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

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

[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]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 + b/(Sqrt[a]*Sqrt[c
]))/4])/(c^(3/4)*Sqrt[-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 = 0.0897272, 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, Rules used = {1197, 1103, 1195} \[ \frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a-b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\frac{\sqrt{c} d}{\sqrt{a}}+e\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (\frac{b}{\sqrt{a} \sqrt{c}}+2\right )\right )}{2 c^{3/4} \sqrt{-a+b x^2-c x^4}}-\frac{\sqrt [4]{a} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a-b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (\frac{b}{\sqrt{a} \sqrt{c}}+2\right )\right )}{c^{3/4} \sqrt{-a+b x^2-c x^4}}-\frac{e x \sqrt{-a+b x^2-c x^4}}{\sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )} \]

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]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 + b/(Sqrt[a]*Sqrt[c
]))/4])/(c^(3/4)*Sqrt[-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])

Rule 1197

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

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1195

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

Rubi steps

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

Mathematica [C]  time = 0.310562, size = 295, normalized size = 1.01 \[ -\frac{i \sqrt{\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}+1} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}} \left (\left (e \left (b-\sqrt{b^2-4 a c}\right )+2 c d\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{-\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )+e \left (\sqrt{b^2-4 a c}-b\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{2 \sqrt{2} c \sqrt{-\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{-a+b x^2-c x^4}} \]

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 + 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 = 0.03, size = 357, normalized size = 1.2 \begin{align*}{ae\sqrt{4+2\,{\frac{ \left ( \sqrt{-4\,ac+{b}^{2}}-b \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{\sqrt{-4\,ac+{b}^{2}}-b}{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{\sqrt{-4\,ac+{b}^{2}}-b}{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{\sqrt{-4\,ac+{b}^{2}}-b}{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 ( \sqrt{-4\,ac+{b}^{2}}-b \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{\sqrt{-4\,ac+{b}^{2}}-b}{a}}}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{-2\,{\frac{\sqrt{-4\,ac+{b}^{2}}-b}{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*((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4+2*((-4*a*c+b^2)^(1/2)-b)/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*((-4*a*c+b^2)^(1/2)-b)/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*((-4*a*c+b^2)^(1/2)-b)/a)^(1/2),1/2*(-4+2*
b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))+1/2*d/(-2*((-4*a*c+b^2)^(1/2)-b)/a)^(1/2)*(4+2*((-4*a*c+b^2)^(1/2)-b)/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*((-4*a*c+b^2)^(
1/2)-b)/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{\sqrt{-c x^{4} + b x^{2} - a}{\left (e x^{2} + d\right )}}{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(-sqrt(-c*x^4 + b*x^2 - a)*(e*x^2 + d)/(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)