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 \[ \frac{e^{3/2} \sqrt{d x^2+c} \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) b^4}{a c (b c-a d)^2 \sqrt{f} (b e-a f)^2 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{f^{3/2} \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right ) b^2}{(b c-a d)^2 \sqrt{e} (b e-a f) (d e-c f) \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{\sqrt{e} \sqrt{f} (2 b d e-b c f-a d f) \sqrt{d x^2+c} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right ) b^2}{c (b c-a d)^2 (b e-a f)^2 (d e-c f) \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{d \sqrt{f} \left (b c \left (5 d^2 e^2-7 c d f e-6 c^2 f^2\right )-a d \left (2 d^2 e^2-7 c d f e-3 c^2 f^2\right )\right ) \sqrt{d x^2+c} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 (b c-a d)^2 \sqrt{e} (d e-c f)^3 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}+\frac{d^2 \sqrt{e} \sqrt{f} (b c (7 d e-15 c f)-a d (d e-9 c f)) \sqrt{d x^2+c} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{3 c^2 (b c-a d)^2 (d e-c f)^3 \sqrt{\frac{e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt{f x^2+e}}-\frac{d^2 (b c (5 d e-9 c f)-2 a d (d e-3 c f)) x}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt{d x^2+c} \sqrt{f x^2+e}}-\frac{d^2 x}{3 c (b c-a d) (d e-c f) \left (d x^2+c\right )^{3/2} \sqrt{f x^2+e}} \]

[Out]

-(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])

________________________________________________________________________________________

Rubi [A]  time = 0.957996, antiderivative size = 814, normalized size of antiderivative = 1., 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 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]

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 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 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 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 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]

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 = 5.94163, 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])

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Maple [B]  time = 0.058, size = 4115, normalized size = 5.1 \begin{align*} \text{output too large to display} \end{align*}

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*(-8*x^3*a^3*c^2*d^4*e*f^3*(-d/c)^(1/2)-6*x^3*a^3*c^3*d^3*f^4*(-d/c)^(1/2)-9*EllipticF(x*(-d/c)^(1/2),(c*f
/d/e)^(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)+11*x^3*a^2*b*c^3*d^3*e*f^3*(-d/c)^(1/
2)-14*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(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)+3*
EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(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*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(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)-14*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(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)+10*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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)-6*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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)+
3*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(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*El
lipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(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(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(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)-6*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(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)+6*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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*E
llipticF(x*(-d/c)^(1/2),(c*f/d/e)^(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)+8*Ellipti
cF(x*(-d/c)^(1/2),(c*f/d/e)^(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(x
*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c
)^(1/2),(c*f/d/e)^(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)-2*EllipticE(x*(-d/c)^(1
/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c
*f/d/e)^(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*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)
^(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)+2*x^3*a^3*d^6*e^3*f*(-d/c)^(1/2)-2*x^3*a^2
*b*d^6*e^4*(-d/c)^(1/2)-3*x*a^3*c^4*d^2*f^4*(-d/c)^(1/2)-3*x*a*b^2*c^6*f^4*(-d/c)^(1/2)-3*x^5*a^3*c^2*d^4*f^4*
(-d/c)^(1/2)+2*x^5*a^3*d^6*e^2*f^2*(-d/c)^(1/2)-6*x^3*a*b^2*c^5*d*f^4*(-d/c)^(1/2)+5*x^3*a*b^2*c*d^5*e^4*(-d/c
)^(1/2)-8*x*a^3*c^2*d^4*e^2*f^2*(-d/c)^(1/2)+3*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(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(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1
/2))*b^3*c^3*d^3*e^4*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*x*a^3*c*d^5*e^3*f*(-d/c)^(1/2)+6*x*a^2*b*c^5*d*
f^4*(-d/c)^(1/2)-3*x*a^2*b*c*d^5*e^4*(-d/c)^(1/2)+6*x*a*b^2*c^2*d^4*e^4*(-d/c)^(1/2)-8*EllipticF(x*(-d/c)^(1/2
),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f
/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(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)+11*x*a^2*b*c^3*d^3*e^2*f^2*(-d/c)^(1/2)+2*x*a^2*
b*c^2*d^4*e^3*f*(-d/c)^(1/2)-11*x*a*b^2*c^3*d^3*e^3*f*(-d/c)^(1/2)-2*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(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)+3*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(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(x*(-d/c)^(1/2),(c*f/d/e)^(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)+12*x^3*a^2*b*c^2*d^4*e^2*f^2*(-d/c)^(1/2)-x^3*a^2*b*c*d^5*e^3*f*(-d/c)^(1/2)-11*x^3*a*b^2*c^3*d^3*e^2*f^
2*(-d/c)^(1/2)-4*x^3*a*b^2*c^2*d^4*e^3*f*(-d/c)^(1/2)+10*x^5*a^2*b*c^2*d^4*e*f^3*(-d/c)^(1/2)+2*x^5*a^2*b*c*d^
5*e^2*f^2*(-d/c)^(1/2)-10*x^5*a*b^2*c^2*d^4*e^2*f^2*(-d/c)^(1/2)+5*x^5*a*b^2*c*d^5*e^3*f*(-d/c)^(1/2)+9*Ellipt
icPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(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)+6*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(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)-4*x^3
*a^3*c*d^5*e^2*f^2*(-d/c)^(1/2)+12*x^3*a^2*b*c^4*d^2*f^4*(-d/c)^(1/2)-7*x^5*a^3*c*d^5*e*f^3*(-d/c)^(1/2)+6*x^5
*a^2*b*c^3*d^3*f^4*(-d/c)^(1/2)-2*x^5*a^2*b*d^6*e^3*f*(-d/c)^(1/2)-3*x^5*a*b^2*c^4*d^2*f^4*(-d/c)^(1/2))/(f*x^
2+e)^(1/2)/(c*f-d*e)^3/(a*d-b*c)^2/(-d/c)^(1/2)/(a*f-b*e)/c^2/a/e/(d*x^2+c)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)