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

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

[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 - S
qrt[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*Sqrt[2]*c^(3/
2)*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]) + (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 = 0.475038, 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, Rules used = {1202, 531, 418, 492, 411} \[ -\frac{e \left (b-\sqrt{4 a c+b^2}\right ) \sqrt{\sqrt{4 a c+b^2}+b} \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{2 \sqrt{2} c^{3/2} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}+\frac{d \sqrt{\sqrt{4 a c+b^2}+b} \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right ) F\left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2+4 a c}}}\right )|-\frac{2 \sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{\frac{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1}{\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}+1}} \sqrt{-a+b x^2+c x^4}}+\frac{e x \left (b-\sqrt{4 a c+b^2}\right ) \left (\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1\right )}{2 c \sqrt{-a+b x^2+c x^4}} \]

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 - S
qrt[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*Sqrt[2]*c^(3/
2)*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]) + (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])

Rule 1202

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[(Sqrt[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)])/Sqrt[a + b*x^2 + c*x^4], Int[(d + e*x^2)/(Sqr
t[1 + (2*c*x^2)/(b - q)]*Sqrt[1 + (2*c*x^2)/(b + q)]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c
, 0] && NegQ[c/a]

Rule 531

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{d+e x^2}{\sqrt{-a+b x^2+c x^4}} \, dx &=\frac{\left (\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}\right ) \int \frac{d+e x^2}{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \, dx}{\sqrt{-a+b x^2+c x^4}}\\ &=\frac{\left (d \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}\right ) \int \frac{1}{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \, dx}{\sqrt{-a+b x^2+c x^4}}+\frac{\left (e \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}\right ) \int \frac{x^2}{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \, dx}{\sqrt{-a+b x^2+c x^4}}\\ &=\frac{\left (b-\sqrt{b^2+4 a c}\right ) e x \left (1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}\right )}{2 c \sqrt{-a+b x^2+c x^4}}+\frac{\sqrt{b+\sqrt{b^2+4 a c}} d \left (1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}\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{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}}{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \sqrt{-a+b x^2+c x^4}}-\frac{\left (\left (b-\sqrt{b^2+4 a c}\right ) e \sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}} \sqrt{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}\right ) \int \frac{\sqrt{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}}}{\left (1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}\right )^{3/2}} \, dx}{2 c \sqrt{-a+b x^2+c x^4}}\\ &=\frac{\left (b-\sqrt{b^2+4 a c}\right ) e x \left (1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}\right )}{2 c \sqrt{-a+b x^2+c x^4}}-\frac{\left (b-\sqrt{b^2+4 a c}\right ) \sqrt{b+\sqrt{b^2+4 a c}} e \left (1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}\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{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}}{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \sqrt{-a+b x^2+c x^4}}+\frac{\sqrt{b+\sqrt{b^2+4 a c}} d \left (1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}\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{1+\frac{2 c x^2}{b-\sqrt{b^2+4 a c}}}{1+\frac{2 c x^2}{b+\sqrt{b^2+4 a c}}}} \sqrt{-a+b x^2+c x^4}}\\ \end{align*}

Mathematica [C]  time = 0.300859, size = 304, normalized size = 0.63 \[ \frac{i \sqrt{\frac{\sqrt{4 a c+b^2}+b+2 c x^2}{\sqrt{4 a c+b^2}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{4 a c+b^2}}+1} \left (\left (e \left (b-\sqrt{4 a c+b^2}\right )-2 c d\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{4 a c+b^2}+b}}\right ),\frac{\sqrt{4 a c+b^2}+b}{b-\sqrt{4 a c+b^2}}\right )+e \left (\sqrt{4 a c+b^2}-b\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )\right )}{2 \sqrt{2} c \sqrt{\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{-a+b x^2+c x^4}} \]

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*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])]))/(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 = 0.028, size = 355, normalized size = 0.7 \begin{align*}{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}}+{\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}}}} \end{align*}

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]

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)))+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))

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

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, 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. \begin{align*}{\rm integral}\left (\frac{e x^{2} + d}{\sqrt{c x^{4} + b x^{2} - a}}, x\right ) \end{align*}

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, 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. \begin{align*} \int \frac{d + e x^{2}}{\sqrt{- a + b x^{2} + c x^{4}}}\, dx \end{align*}

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

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, algorithm="giac")

[Out]

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