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

Optimal. Leaf size=479 \[ -\frac{e \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{2 \sqrt{2} c^{3/2} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}+\frac{d \sqrt{\sqrt{4 a c+b^2}+b} \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}+\frac{e x \left (b-\sqrt{4 a c+b^2}\right ) \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right )}{2 c \sqrt{-a+b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 1.37875, antiderivative size = 479, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{e \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{2 \sqrt{2} c^{3/2} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}+\frac{d \sqrt{\sqrt{4 a c+b^2}+b} \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}+\frac{e x \left (b-\sqrt{4 a c+b^2}\right ) \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right )}{2 c \sqrt{-a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 142.476, size = 420, normalized size = 0.88 \[ \frac{e x \left (b - \sqrt{4 a c + b^{2}}\right ) \left (\frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1\right )}{2 c \sqrt{- a + b x^{2} + c x^{4}}} + \frac{\sqrt{2} d \sqrt{b + \sqrt{4 a c + b^{2}}} \left (\frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{4 a c + b^{2}}}} \right )}\middle | - \frac{2 \sqrt{4 a c + b^{2}}}{b - \sqrt{4 a c + b^{2}}}\right )}{2 \sqrt{c} \sqrt{\frac{\frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1}{\frac{2 c x^{2}}{b + \sqrt{4 a c + b^{2}}} + 1}} \sqrt{- a + b x^{2} + c x^{4}}} - \frac{\sqrt{2} e \left (b - \sqrt{4 a c + b^{2}}\right ) \sqrt{b + \sqrt{4 a c + b^{2}}} \left (\frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{4 a c + b^{2}}}} \right )}\middle | - \frac{2 \sqrt{4 a c + b^{2}}}{b - \sqrt{4 a c + b^{2}}}\right )}{4 c^{\frac{3}{2}} \sqrt{\frac{\frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1}{\frac{2 c x^{2}}{b + \sqrt{4 a c + b^{2}}} + 1}} \sqrt{- a + b x^{2} + c x^{4}}} \]

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]

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

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Mathematica [C]  time = 0.476807, size = 304, normalized size = 0.63 \[ \frac{i \sqrt{\frac{\sqrt{4 a c+b^2}+b+2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \left (\left (e \left (b-\sqrt{4 a c+b^2}\right )-2 c d\right ) F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )+e \left (\sqrt{4 a c+b^2}-b\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )\right )}{2 \sqrt{2} c \sqrt{\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{-a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

((I/2)*Sqrt[(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])]*((-b + Sqrt[b^2 + 4*a*c])*e*EllipticE[I*ArcSi
nh[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])]))/(Sqrt[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 = 0.013, size = 355, normalized size = 0.7 \[{\frac{d}{2}\sqrt{4+2\,{\frac{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4-2\,{\frac{ \left ( b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{x}{2}\sqrt{-2\,{\frac{-b+\sqrt{4\,ac+{b}^{2}}}{a}}}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{-2\,{\frac{-b+\sqrt{4\,ac+{b}^{2}}}{a}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}-a}}}}+{ae\sqrt{4+2\,{\frac{ \left ( -b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4-2\,{\frac{ \left ( b+\sqrt{4\,ac+{b}^{2}} \right ){x}^{2}}{a}}} \left ({\it EllipticF} \left ({\frac{x}{2}\sqrt{-2\,{\frac{-b+\sqrt{4\,ac+{b}^{2}}}{a}}}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) -{\it EllipticE} \left ({\frac{x}{2}\sqrt{-2\,{\frac{-b+\sqrt{4\,ac+{b}^{2}}}{a}}}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) \right ){\frac{1}{\sqrt{-2\,{\frac{-b+\sqrt{4\,ac+{b}^{2}}}{a}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}-a}}} \left ( b+\sqrt{4\,ac+{b}^{2}} \right ) ^{-1}} \]

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)