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

Optimal. Leaf size=359 \[ \frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (a^2 d f+b^2 c e\right ) \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a^2 b^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (a f+b e) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a b^2 \sqrt{c-d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a b \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1}} \]

[Out]

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

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Rubi [A]  time = 1.17052, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (a^2 d f+b^2 c e\right ) \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a^2 b^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (a f+b e) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a b^2 \sqrt{c-d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}+\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a b \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 170.75, size = 308, normalized size = 0.86 \[ \frac{x \sqrt{c - d x^{2}} \sqrt{e + f x^{2}}}{2 a \left (a + b x^{2}\right )} + \frac{\sqrt{c} \sqrt{d} \sqrt{1 - \frac{d x^{2}}{c}} \sqrt{e + f x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e}\right )}{2 a b \sqrt{1 + \frac{f x^{2}}{e}} \sqrt{c - d x^{2}}} - \frac{\sqrt{c} \sqrt{d} \sqrt{1 - \frac{d x^{2}}{c}} \sqrt{1 + \frac{f x^{2}}{e}} \left (a f + b e\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e}\right )}{2 a b^{2} \sqrt{c - d x^{2}} \sqrt{e + f x^{2}}} + \frac{\sqrt{c} \sqrt{1 - \frac{d x^{2}}{c}} \sqrt{1 + \frac{f x^{2}}{e}} \left (a^{2} d f + b^{2} c e\right ) \Pi \left (- \frac{b c}{a d}; \operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e}\right )}{2 a^{2} b^{2} \sqrt{d} \sqrt{c - d x^{2}} \sqrt{e + f x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 4.55728, size = 422, normalized size = 1.18 \[ \frac{-\frac{i c \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} (a f+b e) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{b^2}+\frac{i a c f \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{b^2}+\frac{i d e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{a \left (-\frac{d}{c}\right )^{3/2}}+\frac{c e x}{a+b x^2}+\frac{c f x^3}{a+b x^2}-\frac{d e x^3}{a+b x^2}-\frac{d f x^5}{a+b x^2}+\frac{i c e \sqrt{-\frac{d}{c}} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{b}}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.069, size = 793, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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