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

Optimal. Leaf size=127 \[ \frac{c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{-a-b x^4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{-a-b x^4}}\right )}{2 \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.156468, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{c \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{-a-b x^4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{-a-b x^4}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.8655, size = 112, normalized size = 0.88 \[ \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{- a - b x^{4}}} \right )}}{2 \sqrt{b}} + \frac{c \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt{- a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*atan(sqrt(b)*x**2/sqrt(-a - b*x**4))/(2*sqrt(b)) + c*sqrt((a + b*x**4)/(sqrt(a
) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*elliptic_f(2*atan(b**(1/4)*x/a**(
1/4)), 1/2)/(2*a**(1/4)*b**(1/4)*sqrt(-a - b*x**4))

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Mathematica [C]  time = 0.268889, size = 113, normalized size = 0.89 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{-a-b x^4}}\right )}{2 \sqrt{b}}-\frac{i c \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{-a-b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(d*ArcTan[(Sqrt[b]*x^2)/Sqrt[-a - b*x^4]])/(2*Sqrt[b]) - (I*c*Sqrt[1 + (b*x^4)/a
]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[b])/Sqrt[
a]]*Sqrt[-a - b*x^4])

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Maple [C]  time = 0.023, size = 101, normalized size = 0.8 \[{c\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{-i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{-i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}-a}}}}+{\frac{d}{2}\arctan \left ({{x}^{2}\sqrt{b}{\frac{1}{\sqrt{-b{x}^{4}-a}}}} \right ){\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(-b*x^4-a)^(1/2),x)

[Out]

c/(-I/a^(1/2)*b^(1/2))^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1-I/a^(1/2)*b^(1/2
)*x^2)^(1/2)/(-b*x^4-a)^(1/2)*EllipticF(x*(-I/a^(1/2)*b^(1/2))^(1/2),I)+1/2*d*ar
ctan(x^2*b^(1/2)/(-b*x^4-a)^(1/2))/b^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x + c}{\sqrt{-b x^{4} - a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.92057, size = 66, normalized size = 0.52 \[ - \frac{i d \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} - \frac{i c x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*d*asinh(sqrt(b)*x**2/sqrt(a))/(2*sqrt(b)) - I*c*x*gamma(1/4)*hyper((1/4, 1/2)
, (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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