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

Optimal. Leaf size=321 \[ \frac{e^{3/2} \sqrt{c+d x^2} (b c-a d) \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d e^{3/2} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f x \sqrt{c+d x^2}}{b \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

[Out]

(f*x*Sqrt[c + d*x^2])/(b*Sqrt[e + f*x^2]) - (Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*Ell
ipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*Sqrt[(e*(c + d*x^2))/(c
*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(S
qrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e +
f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)*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*b*c*Sqrt[f]*Sqrt[
(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.644136, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{e^{3/2} \sqrt{c+d x^2} (b c-a d) \Pi \left (1-\frac{b e}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{a b c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d e^{3/2} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b c \sqrt{f} \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{f x \sqrt{c+d x^2}}{b \sqrt{e+f x^2}}-\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{b \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

(f*x*Sqrt[c + d*x^2])/(b*Sqrt[e + f*x^2]) - (Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*Ell
ipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*Sqrt[(e*(c + d*x^2))/(c
*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*e^(3/2)*Sqrt[c + d*x^2]*EllipticF[ArcTan[(S
qrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e +
f*x^2))]*Sqrt[e + f*x^2]) + ((b*c - a*d)*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*b*c*Sqrt[f]*Sqrt[
(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi in Sympy [A]  time = 78.9471, size = 269, normalized size = 0.84 \[ - \frac{\sqrt{c} \sqrt{d} \sqrt{e + f x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{b \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{d x \sqrt{e + f x^{2}}}{b \sqrt{c + d x^{2}}} + \frac{d e^{\frac{3}{2}} \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{b c \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} - \frac{e^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d - b c\right ) \Pi \left (1 - \frac{b e}{a f}; \operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{a b c \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}}} \]

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),x)

[Out]

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

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Mathematica [C]  time = 0.442855, size = 184, normalized size = 0.57 \[ -\frac{i \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (a b d e E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+(b c-a d) \left ((b e-a f) \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )+a f F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )\right )}{a b^2 \sqrt{\frac{d}{c}} \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 340, normalized size = 1.1 \[{\frac{1}{ \left ( df{x}^{4}+cf{x}^{2}+de{x}^{2}+ce \right ){b}^{2}a} \left ( -{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ){a}^{2}df+{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) abcf+{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) abde+{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{bc}{ad}},{1\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ){a}^{2}df-{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{bc}{ad}},{1\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ) abcf-{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{bc}{ad}},{1\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ) abde+{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},{\frac{bc}{ad}},{1\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ){b}^{2}ce \right ) \sqrt{d{x}^{2}+c}\sqrt{f{x}^{2}+e}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

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),x)

[Out]

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

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

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),x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}{a + b x^{2}}\, dx \]

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),x)

[Out]

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

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

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),x, algorithm="giac")

[Out]

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