3.1019 \(\int \frac{a-c x^4}{(a e+c d x^2) (d+e x^2) \sqrt{a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=80 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{\sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}} \]

[Out]

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

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Rubi [A]  time = 0.449093, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {2112, 205} \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{\sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2112

Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coef
f[v, x, 4], d = Coeff[1/u, x, 0], e = Coeff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Dist[A, Subst[Int[1/(d - (b*d -
a*e)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x
^2, 2] && PolyQ[1/u, x^2, 2]

Rule 205

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

Rubi steps

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

Mathematica [C]  time = 0.798444, size = 383, normalized size = 4.79 \[ \frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (-\Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) d}{2 a e};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )-\Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{\sqrt{2} d e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.06, size = 555, normalized size = 6.9 \begin{align*} -{\frac{\sqrt{2}}{4\,de}\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\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{\sqrt{2}}{de}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}-{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},-2\,{\frac{ae}{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) d}},{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}+{\frac{1}{a}\sqrt{-4\,ac+{b}^{2}}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{\sqrt{2}}{de}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}-{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},-2\,{\frac{cd}{e \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }},{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}+{\frac{1}{a}\sqrt{-4\,ac+{b}^{2}}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/e/d*2^(1/2)/(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*(4-2/a*(-b+(-4*a*c+b^2)^(1/2))*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)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2),1/
2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+1/e/d*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a-
1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*E
llipticPi(1/2*x*2^(1/2)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2),-2*a/(-b+(-4*a*c+b^2)^(1/2))*e/d,(-1/2*(b+(-4*a*c+
b^2)^(1/2))/a)^(1/2)*2^(1/2)/(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2))+1/d/e*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^
(1/2)*(1+1/2*b*x^2/a-1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b*x^2/a+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c
*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2),-2/(-b+(-4*a*c+b^2)^(1/2))*c*
d/e,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*x^4+a)/(c*d*x^2+a*e)/(e*x^2+d)/(c*x^4+b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*x^4+a)/(c*d*x^2+a*e)/(e*x^2+d)/(c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*x^4+a)/(c*d*x^2+a*e)/(e*x^2+d)/(c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

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