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

Optimal. Leaf size=435 \[ -\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} (a d (d e-3 c f)-2 b c (2 d e-3 c f)) \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{b^2 \sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)^2}-\frac{d^{3/2} \sqrt{e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{d^2 x \sqrt{e+f x^2}}{3 c \left (c+d x^2\right )^{3/2} (b c-a d) (d e-c f)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.554213, antiderivative size = 435, 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, 538, 537, 527, 525, 418, 411} \[ \frac{b^2 \sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d)^2}-\frac{d^{3/2} \sqrt{e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{c f}{d e}\right )}{3 c^{3/2} \sqrt{c+d x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} (a d (d e-3 c f)-2 b c (2 d e-3 c f)) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{3 c^2 \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{d^2 x \sqrt{e+f x^2}}{3 c \left (c+d x^2\right )^{3/2} (b c-a d) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^2*x*Sqrt[e + f*x^2])/(3*c*(b*c - a*d)*(d*e - c*f)*(c + d*x^2)^(3/2)) - (d^(3/2)*(b*c*(5*d*e - 7*c*f) - 2*a
*d*(d*e - 2*c*f))*Sqrt[e + f*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(3*c^(3/2)*(b*c - a
*d)^2*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) - (d*Sqrt[e]*Sqrt[f]*(a*d*(d*e - 3*
c*f) - 2*b*c*(2*d*e - 3*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)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (b^2*Sqrt[-c]*Sqrt[1 + (d
*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), ArcSin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(a*Sqrt[d]*(b
*c - a*d)^2*Sqrt[c + d*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 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(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]

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]

Rubi steps

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

Mathematica [C]  time = 3.42181, size = 433, normalized size = 1. \[ \frac{i a d \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (c f-d e) (a d (2 d e-3 c f)+b c (6 c f-5 d e)) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )-3 i b^2 c^2 \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (d e-c f)^2 \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+a c d x \left (\frac{d}{c}\right )^{3/2} \left (e+f x^2\right ) \left (a d \left (-5 c^2 f+c d \left (3 e-4 f x^2\right )+2 d^2 e x^2\right )+b c \left (8 c^2 f-6 c d e+7 c d f x^2-5 d^2 e x^2\right )\right )+i a d^2 e \left (c+d x^2\right ) \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 a d (d e-2 c f)+b c (7 c f-5 d e)) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{3 a c^2 \sqrt{\frac{d}{c}} \left (c+d x^2\right )^{3/2} \sqrt{e+f x^2} (b c-a d)^2 (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c*d*(d/c)^(3/2)*x*(e + f*x^2)*(b*c*(-6*c*d*e + 8*c^2*f - 5*d^2*e*x^2 + 7*c*d*f*x^2) + a*d*(-5*c^2*f + 2*d^2
*e*x^2 + c*d*(3*e - 4*f*x^2))) + I*a*d^2*e*(2*a*d*(d*e - 2*c*f) + b*c*(-5*d*e + 7*c*f))*(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)] + I*a*d*(-(d*e) + c*f)*(a*d*(2*d*
e - 3*c*f) + b*c*(-5*d*e + 6*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqr
t[d/c]*x], (c*f)/(d*e)] - (3*I)*b^2*c^2*(d*e - c*f)^2*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Elli
pticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(3*a*c^2*Sqrt[d/c]*(b*c - a*d)^2*(d*e - c*f)^2*(c +
d*x^2)^(3/2)*Sqrt[e + f*x^2])

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Maple [B]  time = 0.043, size = 2062, normalized size = 4.7 \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)^(5/2)/(f*x^2+e)^(1/2),x)

[Out]

1/3*(-4*x^5*a^2*c*d^4*f^2*(-d/c)^(1/2)+2*x^3*a^2*d^5*e^2*(-d/c)^(1/2)-2*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/
2))*a^2*c*d^4*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+7*x^5*a*b*c^2*d^3*f^2*(-d/c)^(1/2)-x^3*a^2*c*d^4*e*f
*(-d/c)^(1/2)+2*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*d^5*e^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*d^5*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+3*Elli
pticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*b^2*c^3*d^2*e^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*c^3*d^2*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*Ellip
ticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*c*d^4*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+8*x^3*a*b*c^3*d^2*f
^2*(-d/c)^(1/2)-5*x^3*a*b*c*d^4*e^2*(-d/c)^(1/2)-6*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b*c^3*d^2*f
^2*((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))*x^2*a*b*c*d^4*e^2*((d*x
^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+4*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*c*d^4*e*f*((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*c*d^4*e^2*((d*x^2+c)/c)^(1/2)*(
(f*x^2+e)/e)^(1/2)+11*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*b*c^3*d^2*e*f*((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*b*c^3*d^2*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+
8*x*a*b*c^3*d^2*e*f*(-d/c)^(1/2)+2*x^5*a^2*d^5*e*f*(-d/c)^(1/2)-5*x^3*a^2*c^2*d^3*f^2*(-d/c)^(1/2)+3*x*a^2*c*d
^4*e^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^2*c^5*f^2*((d*x^2+c)/c)^(
1/2)*((f*x^2+e)/e)^(1/2)-5*x*a^2*c^2*d^3*e*f*(-d/c)^(1/2)-6*x*a*b*c^2*d^3*e^2*(-d/c)^(1/2)-6*EllipticPi(x*(-d/
c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*x^2*b^2*c^3*d^2*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-5*Elli
pticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*c*d^4*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+11*EllipticF(x
*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a*b*c^2*d^3*e*f*((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*b*c^2*d^3*e*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^2*c^4*d*f^2*((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^2*c^2*d^3*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-5
*x^5*a*b*c*d^4*e*f*(-d/c)^(1/2)+x^3*a*b*c^2*d^3*e*f*(-d/c)^(1/2)+3*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x
^2*a^2*c^2*d^3*f^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-6*EllipticPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-
d/c)^(1/2))*b^2*c^4*d*e*f*((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^2*c^2*d^3*e*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*b*c^4*d*
f^2*((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*c^2*d^3*e^2*((d*x^
2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+4*EllipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a^2*c^2*d^3*e*f*((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*c^2*d^3*e^2*((d*x^2+c)/c)^(1/2)*((f*x^
2+e)/e)^(1/2))/(f*x^2+e)^(1/2)/(c*f-d*e)^2/(a*d-b*c)^2/(-d/c)^(1/2)/c^2/a/(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}} \sqrt{f x^{2} + e}}\,{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)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*sqrt(f*x^2 + e)), 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)^(1/2),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{5}{2}} \sqrt{e + f x^{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)**(5/2)/(f*x**2+e)**(1/2),x)

[Out]

Integral(1/((a + b*x**2)*(c + d*x**2)**(5/2)*sqrt(e + f*x**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{5}{2}} \sqrt{f x^{2} + e}}\,{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)^(1/2),x, algorithm="giac")

[Out]

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