3.108 \(\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=484 \[ -\frac{c^{3/2} \sqrt{a+b x^2} (b e-a f) F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e f \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{x \sqrt{a+b x^2} (d e-c f)}{e f \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e f \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

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

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Rubi [A]  time = 2.22447, antiderivative size = 484, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{c^{3/2} \sqrt{a+b x^2} (b e-a f) F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e f \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{x \sqrt{a+b x^2} (d e-c f)}{e f \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e f \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.856013, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{1\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( f{x}^{2}+e \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(f*x^2 + e)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(f*x^2 + e)^(3/2),x, algorithm="giac")

[Out]

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