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

Optimal. Leaf size=148 \[ \frac{\sqrt{e} \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right )|\frac{(b c-a d) e}{c (b e-a f)}\right )}{a \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]

[Out]

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

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Rubi [A]  time = 0.313504, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{\sqrt{e} \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right )|\frac{(b c-a d) e}{c (b e-a f)}\right )}{a \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 165.39, size = 580, normalized size = 3.92 \[ \frac{\sqrt{e} \sqrt{\frac{a \left (e + f x^{2}\right )}{e \left (a + b x^{2}\right )}} \sqrt{1 - \frac{x^{2} \left (- a d + b c\right )}{c \left (a + b x^{2}\right )}} \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{x \sqrt{a f - b e}}{\sqrt{e} \sqrt{a + b x^{2}}} \right )}\middle | \frac{a \left (c f - d e\right )}{c \left (a f - b e\right )}\right )}{a \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{\frac{e \left (c + \frac{x^{2} \left (a d - b c\right )}{a + b x^{2}}\right )}{c \left (e + \frac{x^{2} \left (a f - b e\right )}{a + b x^{2}}\right )}} \sqrt{1 - \frac{x^{2} \left (- a f + b e\right )}{e \left (a + b x^{2}\right )}} \sqrt{e + f x^{2}} \sqrt{a f - b e}} + \frac{e x \sqrt{\frac{a \left (e + f x^{2}\right )}{e \left (a + b x^{2}\right )}} \sqrt{1 - \frac{x^{2} \left (- a f + b e\right )}{e \left (a + b x^{2}\right )}} \sqrt{c + d x^{2}} \left (a d - b c\right )}{a c \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{1 - \frac{x^{2} \left (- a d + b c\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}} \sqrt{e + f x^{2}} \left (a f - b e\right )} - \frac{e \sqrt{\frac{a \left (e + f x^{2}\right )}{e \left (a + b x^{2}\right )}} \sqrt{1 - \frac{x^{2} \left (- a f + b e\right )}{e \left (a + b x^{2}\right )}} \sqrt{c + d x^{2}} \sqrt{a d - b c} E\left (\operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}\middle | - \frac{a \left (c f - d e\right )}{e \left (a d - b c\right )}\right )}{a \sqrt{c} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{\frac{c \left (e + \frac{x^{2} \left (a f - b e\right )}{a + b x^{2}}\right )}{e \left (c + \frac{x^{2} \left (a d - b c\right )}{a + b x^{2}}\right )}} \sqrt{1 - \frac{x^{2} \left (- a d + b c\right )}{c \left (a + b x^{2}\right )}} \sqrt{e + f x^{2}} \left (a f - b e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.19262, size = 0, normalized size = 0. \[ \int \frac{\sqrt{c+d x^2}}{\left (a+b x^2\right )^{3/2} \sqrt{e+f x^2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.099, size = 0, normalized size = 0. \[ \int{1\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{f{x}^{2}+e}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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