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

Optimal. Leaf size=116 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x \left (\sqrt{a}-\sqrt{b} x^2\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c} \]

[Out]

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

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Rubi [A]  time = 0.0234927, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {405} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x \left (\sqrt{a}-\sqrt{b} x^2\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 405

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

Rubi steps

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

Mathematica [C]  time = 0.154231, size = 155, normalized size = 1.34 \[ \frac{5 a x \sqrt{a-b x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right )}{c \left (a+b x^4\right ) \left (5 a F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right )-2 b x^4 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right )+F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(5*a*x*Sqrt[a - b*x^4]*AppellF1[1/4, -1/2, 1, 5/4, (b*x^4)/a, -((b*x^4)/a)])/(c*(a + b*x^4)*(5*a*AppellF1[1/4,
-1/2, 1, 5/4, (b*x^4)/a, -((b*x^4)/a)] - 2*b*x^4*(2*AppellF1[5/4, -1/2, 2, 9/4, (b*x^4)/a, -((b*x^4)/a)] + Ap
pellF1[5/4, 1/2, 1, 9/4, (b*x^4)/a, -((b*x^4)/a)])))

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Maple [A]  time = 0.018, size = 158, normalized size = 1.4 \begin{align*} -{\frac{1}{8\,c}\ln \left ({ \left ({\frac{-b{x}^{4}+a}{2\,{x}^{2}}}-{\frac{1}{x}\sqrt [4]{ab}\sqrt{-b{x}^{4}+a}}+\sqrt{ab} \right ) \left ({\frac{-b{x}^{4}+a}{2\,{x}^{2}}}+{\frac{1}{x}\sqrt [4]{ab}\sqrt{-b{x}^{4}+a}}+\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{ab}}}}-{\frac{1}{4\,c}\arctan \left ({\frac{1}{x}\sqrt{-b{x}^{4}+a}{\frac{1}{\sqrt [4]{ab}}}}+1 \right ){\frac{1}{\sqrt [4]{ab}}}}+{\frac{1}{4\,c}\arctan \left ( -{\frac{1}{x}\sqrt{-b{x}^{4}+a}{\frac{1}{\sqrt [4]{ab}}}}+1 \right ){\frac{1}{\sqrt [4]{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/c/(a*b)^(1/4)*ln((1/2*(-b*x^4+a)/x^2-(a*b)^(1/4)*(-b*x^4+a)^(1/2)/x+(a*b)^(1/2))/(1/2*(-b*x^4+a)/x^2+(a*b
)^(1/4)*(-b*x^4+a)^(1/2)/x+(a*b)^(1/2)))-1/4/c/(a*b)^(1/4)*arctan(1/(a*b)^(1/4)*(-b*x^4+a)^(1/2)/x+1)+1/4/c/(a
*b)^(1/4)*arctan(-1/(a*b)^(1/4)*(-b*x^4+a)^(1/2)/x+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 5.00008, size = 822, normalized size = 7.09 \begin{align*} -\left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{2 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} \sqrt{-\frac{1}{b}} \left (-\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + \left (\frac{1}{4}\right )^{\frac{1}{4}}{\left (b c x^{2} \sqrt{-\frac{1}{b}} + \sqrt{-b x^{4} + a} c\right )} \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}}}{x}\right ) - \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (-\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + \sqrt{-b x^{4} + a} a c^{2} \sqrt{-\frac{1}{a b c^{4}}} - 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) + \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (-\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} - \sqrt{-b x^{4} + a} a c^{2} \sqrt{-\frac{1}{a b c^{4}}} - 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} - \sqrt{-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1/4)^(1/4)*(-1/(a*b*c^4))^(1/4)*arctan((2*(1/4)^(3/4)*a*b*c^3*sqrt(-1/b)*(-1/(a*b*c^4))^(3/4) + (1/4)^(1/4)*
(b*c*x^2*sqrt(-1/b) + sqrt(-b*x^4 + a)*c)*(-1/(a*b*c^4))^(1/4))/x) - 1/4*(1/4)^(1/4)*(-1/(a*b*c^4))^(1/4)*log(
-(4*(1/4)^(3/4)*a*b*c^3*x^3*(-1/(a*b*c^4))^(3/4) + sqrt(-b*x^4 + a)*a*c^2*sqrt(-1/(a*b*c^4)) - 2*(1/4)^(1/4)*a
*c*x*(-1/(a*b*c^4))^(1/4) + sqrt(-b*x^4 + a)*x^2)/(b*x^4 + a)) + 1/4*(1/4)^(1/4)*(-1/(a*b*c^4))^(1/4)*log((4*(
1/4)^(3/4)*a*b*c^3*x^3*(-1/(a*b*c^4))^(3/4) - sqrt(-b*x^4 + a)*a*c^2*sqrt(-1/(a*b*c^4)) - 2*(1/4)^(1/4)*a*c*x*
(-1/(a*b*c^4))^(1/4) - sqrt(-b*x^4 + a)*x^2)/(b*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^{4}}}{a + b x^{4}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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