e^2*f + 3*c^3*f^3))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh
[Sqrt[d/c]*x], (c*f)/(d*e)] + (Sqrt[d/c]*(6*a*b^2*c^2*d^5*e^4*x - 3*a^2*b*c*d^6*e^4*x - 11*a*b^2*c^3*d^4*e^3*f
*x + 2*a^2*b*c^2*d^5*e^3*f*x + 3*a^3*c*d^6*e^3*f*x + 11*a^2*b*c^3*d^4*e^2*f^2*x - 8*a^3*c^2*d^5*e^2*f^2*x - 3*
a*b^2*c^6*d*f^4*x + 6*a^2*b*c^5*d^2*f^4*x - 3*a^3*c^4*d^3*f^4*x + 5*a*b^2*c*d^6*e^4*x^3 - 2*a^2*b*d^7*e^4*x^3
- 4*a*b^2*c^2*d^5*e^3*f*x^3 - a^2*b*c*d^6*e^3*f*x^3 + 2*a^3*d^7*e^3*f*x^3 - 11*a*b^2*c^3*d^4*e^2*f^2*x^3 + 12*
a^2*b*c^2*d^5*e^2*f^2*x^3 - 4*a^3*c*d^6*e^2*f^2*x^3 + 11*a^2*b*c^3*d^4*e*f^3*x^3 - 8*a^3*c^2*d^5*e*f^3*x^3 - 6
*a*b^2*c^5*d^2*f^4*x^3 + 12*a^2*b*c^4*d^3*f^4*x^3 - 6*a^3*c^3*d^4*f^4*x^3 + 5*a*b^2*c*d^6*e^3*f*x^5 - 2*a^2*b*
d^7*e^3*f*x^5 - 10*a*b^2*c^2*d^5*e^2*f^2*x^5 + 2*a^2*b*c*d^6*e^2*f^2*x^5 + 2*a^3*d^7*e^2*f^2*x^5 + 10*a^2*b*c^
2*d^5*e*f^3*x^5 - 7*a^3*c*d^6*e*f^3*x^5 - 3*a*b^2*c^4*d^3*f^4*x^5 + 6*a^2*b*c^3*d^4*f^4*x^5 - 3*a^3*c^2*d^5*f^
4*x^5 - I*a*c*d^2*Sqrt[d/c]*e*(b*e - a*f)*(-(d*e) + c*f)*(2*a*d*(d*e - 3*c*f) + b*c*(-5*d*e + 9*c*f))*(c + d*x
^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (3*I)*b^3*c^4*d^3
*Sqrt[d/c]*e^4*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(
d*e)] - (9*I)*b^3*c^7*(d/c)^(5/2)*e^3*f*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcS
inh[Sqrt[d/c]*x], (c*f)/(d*e)] + (9*I)*b^3*c^7*(d/c)^(3/2)*e^2*f^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Ell
ipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b^3*c^7*Sqrt[d/c]*e*f^3*Sqrt[1 + (d*x^2)/c]*
Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + (3*I)*b^3*c^3*d^4*Sqrt[d/c]
*e^4*x^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]
- (9*I)*b^3*c^4*d^3*Sqrt[d/c]*e^3*f*x^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcS
inh[Sqrt[d/c]*x], (c*f)/(d*e)] + (9*I)*b^3*c^7*(d/c)^(5/2)*e^2*f^2*x^2*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]
*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - (3*I)*b^3*c^7*(d/c)^(3/2)*e*f^3*x^2*Sqrt[1 + (
d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/d)/(3*a*c^2*Sqrt[
d/c]*(b*c - a*d)^2*e*(b*e - a*f)*(-(d*e) + c*f)^3*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)), x)

________________________________________________________________________________________

maple [B]  time = 0.07, size = 4115, normalized size = 5.06 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x)

[Out]

-1/3*(-6*x^3*a^3*c^3*d^3*f^4*(-1/c*d)^(1/2)+2*x^3*a^3*d^6*e^3*f*(-1/c*d)^(1/2)-2*x^3*a^2*b*d^6*e^4*(-1/c*d)^(1
/2)-3*x*a^3*c^4*d^2*f^4*(-1/c*d)^(1/2)-3*x*a*b^2*c^6*f^4*(-1/c*d)^(1/2)-3*x^5*a^3*c^2*d^4*f^4*(-1/c*d)^(1/2)+2
*x^5*a^3*d^6*e^2*f^2*(-1/c*d)^(1/2)+2*x^5*a^2*b*c*d^5*e^2*f^2*(-1/c*d)^(1/2)+5*x^5*a*b^2*c*d^5*e^3*f*(-1/c*d)^
(1/2)+11*x^3*a^2*b*c^3*d^3*e*f^3*(-1/c*d)^(1/2)+12*x^3*a^2*b*c^2*d^4*e^2*f^2*(-1/c*d)^(1/2)-x^3*a^2*b*c*d^5*e^
3*f*(-1/c*d)^(1/2)-11*x^3*a*b^2*c^3*d^3*e^2*f^2*(-1/c*d)^(1/2)-4*x^3*a*b^2*c^2*d^4*e^3*f*(-1/c*d)^(1/2)+11*x*a
^2*b*c^3*d^3*e^2*f^2*(-1/c*d)^(1/2)+2*x*a^2*b*c^2*d^4*e^3*f*(-1/c*d)^(1/2)+10*x^5*a^2*b*c^2*d^4*e*f^3*(-1/c*d)
^(1/2)-11*x*a*b^2*c^3*d^3*e^3*f*(-1/c*d)^(1/2)+2*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^3*d^6*e^3*f
*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^2*b*d^6*e^4*((d*x
^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^3*d^6*e^3*f*((d*x^2+c)/
c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^2*b*d^6*e^4*((d*x^2+c)/c)^(1/
2)*((f*x^2+e)/e)^(1/2)-3*EllipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*x^2*b^3*c^2*d^4*
e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+6*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^3*c^3*d^3*e*f^3*((
d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^3*c^2*d^4*e^2*f^2*((d*x^
2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^3*c*d^5*e^3*f*((d*x^2+c)/c)^
(1/2)*((f*x^2+e)/e)^(1/2)-2*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^2*b*c*d^5*e^4*((d*x^2+c)/c)^(1/2)*((
f*x^2+e)/e)^(1/2)+5*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a*b^2*c^2*d^4*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+
e)/e)^(1/2)+3*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^3*c^3*d^3*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^
(1/2)+7*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^3*c^2*d^4*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2
)-2*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^3*c*d^5*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*Elli
pticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^2*b*c*d^5*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-5*EllipticE((-
1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a*b^2*c^2*d^4*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-9*EllipticPi((-1/c*d
)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*b^3*c^5*d*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+9
*EllipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*b^3*c^4*d^2*e^3*f*((d*x^2+c)/c)^(1/2)*((
f*x^2+e)/e)^(1/2)-3*x^5*a*b^2*c^4*d^2*f^4*(-1/c*d)^(1/2)-8*x^3*a^3*c^2*d^4*e*f^3*(-1/c*d)^(1/2)-4*x^3*a^3*c*d^
5*e^2*f^2*(-1/c*d)^(1/2)+12*x^3*a^2*b*c^4*d^2*f^4*(-1/c*d)^(1/2)-6*x^3*a*b^2*c^5*d*f^4*(-1/c*d)^(1/2)+5*x^3*a*
b^2*c*d^5*e^4*(-1/c*d)^(1/2)-8*x*a^3*c^2*d^4*e^2*f^2*(-1/c*d)^(1/2)+3*x*a^3*c*d^5*e^3*f*(-1/c*d)^(1/2)+6*x*a^2
*b*c^5*d*f^4*(-1/c*d)^(1/2)-3*x*a^2*b*c*d^5*e^4*(-1/c*d)^(1/2)+6*x*a*b^2*c^2*d^4*e^4*(-1/c*d)^(1/2)+3*Elliptic
Pi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*b^3*c^6*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(
1/2)-3*EllipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*b^3*c^3*d^3*e^4*((d*x^2+c)/c)^(1/2
)*((f*x^2+e)/e)^(1/2)-7*x^5*a^3*c*d^5*e*f^3*(-1/c*d)^(1/2)+6*x^5*a^2*b*c^3*d^3*f^4*(-1/c*d)^(1/2)-2*x^5*a^2*b*
d^6*e^3*f*(-1/c*d)^(1/2)-10*x^5*a*b^2*c^2*d^4*e^2*f^2*(-1/c*d)^(1/2)-5*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1
/2))*x^2*a*b^2*c*d^5*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e
*f)^(1/2)/(-1/c*d)^(1/2))*x^2*b^3*c^5*d*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-9*EllipticPi((-1/c*d)^(1
/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*x^2*b^3*c^4*d^2*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)
+9*EllipticPi((-1/c*d)^(1/2)*x,1/a*b*c/d,(-1/e*f)^(1/2)/(-1/c*d)^(1/2))*x^2*b^3*c^3*d^3*e^3*f*((d*x^2+c)/c)^(1
/2)*((f*x^2+e)/e)^(1/2)-9*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^2*b*c^4*d^2*e*f^3*((d*x^2+c)/c)^(1/2)*
((f*x^2+e)/e)^(1/2)+8*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^2*b*c^3*d^3*e^2*f^2*((d*x^2+c)/c)^(1/2)*((
f*x^2+e)/e)^(1/2)+3*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^2*b*c^2*d^4*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^
2+e)/e)^(1/2)+9*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a*b^2*c^4*d^2*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+
e)/e)^(1/2)-14*EllipticF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a*b^2*c^3*d^3*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/
e)^(1/2)-6*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^2*b*c^4*d^2*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(
1/2)-10*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^2*b*c^3*d^3*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1
/2)-2*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a^2*b*c^2*d^4*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+
3*EllipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a*b^2*c^5*d*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+10*Ell
ipticE((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*a*b^2*c^3*d^3*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+6*Ellipti
cF((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^3*c^2*d^4*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-8*EllipticF
((-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^3*c*d^5*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+5*EllipticF((
-1/c*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a*b^2*c*d^5*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticE((-1/c
*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^3*c^2*d^4*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+7*EllipticE((-1/c*d
)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^3*c*d^5*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-14*EllipticF((-1/c*d)
^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a*b^2*c^2*d^4*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-6*EllipticE((-1/c*d)
^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^2*b*c^3*d^3*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-10*EllipticE((-1/c*d
)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^2*b*c^2*d^4*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-2*EllipticE((-1/c
*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^2*b*c*d^5*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticE((-1/c*d
)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a*b^2*c^4*d^2*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+10*EllipticE((-1/c*
d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a*b^2*c^2*d^4*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-9*EllipticF((-1/c*
d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^2*b*c^3*d^3*e*f^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*EllipticF((-1/c*
d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^2*b*c^2*d^4*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*EllipticF((-1/
c*d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a^2*b*c*d^5*e^3*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+9*EllipticF((-1/c*
d)^(1/2)*x,(c/d/e*f)^(1/2))*x^2*a*b^2*c^3*d^3*e^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2))/(f*x^2+e)^(1/2)
/(a*d-b*c)^2/(a*f-b*e)/(-1/c*d)^(1/2)/a/(c*f-d*e)^3/e/c^2/(d*x^2+c)^(3/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^2)*(c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x)

[Out]

int(1/((a + b*x^2)*(c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+a)/(d*x**2+c)**(5/2)/(f*x**2+e)**(3/2),x)

[Out]

Integral(1/((a + b*x**2)*(c + d*x**2)**(5/2)*(e + f*x**2)**(3/2)), x)

________________________________________________________________________________________