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

Optimal. Leaf size=239 \[ \frac{\sqrt{\sqrt{4 a c+b^2}-b} \tanh ^{-1}\left (\frac{x \sqrt{\sqrt{4 a c+b^2}-b} \left (\sqrt{4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt{2} \sqrt{a} \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{2 \sqrt{2} \sqrt{a} \sqrt{c} d}-\frac{\sqrt{\sqrt{4 a c+b^2}+b} \tan ^{-1}\left (\frac{x \sqrt{\sqrt{4 a c+b^2}+b} \left (-\sqrt{4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt{2} \sqrt{a} \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{2 \sqrt{2} \sqrt{a} \sqrt{c} d} \]

[Out]

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

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Rubi [A]  time = 0.184369, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {2072} \[ \frac{\sqrt{\sqrt{4 a c+b^2}-b} \tanh ^{-1}\left (\frac{x \sqrt{\sqrt{4 a c+b^2}-b} \left (\sqrt{4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt{2} \sqrt{a} \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{2 \sqrt{2} \sqrt{a} \sqrt{c} d}-\frac{\sqrt{\sqrt{4 a c+b^2}+b} \tan ^{-1}\left (\frac{x \sqrt{\sqrt{4 a c+b^2}+b} \left (-\sqrt{4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt{2} \sqrt{a} \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{2 \sqrt{2} \sqrt{a} \sqrt{c} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2072

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

Rubi steps

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

Mathematica [C]  time = 0.693984, size = 432, normalized size = 1.81 \[ \frac{\sqrt{\frac{4 c x^2}{\sqrt{4 a c+b^2}-b}+2} \sqrt{1-\frac{2 c x^2}{\sqrt{4 a c+b^2}+b}} \left (2 i \sqrt{a} \sqrt{c} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}}\right ),\frac{\sqrt{4 a c+b^2}+b}{b-\sqrt{4 a c+b^2}}\right )+\left (b-2 i \sqrt{a} \sqrt{c}\right ) \Pi \left (-\frac{i \left (b+\sqrt{b^2+4 a c}\right )}{2 \sqrt{a} \sqrt{c}};i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )-\left (b+2 i \sqrt{a} \sqrt{c}\right ) \Pi \left (\frac{i \left (b+\sqrt{b^2+4 a c}\right )}{2 \sqrt{a} \sqrt{c}};i \sinh ^{-1}\left (\sqrt{2} \sqrt{-\frac{c}{b+\sqrt{b^2+4 a c}}} x\right )|\frac{b+\sqrt{b^2+4 a c}}{b-\sqrt{b^2+4 a c}}\right )\right )}{4 \sqrt{a} \sqrt{c} d \sqrt{-\frac{c}{\sqrt{4 a c+b^2}+b}} \sqrt{a+b x^2-c x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + (4*c*x^2)/(-b + Sqrt[b^2 + 4*a*c])]*Sqrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*((2*I)*Sqrt[a]*Sqrt[
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])] + (b - (2*I)*Sqrt[a]*Sqrt[c])*EllipticPi[((-I/2)*(b + Sqrt[b^2 + 4*a*c]))/(Sqrt[a]*Sqrt[c]), I*ArcSin
h[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])] - (b + (2*I)
*Sqrt[a]*Sqrt[c])*EllipticPi[((I/2)*(b + Sqrt[b^2 + 4*a*c]))/(Sqrt[a]*Sqrt[c]), 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])]))/(4*Sqrt[a]*Sqrt[c]*Sqrt[-(c/(b +
Sqrt[b^2 + 4*a*c]))]*d*Sqrt[a + b*x^2 - c*x^4])

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Maple [B]  time = 0.072, size = 568, normalized size = 2.4 \begin{align*} -{\frac{\sqrt{2}b}{32\,acd}\sqrt{b+\sqrt{4\,ac+{b}^{2}}}\ln \left ({\frac{-c{x}^{4}+b{x}^{2}+a}{{x}^{2}}}+{\frac{\sqrt{2}}{x}\sqrt{-c{x}^{4}+b{x}^{2}+a}\sqrt{b+\sqrt{4\,ac+{b}^{2}}}}+\sqrt{4\,ac+{b}^{2}} \right ) }+{\frac{\sqrt{2}}{32\,acd}\sqrt{b+\sqrt{4\,ac+{b}^{2}}}\sqrt{4\,ac+{b}^{2}}\ln \left ({\frac{-c{x}^{4}+b{x}^{2}+a}{{x}^{2}}}+{\frac{\sqrt{2}}{x}\sqrt{-c{x}^{4}+b{x}^{2}+a}\sqrt{b+\sqrt{4\,ac+{b}^{2}}}}+\sqrt{4\,ac+{b}^{2}} \right ) }-{\frac{\sqrt{2}}{4\,d}\arctan \left ({\frac{1}{2} \left ( 2\,{\frac{\sqrt{-c{x}^{4}+b{x}^{2}+a}\sqrt{2}}{x}}+2\,\sqrt{b+\sqrt{4\,ac+{b}^{2}}} \right ){\frac{1}{\sqrt{-b+\sqrt{4\,ac+{b}^{2}}}}}} \right ){\frac{1}{\sqrt{-b+\sqrt{4\,ac+{b}^{2}}}}}}+{\frac{\sqrt{2}b}{32\,acd}\sqrt{b+\sqrt{4\,ac+{b}^{2}}}\ln \left ({\frac{\sqrt{2}}{x}\sqrt{-c{x}^{4}+b{x}^{2}+a}\sqrt{b+\sqrt{4\,ac+{b}^{2}}}}-{\frac{-c{x}^{4}+b{x}^{2}+a}{{x}^{2}}}-\sqrt{4\,ac+{b}^{2}} \right ) }-{\frac{\sqrt{2}}{32\,acd}\sqrt{b+\sqrt{4\,ac+{b}^{2}}}\sqrt{4\,ac+{b}^{2}}\ln \left ({\frac{\sqrt{2}}{x}\sqrt{-c{x}^{4}+b{x}^{2}+a}\sqrt{b+\sqrt{4\,ac+{b}^{2}}}}-{\frac{-c{x}^{4}+b{x}^{2}+a}{{x}^{2}}}-\sqrt{4\,ac+{b}^{2}} \right ) }+{\frac{\sqrt{2}}{4\,d}\arctan \left ({\frac{1}{2} \left ( 2\,\sqrt{b+\sqrt{4\,ac+{b}^{2}}}-2\,{\frac{\sqrt{-c{x}^{4}+b{x}^{2}+a}\sqrt{2}}{x}} \right ){\frac{1}{\sqrt{-b+\sqrt{4\,ac+{b}^{2}}}}}} \right ){\frac{1}{\sqrt{-b+\sqrt{4\,ac+{b}^{2}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32/d*2^(1/2)*(b+(4*a*c+b^2)^(1/2))^(1/2)/a/c*b*ln((-c*x^4+b*x^2+a)/x^2+(-c*x^4+b*x^2+a)^(1/2)*2^(1/2)/x*(b+
(4*a*c+b^2)^(1/2))^(1/2)+(4*a*c+b^2)^(1/2))+1/32/d*2^(1/2)*(b+(4*a*c+b^2)^(1/2))^(1/2)/a/c*(4*a*c+b^2)^(1/2)*l
n((-c*x^4+b*x^2+a)/x^2+(-c*x^4+b*x^2+a)^(1/2)*2^(1/2)/x*(b+(4*a*c+b^2)^(1/2))^(1/2)+(4*a*c+b^2)^(1/2))-1/4/d*2
^(1/2)/(-b+(4*a*c+b^2)^(1/2))^(1/2)*arctan(1/2*(2*(-c*x^4+b*x^2+a)^(1/2)*2^(1/2)/x+2*(b+(4*a*c+b^2)^(1/2))^(1/
2))/(-b+(4*a*c+b^2)^(1/2))^(1/2))+1/32/d*2^(1/2)*(b+(4*a*c+b^2)^(1/2))^(1/2)/a/c*b*ln((-c*x^4+b*x^2+a)^(1/2)*2
^(1/2)/x*(b+(4*a*c+b^2)^(1/2))^(1/2)-(-c*x^4+b*x^2+a)/x^2-(4*a*c+b^2)^(1/2))-1/32/d*2^(1/2)*(b+(4*a*c+b^2)^(1/
2))^(1/2)/a/c*(4*a*c+b^2)^(1/2)*ln((-c*x^4+b*x^2+a)^(1/2)*2^(1/2)/x*(b+(4*a*c+b^2)^(1/2))^(1/2)-(-c*x^4+b*x^2+
a)/x^2-(4*a*c+b^2)^(1/2))+1/4/d*2^(1/2)/(-b+(4*a*c+b^2)^(1/2))^(1/2)*arctan(1/2*(2*(b+(4*a*c+b^2)^(1/2))^(1/2)
-2*(-c*x^4+b*x^2+a)^(1/2)*2^(1/2)/x)/(-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{\sqrt{-c x^{4} + b x^{2} + a}}{c d x^{4} + a d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 22.6723, size = 1364, normalized size = 5.71 \begin{align*} -\frac{1}{8} \, \sqrt{\frac{2 \, a c d^{2} \sqrt{-\frac{1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac{\sqrt{-c x^{4} + b x^{2} + a} a d^{2} \sqrt{-\frac{1}{a c d^{4}}} + \sqrt{-c x^{4} + b x^{2} + a} x^{2} +{\left (a c d^{3} x^{3} \sqrt{-\frac{1}{a c d^{4}}} - a d x\right )} \sqrt{\frac{2 \, a c d^{2} \sqrt{-\frac{1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} + a}\right ) + \frac{1}{8} \, \sqrt{\frac{2 \, a c d^{2} \sqrt{-\frac{1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac{\sqrt{-c x^{4} + b x^{2} + a} a d^{2} \sqrt{-\frac{1}{a c d^{4}}} + \sqrt{-c x^{4} + b x^{2} + a} x^{2} -{\left (a c d^{3} x^{3} \sqrt{-\frac{1}{a c d^{4}}} - a d x\right )} \sqrt{\frac{2 \, a c d^{2} \sqrt{-\frac{1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} + a}\right ) - \frac{1}{8} \, \sqrt{-\frac{2 \, a c d^{2} \sqrt{-\frac{1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac{\sqrt{-c x^{4} + b x^{2} + a} a d^{2} \sqrt{-\frac{1}{a c d^{4}}} - \sqrt{-c x^{4} + b x^{2} + a} x^{2} +{\left (a c d^{3} x^{3} \sqrt{-\frac{1}{a c d^{4}}} + a d x\right )} \sqrt{-\frac{2 \, a c d^{2} \sqrt{-\frac{1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} + a}\right ) + \frac{1}{8} \, \sqrt{-\frac{2 \, a c d^{2} \sqrt{-\frac{1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac{\sqrt{-c x^{4} + b x^{2} + a} a d^{2} \sqrt{-\frac{1}{a c d^{4}}} - \sqrt{-c x^{4} + b x^{2} + a} x^{2} -{\left (a c d^{3} x^{3} \sqrt{-\frac{1}{a c d^{4}}} + a d x\right )} \sqrt{-\frac{2 \, a c d^{2} \sqrt{-\frac{1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} + a}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt((2*a*c*d^2*sqrt(-1/(a*c*d^4)) - b)/(a*c*d^2))*log(-(sqrt(-c*x^4 + b*x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)
) + sqrt(-c*x^4 + b*x^2 + a)*x^2 + (a*c*d^3*x^3*sqrt(-1/(a*c*d^4)) - a*d*x)*sqrt((2*a*c*d^2*sqrt(-1/(a*c*d^4))
- b)/(a*c*d^2)))/(c*x^4 + a)) + 1/8*sqrt((2*a*c*d^2*sqrt(-1/(a*c*d^4)) - b)/(a*c*d^2))*log(-(sqrt(-c*x^4 + b*
x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)) + sqrt(-c*x^4 + b*x^2 + a)*x^2 - (a*c*d^3*x^3*sqrt(-1/(a*c*d^4)) - a*d*x)*sq
rt((2*a*c*d^2*sqrt(-1/(a*c*d^4)) - b)/(a*c*d^2)))/(c*x^4 + a)) - 1/8*sqrt(-(2*a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/
(a*c*d^2))*log((sqrt(-c*x^4 + b*x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)) - sqrt(-c*x^4 + b*x^2 + a)*x^2 + (a*c*d^3*x^
3*sqrt(-1/(a*c*d^4)) + a*d*x)*sqrt(-(2*a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/(a*c*d^2)))/(c*x^4 + a)) + 1/8*sqrt(-(2
*a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/(a*c*d^2))*log((sqrt(-c*x^4 + b*x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)) - sqrt(-c*x
^4 + b*x^2 + a)*x^2 - (a*c*d^3*x^3*sqrt(-1/(a*c*d^4)) + a*d*x)*sqrt(-(2*a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/(a*c*d
^2)))/(c*x^4 + a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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