3.12 \(\int \frac{e+f x^2}{\left (d+e x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=424 \[ \frac{\left (e^2-d f\right ) \tan ^{-1}\left (\frac{x \sqrt{\frac{a e}{d}-b+\frac{c d}{e}}}{\sqrt{a+b x^2+c x^4}}\right )}{2 d e \sqrt{\frac{a e}{d}-b+\frac{c d}{e}}}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \left (e^2-d f\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \sqrt{a+b x^2+c x^4} \left (\frac{\sqrt{c} d}{\sqrt{a}}-e\right )}+\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} e-\sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a+b x^2+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )} \]

[Out]

((e^2 - d*f)*ArcTan[(Sqrt[-b + (c*d)/e + (a*e)/d]*x)/Sqrt[a + b*x^2 + c*x^4]])/(
2*d*e*Sqrt[-b + (c*d)/e + (a*e)/d]) + ((Sqrt[c]*e - Sqrt[a]*f)*(Sqrt[a] + Sqrt[c
]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c
^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*c^(1/4)*(Sqrt[c]*d
- Sqrt[a]*e)*Sqrt[a + b*x^2 + c*x^4]) - (((Sqrt[c]*d)/Sqrt[a] + e)*(e^2 - d*f)*(
Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellip
ticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a
^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1/4)*c^(1/4)*d*((Sqrt[c]*d)/Sqrt[a]
- e)*e*Sqrt[a + b*x^2 + c*x^4])

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Rubi [A]  time = 0.846639, antiderivative size = 424, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (e^2-d f\right ) \tan ^{-1}\left (\frac{x \sqrt{\frac{a e}{d}-b+\frac{c d}{e}}}{\sqrt{a+b x^2+c x^4}}\right )}{2 d e \sqrt{\frac{a e}{d}-b+\frac{c d}{e}}}-\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \left (e^2-d f\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} e+\sqrt{c} d\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \sqrt{a+b x^2+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )}+\frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} e-\sqrt{a} f\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt{a+b x^2+c x^4} \left (\sqrt{c} d-\sqrt{a} e\right )} \]

Warning: Unable to verify antiderivative.

[In]  Int[(e + f*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

((e^2 - d*f)*ArcTan[(Sqrt[-b + (c*d)/e + (a*e)/d]*x)/Sqrt[a + b*x^2 + c*x^4]])/(
2*d*e*Sqrt[-b + (c*d)/e + (a*e)/d]) + ((Sqrt[c]*e - Sqrt[a]*f)*(Sqrt[a] + Sqrt[c
]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c
^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*c^(1/4)*(Sqrt[c]*d
- Sqrt[a]*e)*Sqrt[a + b*x^2 + c*x^4]) - ((Sqrt[c]*d + Sqrt[a]*e)*(e^2 - d*f)*(Sq
rt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipti
cPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(
1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1/4)*c^(1/4)*d*e*(Sqrt[c]*d - Sqrt[a]
*e)*Sqrt[a + b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 57.731, size = 464, normalized size = 1.09 \[ \frac{\left (- d f + e^{2}\right ) \operatorname{atan}{\left (\frac{x \sqrt{\frac{a e}{d} - b + \frac{c d}{e}}}{\sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 d e \sqrt{\frac{a e}{d} - b + \frac{c d}{e}}} - \frac{\sqrt [4]{c} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (- d f + e^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{2 \sqrt [4]{a} e \left (\sqrt{a} e - \sqrt{c} d\right ) \sqrt{a + b x^{2} + c x^{4}}} + \frac{f \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} e \sqrt{a + b x^{2} + c x^{4}}} + \frac{\sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} e + \sqrt{c} d\right ) \left (- d f + e^{2}\right ) \Pi \left (- \frac{\sqrt{a} \left (e - \frac{\sqrt{c} d}{\sqrt{a}}\right )^{2}}{4 \sqrt{c} d e}; 2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \left (\sqrt{a} e - \sqrt{c} d\right ) \sqrt{a + b x^{2} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**2+e)/(e*x**2+d)/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

(-d*f + e**2)*atan(x*sqrt(a*e/d - b + c*d/e)/sqrt(a + b*x**2 + c*x**4))/(2*d*e*s
qrt(a*e/d - b + c*d/e)) - c**(1/4)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)
*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*(-d*f + e**2)*elliptic_f(2*atan(c**(1/4)*x/a
**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(2*a**(1/4)*e*(sqrt(a)*e - sqrt(c)*d)*sqr
t(a + b*x**2 + c*x**4)) + f*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)*
*2)*(sqrt(a) + sqrt(c)*x**2)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*
sqrt(a)*sqrt(c)))/(2*a**(1/4)*c**(1/4)*e*sqrt(a + b*x**2 + c*x**4)) + sqrt((a +
b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*(sqrt(a)*
e + sqrt(c)*d)*(-d*f + e**2)*elliptic_pi(-sqrt(a)*(e - sqrt(c)*d/sqrt(a))**2/(4*
sqrt(c)*d*e), 2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(4*a**(1
/4)*c**(1/4)*d*e*(sqrt(a)*e - sqrt(c)*d)*sqrt(a + b*x**2 + c*x**4))

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Mathematica [C]  time = 0.5956, size = 298, normalized size = 0.7 \[ -\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (\left (e^2-d f\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+d f F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{\sqrt{2} d e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(e + f*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

((-I)*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (
2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d*f*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + S
qrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (e^2 -
d*f)*EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(
b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(
Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*Sqrt[a + b*x^2 + c*x^4])

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Maple [A]  time = 0.015, size = 359, normalized size = 0.9 \[{\frac{f\sqrt{2}}{4\,e}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{ \left ( -df+{e}^{2} \right ) \sqrt{2}}{de}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}-{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},-2\,{\frac{ae}{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) d}},{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}+{\frac{1}{a}\sqrt{-4\,ac+{b}^{2}}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^2+e)/(e*x^2+d)/(c*x^4+b*x^2+a)^(1/2),x)

[Out]

1/4*f/e*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a
*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*Ellip
ticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)
^(1/2))/a/c)^(1/2))+(-d*f+e^2)/e/d*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(
1+1/2*b*x^2/a-1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a+1/2/a*x^2*(-4*a
*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+(-4*a*c
+b^2)^(1/2))/a)^(1/2),-2/(-b+(-4*a*c+b^2)^(1/2))*a*e/d,(-1/2*(b+(-4*a*c+b^2)^(1/
2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f x^{2} + e}{\sqrt{c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="maxima")

[Out]

integrate((f*x^2 + e)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{f x^{2} + e}{\sqrt{c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="fricas")

[Out]

integral((f*x^2 + e)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e + f x^{2}}{\left (d + e x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**2+e)/(e*x**2+d)/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

Integral((e + f*x**2)/((d + e*x**2)*sqrt(a + b*x**2 + c*x**4)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^2 + e)/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="giac")

[Out]

Exception raised: TypeError