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

Optimal. Leaf size=273 \[ \frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} e+\sqrt{c} d\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \sqrt{a+b x^2+c x^4}}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}} \]

[Out]

-((Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(
2*Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 - b*d*e + a*e^2]) + ((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]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTa
n[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1/4)*c^(1/4)*d*e*Sqrt[a + b*x^2 + c*x^4])

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Rubi [A]  time = 0.135684, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.024, Rules used = {1706} \[ \frac{\left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{a} e+\sqrt{c} d\right ) \Pi \left (-\frac{\left (\sqrt{c} d-\sqrt{a} e\right )^2}{4 \sqrt{a} \sqrt{c} d e};2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \sqrt{a+b x^2+c x^4}}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(
2*Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 - b*d*e + a*e^2]) + ((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]*EllipticPi[-(Sqrt[c]*d - Sqrt[a]*e)^2/(4*Sqrt[a]*Sqrt[c]*d*e), 2*ArcTa
n[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1/4)*c^(1/4)*d*e*Sqrt[a + b*x^2 + c*x^4])

Rule 1706

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

Rubi steps

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

Mathematica [C]  time = 0.53715, size = 310, normalized size = 1.14 \[ -\frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (\sqrt{c} d \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )+\left (\sqrt{a} e-\sqrt{c} d\right ) \Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{\sqrt{2} d e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.056, size = 365, normalized size = 1.3 \begin{align*}{\frac{\sqrt{2}}{4\,e}\sqrt{c}\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\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( \sqrt{-4\,ac+{b}^{2}}-b \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 ( \sqrt{-4\,ac+{b}^{2}}-b \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{\sqrt{2}}{ed} \left ( e\sqrt{a}-d\sqrt{c} \right ) \sqrt{1-{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}+{\frac{b{x}^{2}}{2\,a}}}\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 ( \sqrt{-4\,ac+{b}^{2}}-b \right ) }}},-2\,{\frac{ae}{ \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) d}},{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) }}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a}\sqrt{-4\,ac+{b}^{2}}}-{\frac{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((a^(1/2)+x^2*c^(1/2))/(e*x^2+d)/(c*x^4+b*x^2+a)^(1/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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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((a^(1/2)+x^2*c^(1/2))/(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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a} + \sqrt{c} x^{2}}{\left (d + e x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((sqrt(a) + sqrt(c)*x**2)/((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{\sqrt{c} x^{2} + \sqrt{a}}{\sqrt{c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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