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

Optimal. Leaf size=541 \[ \frac{\sqrt{e} \sqrt{c+d x^2} (b c-a d) (2 b e-a f) \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right )|\frac{(b c-a d) e}{c (b e-a f)}\right )}{2 b^2 c \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} (a d f-b (c f+d e)) \Pi \left (\frac{b c}{b c-a d};\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{2 b^2 \sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 \sqrt{a+b x^2}}-\frac{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{2 b \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}} \]

[Out]

(x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*Sqrt[a + b*x^2]) - (Sqrt[c]*Sqrt[b*c - a*
d]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2]*EllipticE[ArcSin[(Sqrt[
b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])], (c*(b*e - a*f))/((b*c - a*d)*e)])/(2*b
*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]) + ((b*c - a*d)*Sqrt[e]*(
2*b*e - a*f)*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))]*EllipticF[Arc
Sin[(Sqrt[b*e - a*f]*x)/(Sqrt[e]*Sqrt[a + b*x^2])], ((b*c - a*d)*e)/(c*(b*e - a*
f))])/(2*b^2*c*Sqrt[b*e - a*f]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*
x^2]) - (a*(a*d*f - b*(d*e + c*f))*Sqrt[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a +
b*x^2))]*EllipticPi[(b*c)/(b*c - a*d), ArcSin[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[
a + b*x^2])], (c*(b*e - a*f))/((b*c - a*d)*e)])/(2*b^2*Sqrt[c]*Sqrt[b*c - a*d]*S
qrt[(a*(c + d*x^2))/(c*(a + b*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 1.58281, antiderivative size = 541, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.206 \[ \frac{\sqrt{e} \sqrt{c+d x^2} (b c-a d) (2 b e-a f) \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{b e-a f} x}{\sqrt{e} \sqrt{b x^2+a}}\right )|\frac{(b c-a d) e}{c (b e-a f)}\right )}{2 b^2 c \sqrt{e+f x^2} \sqrt{b e-a f} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac{a \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} (a d f-b (c f+d e)) \Pi \left (\frac{b c}{b c-a d};\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{2 b^2 \sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 \sqrt{a+b x^2}}-\frac{\sqrt{c} \sqrt{e+f x^2} \sqrt{b c-a d} \sqrt{\frac{a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\sin ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{c} \sqrt{b x^2+a}}\right )|\frac{c (b e-a f)}{(b c-a d) e}\right )}{2 b \sqrt{c+d x^2} \sqrt{\frac{a \left (e+f x^2\right )}{e \left (a+b x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.125541, size = 0, normalized size = 0. \[ \int \frac{\sqrt{c+d x^2} \sqrt{e+f x^2}}{\sqrt{a+b x^2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{1\sqrt{d{x}^{2}+c}\sqrt{f{x}^{2}+e}{\frac{1}{\sqrt{b{x}^{2}+a}}}}\, dx \]

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)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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