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

Optimal. Leaf size=103 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c} \]

[Out]

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

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Rubi [A]  time = 0.141018, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.7375, size = 94, normalized size = 0.91 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a + b x^{4}}} \right )}}{4 \sqrt [4]{a} \sqrt [4]{b} c} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a + b x^{4}}} \right )}}{4 \sqrt [4]{a} \sqrt [4]{b} c} \]

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]

sqrt(2)*atan(sqrt(2)*a**(1/4)*b**(1/4)*x/sqrt(a + b*x**4))/(4*a**(1/4)*b**(1/4)*
c) + sqrt(2)*atanh(sqrt(2)*a**(1/4)*b**(1/4)*x/sqrt(a + b*x**4))/(4*a**(1/4)*b**
(1/4)*c)

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Mathematica [C]  time = 0.242136, size = 155, normalized size = 1.5 \[ \frac{5 a x \sqrt{a+b x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )}{c \left (a-b x^4\right ) \left (2 b x^4 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )+F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )\right )+5 a F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )\right )} \]

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 = 0.024, size = 103, normalized size = 1. \[ -{\frac{\sqrt{2}}{4\,c}\arctan \left ({\frac{\sqrt{2}}{2\,x}\sqrt{b{x}^{4}+a}{\frac{1}{\sqrt [4]{ab}}}} \right ){\frac{1}{\sqrt [4]{ab}}}}+{\frac{\sqrt{2}}{8\,c}\ln \left ({1 \left ({\frac{\sqrt{2}}{2\,x}\sqrt{b{x}^{4}+a}}+\sqrt [4]{ab} \right ) \left ({\frac{\sqrt{2}}{2\,x}\sqrt{b{x}^{4}+a}}-\sqrt [4]{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{ab}}}} \]

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]

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

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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 = 0.592124, size = 450, normalized size = 4.37 \[ -\left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{2 \,{\left (2 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} - \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}}\right )}}{\sqrt{b x^{4} + a}{\left (a c^{2} \sqrt{\frac{1}{a b c^{4}}} - x^{2}\right )} - \frac{b x^{4} - a}{\sqrt{b}}}\right ) + \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a}{\left (a c^{2} \sqrt{\frac{1}{a b c^{4}}} + x^{2}\right )}}{b x^{4} - a}\right ) - \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} - \sqrt{b x^{4} + a}{\left (a c^{2} \sqrt{\frac{1}{a b c^{4}}} + x^{2}\right )}}{b x^{4} - a}\right ) \]

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)) - x^2) - (b*x^4 - a)/sqrt(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) + 2*(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)) + 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) + 2*(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)) + 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)