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

Optimal. Leaf size=454 \[ -\frac{e \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} (b e+3 c d) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} c^{5/2} \sqrt{a+b x^2-c x^4}}+\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (c e \left (-3 d \sqrt{4 a c+b^2}+a e+3 b d\right )+b e^2 \left (b-\sqrt{4 a c+b^2}\right )+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} c^{5/2} \sqrt{a+b x^2-c x^4}}-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{3 c} \]

[Out]

-(e^2*x*Sqrt[a + b*x^2 - c*x^4])/(3*c) - ((b - Sqrt[b^2 + 4*a*c])*Sqrt[b + Sqrt[
b^2 + 4*a*c]]*e*(3*c*d + b*e)*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1
- (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[
b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(3*Sq
rt[2]*c^(5/2)*Sqrt[a + b*x^2 - c*x^4]) + (Sqrt[b + Sqrt[b^2 + 4*a*c]]*(3*c^2*d^2
+ b*(b - Sqrt[b^2 + 4*a*c])*e^2 + c*e*(3*b*d - 3*Sqrt[b^2 + 4*a*c]*d + a*e))*Sq
rt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a
*c])]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sq
rt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(3*Sqrt[2]*c^(5/2)*Sqrt[a + b*x^2 - c
*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 1.81206, antiderivative size = 454, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{e \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} (b e+3 c d) E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} c^{5/2} \sqrt{a+b x^2-c x^4}}+\frac{\sqrt{\sqrt{4 a c+b^2}+b} \sqrt{1-\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (c e \left (-3 d \sqrt{4 a c+b^2}+a e+3 b d\right )+b e^2 \left (b-\sqrt{4 a c+b^2}\right )+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{3 \sqrt{2} c^{5/2} \sqrt{a+b x^2-c x^4}}-\frac{e^2 x \sqrt{a+b x^2-c x^4}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

-(e^2*x*Sqrt[a + b*x^2 - c*x^4])/(3*c) - ((b - Sqrt[b^2 + 4*a*c])*Sqrt[b + Sqrt[
b^2 + 4*a*c]]*e*(3*c*d + b*e)*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1
- (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[
b + Sqrt[b^2 + 4*a*c]]], (b + Sqrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(3*Sq
rt[2]*c^(5/2)*Sqrt[a + b*x^2 - c*x^4]) + (Sqrt[b + Sqrt[b^2 + 4*a*c]]*(3*c^2*d^2
+ b*(b - Sqrt[b^2 + 4*a*c])*e^2 + c*e*(3*b*d - 3*Sqrt[b^2 + 4*a*c]*d + a*e))*Sq
rt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a
*c])]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 + 4*a*c]]], (b + Sq
rt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(3*Sqrt[2]*c^(5/2)*Sqrt[a + b*x^2 - c
*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 142.436, size = 405, normalized size = 0.89 \[ - \frac{e^{2} x \sqrt{a + b x^{2} - c x^{4}}}{3 c} - \frac{\sqrt{2} e \left (b - \sqrt{4 a c + b^{2}}\right ) \sqrt{b + \sqrt{4 a c + b^{2}}} \left (b e + 3 c d\right ) \sqrt{- \frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1} \sqrt{- \frac{2 c x^{2}}{b + \sqrt{4 a c + b^{2}}} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{4 a c + b^{2}}}} \right )}\middle | \frac{b + \sqrt{4 a c + b^{2}}}{b - \sqrt{4 a c + b^{2}}}\right )}{6 c^{\frac{5}{2}} \sqrt{a + b x^{2} - c x^{4}}} + \frac{\sqrt{2} \sqrt{b + \sqrt{4 a c + b^{2}}} \sqrt{- \frac{2 c x^{2}}{b - \sqrt{4 a c + b^{2}}} + 1} \sqrt{- \frac{2 c x^{2}}{b + \sqrt{4 a c + b^{2}}} + 1} \left (b e^{2} \left (b - \sqrt{4 a c + b^{2}}\right ) + 3 c d e \left (b - \sqrt{4 a c + b^{2}}\right ) + c \left (a e^{2} + 3 c d^{2}\right )\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{4 a c + b^{2}}}} \right )}\middle | \frac{b + \sqrt{4 a c + b^{2}}}{b - \sqrt{4 a c + b^{2}}}\right )}{6 c^{\frac{5}{2}} \sqrt{a + b x^{2} - c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**2*x*sqrt(a + b*x**2 - c*x**4)/(3*c) - sqrt(2)*e*(b - sqrt(4*a*c + b**2))*sqr
t(b + sqrt(4*a*c + b**2))*(b*e + 3*c*d)*sqrt(-2*c*x**2/(b - sqrt(4*a*c + b**2))
+ 1)*sqrt(-2*c*x**2/(b + sqrt(4*a*c + b**2)) + 1)*elliptic_e(asin(sqrt(2)*sqrt(c
)*x/sqrt(b + sqrt(4*a*c + b**2))), (b + sqrt(4*a*c + b**2))/(b - sqrt(4*a*c + b*
*2)))/(6*c**(5/2)*sqrt(a + b*x**2 - c*x**4)) + sqrt(2)*sqrt(b + sqrt(4*a*c + b**
2))*sqrt(-2*c*x**2/(b - sqrt(4*a*c + b**2)) + 1)*sqrt(-2*c*x**2/(b + sqrt(4*a*c
+ b**2)) + 1)*(b*e**2*(b - sqrt(4*a*c + b**2)) + 3*c*d*e*(b - sqrt(4*a*c + b**2)
) + c*(a*e**2 + 3*c*d**2))*elliptic_f(asin(sqrt(2)*sqrt(c)*x/sqrt(b + sqrt(4*a*c
+ b**2))), (b + sqrt(4*a*c + b**2))/(b - sqrt(4*a*c + b**2)))/(6*c**(5/2)*sqrt(
a + b*x**2 - c*x**4))

_______________________________________________________________________________________

Mathematica [C]  time = 2.44266, size = 503, normalized size = 1.11 \[ \frac{i \sqrt{2} \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} \left (-c e \left (-3 d \sqrt{4 a c+b^2}+a e+3 b d\right )+b e^2 \left (\sqrt{4 a c+b^2}-b\right )-3 c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )-i \sqrt{2} e \left (\sqrt{4 a c+b^2}-b\right ) \sqrt{\frac{\sqrt{4 a c+b^2}+b-2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{\sqrt{4 a c+b^2}-b+2 c x^2}{\sqrt{4 a c+b^2}-b}} (b e+3 c d) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )+2 c e^2 x \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \left (-a-b x^2+c x^4\right )}{6 c^2 \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*c*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*e^2*x*(-a - b*x^2 + c*x^4) - I*Sqrt[2]*(
-b + Sqrt[b^2 + 4*a*c])*e*(3*c*d + b*e)*Sqrt[(b + Sqrt[b^2 + 4*a*c] - 2*c*x^2)/(
b + Sqrt[b^2 + 4*a*c])]*Sqrt[(-b + Sqrt[b^2 + 4*a*c] + 2*c*x^2)/(-b + Sqrt[b^2 +
4*a*c])]*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])] + I*Sqrt[2]*(-3*c^2*d^2 + b*(-b +
Sqrt[b^2 + 4*a*c])*e^2 - c*e*(3*b*d - 3*Sqrt[b^2 + 4*a*c]*d + a*e))*Sqrt[(b + Sq
rt[b^2 + 4*a*c] - 2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*Sqrt[(-b + Sqrt[b^2 + 4*a*c]
+ 2*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*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])])/(6*c^
2*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*Sqrt[a + b*x^2 - c*x^4])

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 761, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*d^2*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)*Ellipti
cF(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))+e^2*(-1/3*x*(-c*x^4+b*x^2+a)^(1/2)/c+1/12/c*a*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)*EllipticF(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))-1/3*b/c*a*
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)/(b+(4*a*c+b^2)^
(1/2))*(EllipticF(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))-EllipticE(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*e*a*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)/(b+(4*a*c+b^2)^(1/2))*(EllipticF(
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))-EllipticE(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)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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