3.102 \(\int \frac{1}{\left (a+b x^2\right )^2 \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx\)

Optimal. Leaf size=485 \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (3 a^2 d f-2 a b (c f+d e)+b^2 c e\right ) \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{2 a^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (b e-a f)}+\frac{b^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}-\frac{b f x \sqrt{c+d x^2}}{2 a \sqrt{e+f x^2} (b c-a d) (b e-a f)}-\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 c \sqrt{e+f x^2} (b c-a d) (b e-a f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{b \sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 a \sqrt{e+f x^2} (b c-a d) (b e-a f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

[Out]

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

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Rubi [A]  time = 1.22597, antiderivative size = 485, 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 \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (3 a^2 d f-2 a b (c f+d e)+b^2 c e\right ) \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{2 a^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2} (b c-a d) (b e-a f)}+\frac{b^2 x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (b c-a d) (b e-a f)}-\frac{b f x \sqrt{c+d x^2}}{2 a \sqrt{e+f x^2} (b c-a d) (b e-a f)}-\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 c \sqrt{e+f x^2} (b c-a d) (b e-a f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{b \sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 a \sqrt{e+f x^2} (b c-a d) (b e-a f) \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 160.607, size = 527, normalized size = 1.09 \[ - \frac{\sqrt{c} \sqrt{d} f \sqrt{e + f x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{2 e \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a d - b c\right ) \left (a f - b e\right )} + \frac{b^{2} x \sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}{2 a \left (a + b x^{2}\right ) \left (a d - b c\right ) \left (a f - b e\right )} + \frac{b \sqrt{c} \sqrt{d} \sqrt{e + f x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{2 a \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a d - b c\right ) \left (a f - b e\right )} - \frac{b d x \sqrt{e + f x^{2}}}{2 a \sqrt{c + d x^{2}} \left (a d - b c\right ) \left (a f - b e\right )} - \frac{\sqrt{c} f \sqrt{e + f x^{2}} \left (- 3 a^{2} d f + 2 a b \left (c f + d e\right ) - b^{2} c e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{2 a \sqrt{d} e \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a d - b c\right ) \left (a f - b e\right )^{2}} + \frac{b e^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (- 3 a^{2} d f + 2 a b \left (c f + d e\right ) - b^{2} c e\right ) \Pi \left (1 - \frac{b e}{a f}; \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{2 a^{2} c \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}} \left (a d - b c\right ) \left (a f - b e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**2/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)

[Out]

-sqrt(c)*sqrt(d)*f*sqrt(e + f*x**2)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), -c*f/(d*
e) + 1)/(2*e*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)*(a*d - b*c)*
(a*f - b*e)) + b**2*x*sqrt(c + d*x**2)*sqrt(e + f*x**2)/(2*a*(a + b*x**2)*(a*d -
b*c)*(a*f - b*e)) + b*sqrt(c)*sqrt(d)*sqrt(e + f*x**2)*elliptic_e(atan(sqrt(d)*
x/sqrt(c)), -c*f/(d*e) + 1)/(2*a*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c +
d*x**2)*(a*d - b*c)*(a*f - b*e)) - b*d*x*sqrt(e + f*x**2)/(2*a*sqrt(c + d*x**2)*
(a*d - b*c)*(a*f - b*e)) - sqrt(c)*f*sqrt(e + f*x**2)*(-3*a**2*d*f + 2*a*b*(c*f
+ d*e) - b**2*c*e)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), -c*f/(d*e) + 1)/(2*a*sqrt
(d)*e*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)*(a*d - b*c)*(a*f -
b*e)**2) + b*e**(3/2)*sqrt(c + d*x**2)*(-3*a**2*d*f + 2*a*b*(c*f + d*e) - b**2*c
*e)*elliptic_pi(1 - b*e/(a*f), atan(sqrt(f)*x/sqrt(e)), 1 - d*e/(c*f))/(2*a**2*c
*sqrt(f)*sqrt(e*(c + d*x**2)/(c*(e + f*x**2)))*sqrt(e + f*x**2)*(a*d - b*c)*(a*f
- b*e)**2)

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Mathematica [C]  time = 6.16083, size = 587, normalized size = 1.21 \[ \frac{-\frac{i b^2 c e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{a \sqrt{\frac{d}{c}}}+\frac{b^2 c e x}{a+b x^2}+\frac{b^2 c f x^3}{a+b x^2}+\frac{b^2 d e x^3}{a+b x^2}+\frac{b^2 d f x^5}{a+b x^2}-i c \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b e-a f) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+2 i b c e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-3 i a c f \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+\frac{2 i b c f \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{\sqrt{\frac{d}{c}}}+i b c e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{2 a \sqrt{c+d x^2} \sqrt{e+f x^2} (a d-b c) (a f-b e)} \]

Antiderivative was successfully verified.

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

[Out]

((b^2*c*e*x)/(a + b*x^2) + (b^2*d*e*x^3)/(a + b*x^2) + (b^2*c*f*x^3)/(a + b*x^2)
+ (b^2*d*f*x^5)/(a + b*x^2) + I*b*c*Sqrt[d/c]*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f
*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*c*Sqrt[d/c]*(b*e - a
*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*e*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*Sqrt[d/c]) + (2*I)*b*c*Sqrt[d/c
]*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sq
rt[d/c]*x], (c*f)/(d*e)] + ((2*I)*b*c*f*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)])/Sqrt[d/c] - (3*I)*
a*c*Sqrt[d/c]*f*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)])/(2*a*(-(b*c) + a*d)*(-(b*e) + a*f)*Sqrt[c
+ d*x^2]*Sqrt[e + f*x^2])

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Maple [B]  time = 0.043, size = 1078, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-(-d/c)^(1/2)*x^5*a*b^2*d*f+((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellipt
icF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*b*d*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))*x^2*a*b^2*d*e+((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))*x^2*a*b^2*d*
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))*x^2*a^2*b*d*f+2*((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))*x^2*a*b^2*c*f+2*((
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))*x^2*a*b^2*d*e-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellipti
cPi(x*(-d/c)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-d/c)^(1/2))*x^2*b^3*c*e-(-d/c)^(1/2)*x
^3*a*b^2*c*f-(-d/c)^(1/2)*x^3*a*b^2*d*e+((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^3*d*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^2*b*d*e+((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*b*d*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))*a^3*d*f+2*((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))*a^2*b*c*f+2*((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))
*a^2*b*d*e-((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))*a*b^2*c*e-(-d/c)^(1/2)*x*a*b^2*c*e)*(f*x^2+e)^(1
/2)*(d*x^2+c)^(1/2)/(-d/c)^(1/2)/(b*x^2+a)/a^2/(a*d-b*c)/(a*f-b*e)/(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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="giac")

[Out]

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