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

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 = 0.0790384, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x \left (\sqrt{a}-\sqrt{b} x^2\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c} \]

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)

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Rubi in Sympy [A]  time = 12.6542, size = 100, normalized size = 0.86 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{b} x \left (\sqrt{a} + \sqrt{b} x^{2}\right )}{\sqrt [4]{a} \sqrt{a - b x^{4}}} \right )}}{2 \sqrt [4]{a} \sqrt [4]{b} c} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{b} x \left (\sqrt{a} - \sqrt{b} x^{2}\right )}{\sqrt [4]{a} \sqrt{a - b x^{4}}} \right )}}{2 \sqrt [4]{a} \sqrt [4]{b} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**4+a)**(1/2)/(b*c*x**4+a*c),x)

[Out]

atan(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) + atanh(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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Mathematica [C]  time = 0.271918, 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)] + AppellF1[5/4, 1/2, 1,
9/4, (b*x^4)/a, -((b*x^4)/a)])))

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Maple [A]  time = 0.025, size = 158, normalized size = 1.4 \[ -{\frac{1}{8\,c}\ln \left ({1 \left ({\frac{-b{x}^{4}+a}{2\,{x}^{2}}}-{\frac{1}{x}\sqrt [4]{ab}\sqrt{-b{x}^{4}+a}}+\sqrt{ab} \right ) \left ({\frac{-b{x}^{4}+a}{2\,{x}^{2}}}+{\frac{1}{x}\sqrt [4]{ab}\sqrt{-b{x}^{4}+a}}+\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{ab}}}}-{\frac{1}{4\,c}\arctan \left ({\frac{1}{x}\sqrt{-b{x}^{4}+a}{\frac{1}{\sqrt [4]{ab}}}}+1 \right ){\frac{1}{\sqrt [4]{ab}}}}+{\frac{1}{4\,c}\arctan \left ( -{\frac{1}{x}\sqrt{-b{x}^{4}+a}{\frac{1}{\sqrt [4]{ab}}}}+1 \right ){\frac{1}{\sqrt [4]{ab}}}} \]

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. \[ \int \frac{\sqrt{-b x^{4} + a}}{b c x^{4} + a c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*x^4 + a)/(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 [A]  time = 0.586726, size = 500, normalized size = 4.31 \[ \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{2 \,{\left (2 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (-\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}}\right )}}{\sqrt{-b x^{4} + a} a c^{2} \sqrt{-\frac{1}{a b c^{4}}} - \sqrt{-b x^{4} + a} x^{2} +{\left (b x^{4} + a\right )} \sqrt{-\frac{1}{b}}}\right ) - \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (-\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + \sqrt{-b x^{4} + a} a c^{2} \sqrt{-\frac{1}{a b c^{4}}} - 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) + \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (-\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} - \sqrt{-b x^{4} + a} a c^{2} \sqrt{-\frac{1}{a b c^{4}}} - 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} - \sqrt{-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*x^4 + a)/(b*c*x^4 + a*c),x, algorithm="fricas")

[Out]

(1/4)^(1/4)*(-1/(a*b*c^4))^(1/4)*arctan(2*(2*(1/4)^(3/4)*a*b*c^3*x^3*(-1/(a*b*c^
4))^(3/4) + (1/4)^(1/4)*a*c*x*(-1/(a*b*c^4))^(1/4))/(sqrt(-b*x^4 + a)*a*c^2*sqrt
(-1/(a*b*c^4)) - sqrt(-b*x^4 + a)*x^2 + (b*x^4 + a)*sqrt(-1/b))) - 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) + s
qrt(-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*sqr
t(-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. \[ \frac{\int \frac{\sqrt{a - b x^{4}}}{a + b x^{4}}\, dx}{c} \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-b x^{4} + a}}{b c x^{4} + a c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*x^4 + a)/(b*c*x^4 + a*c),x, algorithm="giac")

[Out]

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