3.396 \(\int \frac{d+e x^2}{\sqrt{-a+b x^2-c x^4}} \, dx\)

Optimal. Leaf size=293 \[ \frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a-b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (\frac{b}{\sqrt{a} \sqrt{c}}+2\right )\right )}{2 c^{3/4} \sqrt{-a+b x^2-c x^4}}-\frac{\sqrt [4]{a} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a-b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (\frac{b}{\sqrt{a} \sqrt{c}}+2\right )\right )}{c^{3/4} \sqrt{-a+b x^2-c x^4}}-\frac{e x \sqrt{-a+b x^2-c x^4}}{\sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )} \]

[Out]

-((e*x*Sqrt[-a + b*x^2 - c*x^4])/(Sqrt[c]*(Sqrt[a] + Sqrt[c]*x^2))) - (a^(1/4)*e
*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a - b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ell
ipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 + b/(Sqrt[a]*Sqrt[c]))/4])/(c^(3/4)*Sqr
t[-a + b*x^2 - c*x^4]) + (a^(1/4)*((Sqrt[c]*d)/Sqrt[a] + e)*(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*c^(3/4)*Sqrt[-a + b*x^2 - c*x^
4])

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Rubi [A]  time = 0.235354, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a-b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (\frac{b}{\sqrt{a} \sqrt{c}}+2\right )\right )}{2 c^{3/4} \sqrt{-a+b x^2-c x^4}}-\frac{\sqrt [4]{a} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a-b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (\frac{b}{\sqrt{a} \sqrt{c}}+2\right )\right )}{c^{3/4} \sqrt{-a+b x^2-c x^4}}-\frac{e x \sqrt{-a+b x^2-c x^4}}{\sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((e*x*Sqrt[-a + b*x^2 - c*x^4])/(Sqrt[c]*(Sqrt[a] + Sqrt[c]*x^2))) - (a^(1/4)*e
*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a - b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ell
ipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 + b/(Sqrt[a]*Sqrt[c]))/4])/(c^(3/4)*Sqr
t[-a + b*x^2 - c*x^4]) + (a^(1/4)*((Sqrt[c]*d)/Sqrt[a] + e)*(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*c^(3/4)*Sqrt[-a + b*x^2 - c*x^
4])

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Rubi in Sympy [A]  time = 33.8636, size = 260, normalized size = 0.89 \[ - \frac{\sqrt [4]{a} e \sqrt{- \frac{- a + b x^{2} - c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} + \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{c^{\frac{3}{4}} \sqrt{- a + b x^{2} - c x^{4}}} - \frac{e x \sqrt{- a + b x^{2} - c x^{4}}}{\sqrt{c} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} + \frac{\sqrt{- \frac{- a + b x^{2} - c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} e + \sqrt{c} d\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} + \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{2 \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{- a + b x^{2} - c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/4)*e*sqrt(-(-a + b*x**2 - c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) +
sqrt(c)*x**2)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/2 + b/(4*sqrt(a)*sqrt(c
)))/(c**(3/4)*sqrt(-a + b*x**2 - c*x**4)) - e*x*sqrt(-a + b*x**2 - c*x**4)/(sqrt
(c)*(sqrt(a) + sqrt(c)*x**2)) + 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)*elliptic_f(2*atan(c**
(1/4)*x/a**(1/4)), 1/2 + b/(4*sqrt(a)*sqrt(c)))/(2*a**(1/4)*c**(3/4)*sqrt(-a + b
*x**2 - c*x**4))

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Mathematica [C]  time = 0.448899, size = 295, normalized size = 1.01 \[ -\frac{i \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}} \left (\left (e \left (b-\sqrt{b^2-4 a c}\right )+2 c d\right ) F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+e \left (\sqrt{b^2-4 a c}-b\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{2 \sqrt{2} c \sqrt{-\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{-a+b x^2-c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

((-I/2)*Sqrt[1 + (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqr
t[b^2 - 4*a*c])]*((-b + Sqrt[b^2 - 4*a*c])*e*EllipticE[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])]
+ (2*c*d + (b - Sqrt[b^2 - 4*a*c])*e)*EllipticF[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])]))/(Sq
rt[2]*c*Sqrt[-(c/(b + Sqrt[b^2 - 4*a*c]))]*Sqrt[-a + b*x^2 - c*x^4])

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Maple [A]  time = 0.037, size = 357, normalized size = 1.2 \[{\frac{d}{2}\sqrt{4+2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4-2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{x}{2}\sqrt{-2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{a}}}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{-2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{a}}}}}{\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}-a}}}}+{ae\sqrt{4+2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4-2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}} \left ({\it EllipticF} \left ({\frac{x}{2}\sqrt{-2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{a}}}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) -{\it EllipticE} \left ({\frac{x}{2}\sqrt{-2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{a}}}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) \right ){\frac{1}{\sqrt{-2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{a}}}}}{\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}-a}}} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d/(-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)*EllipticF(1
/2*x*(-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))+e*a/(-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)/(b+(
-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*(-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))-EllipticE(1/2*x*(-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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{2} + d}{\sqrt{-c x^{4} + b x^{2} - a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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