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

Optimal. Leaf size=539 \[ -\frac{d^2 \sqrt{e} \sqrt{c+d x^2} (2 a d f-3 b c f+b d e) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{b^3 c^{3/2} \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a \sqrt{d} e \sqrt{c+d x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{b^2 \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{\sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d \sqrt{f} \sqrt{c+d x^2} \left (2 b c^2 f-a d (c f+d e)\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d^2 x}{c \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (d e-c f)} \]

[Out]

-((d^2*x)/(c*(b*c - a*d)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) - (b^2*Sqrt[f]*Sqrt[c + d*x^2]*Elliptic
E[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/((b*c - a*d)^2*Sqrt[e]*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e
+ f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[f]*(2*b*c^2*f - a*d*(d*e + c*f))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt
[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[e]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))
]*Sqrt[e + f*x^2]) - (d^2*Sqrt[e]*(b*d*e - 3*b*c*f + 2*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqr
t[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[f]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e +
f*x^2]) + (b^3*c^(3/2)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*
e)])/(a*Sqrt[d]*(b*c - a*d)^2*e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))])

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Rubi [A]  time = 0.479277, antiderivative size = 539, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {546, 541, 539, 411, 527, 525, 418} \[ \frac{b^3 c^{3/2} \sqrt{e+f x^2} \Pi \left (1-\frac{b c}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{a \sqrt{d} e \sqrt{c+d x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{b^2 \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{\sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (b e-a f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d \sqrt{f} \sqrt{c+d x^2} \left (2 b c^2 f-a d (c f+d e)\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{e} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d^2 x}{c \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (d e-c f)}-\frac{d^2 \sqrt{e} \sqrt{c+d x^2} (2 a d f-3 b c f+b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{f} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^2*x)/(c*(b*c - a*d)*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) - (b^2*Sqrt[f]*Sqrt[c + d*x^2]*Elliptic
E[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/((b*c - a*d)^2*Sqrt[e]*(b*e - a*f)*Sqrt[(e*(c + d*x^2))/(c*(e
+ f*x^2))]*Sqrt[e + f*x^2]) - (d*Sqrt[f]*(2*b*c^2*f - a*d*(d*e + c*f))*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt
[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[e]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))
]*Sqrt[e + f*x^2]) - (d^2*Sqrt[e]*(b*d*e - 3*b*c*f + 2*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqr
t[e]], 1 - (d*e)/(c*f)])/(c*(b*c - a*d)^2*Sqrt[f]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e +
f*x^2]) + (b^3*c^(3/2)*Sqrt[e + f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*
e)])/(a*Sqrt[d]*(b*c - a*d)^2*e*(b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*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 541

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

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 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]

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]

Rubi steps

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

Mathematica [C]  time = 3.85599, size = 418, normalized size = 0.78 \[ \frac{i a c d e \left (\frac{d}{c}\right )^{3/2} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) (c f-d e) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )+i b^2 c e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (d e-c f)^2 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+\frac{a d x \left (b \left (c^2 d f^3 x^2+c^3 f^3+d^3 e^2 \left (e+f x^2\right )\right )-a d f \left (c^2 f^2+c d f^2 x^2+d^2 e \left (e+f x^2\right )\right )\right )}{c}+i a d e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (b \left (c^2 f^2+d^2 e^2\right )-a d f (c f+d e)\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{a d e \sqrt{c+d x^2} \sqrt{e+f x^2} (a d-b c) (b e-a f) (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*d*x*(-(a*d*f*(c^2*f^2 + c*d*f^2*x^2 + d^2*e*(e + f*x^2))) + b*(c^3*f^3 + c^2*d*f^3*x^2 + d^3*e^2*(e + f*x^
2))))/c + I*a*d*Sqrt[d/c]*e*(-(a*d*f*(d*e + c*f)) + b*(d^2*e^2 + 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*a*c*d*(d/c)^(3/2)*e*(b*e - a*f)*(-(d*e) + c*f)*Sqrt[1
+ (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*b^2*c*Sqrt[d/c]*e*(d*e - c
*f)^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)])/(a
*d*(-(b*c) + a*d)*e*(b*e - a*f)*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]  time = 0.035, size = 956, normalized size = 1.8 \begin{align*} \text{result 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)^(3/2)/(f*x^2+e)^(3/2),x)

[Out]

(x^3*a^2*c*d^2*f^3*(-d/c)^(1/2)+x^3*a^2*d^3*e*f^2*(-d/c)^(1/2)-x^3*a*b*c^2*d*f^3*(-d/c)^(1/2)-x^3*a*b*d^3*e^2*
f*(-d/c)^(1/2)-EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*c*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/
2)+EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*d^3*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+EllipticF(x
*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c*d^2*e^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-EllipticF(x*(-d/c)^(1/2
),(c*f/d/e)^(1/2))*a*b*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2
))*a^2*c*d^2*e*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*d^3*e
^2*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c^2*d*e*f^2*((d*x^2
+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*d^3*e^3*((d*x^2+c)/c)^(1/2)*((f
*x^2+e)/e)^(1/2)+EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^2*c^3*e*f^2*((d*x^2+c)/c)^(1/2
)*((f*x^2+e)/e)^(1/2)-2*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^2*c^2*d*e^2*f*((d*x^2+c
)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^2*c*d^2*e^3*((d*
x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+x*a^2*c^2*d*f^3*(-d/c)^(1/2)+x*a^2*d^3*e^2*f*(-d/c)^(1/2)-x*a*b*c^3*f^3*(-
d/c)^(1/2)-x*a*b*d^3*e^3*(-d/c)^(1/2))*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/e/a/c/(a*f-b*e)/(-d/c)^(1/2)/(a*d-b*c)/
(c*f-d*e)^2/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)

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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{3}{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)^(3/2)/(f*x^2+e)^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}} \left (e + f x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{3}{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)^(3/2)/(f*x^2+e)^(3/2),x, algorithm="giac")

[Out]

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