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

Optimal. Leaf size=87 \[ \frac{\sqrt [4]{a} c \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{a-b x^4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.137483, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt [4]{a} c \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{a-b x^4}}+\frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]

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

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Rubi in Sympy [A]  time = 14.9437, size = 78, normalized size = 0.9 \[ \frac{\sqrt [4]{a} c \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{\sqrt [4]{b} \sqrt{a - b x^{4}}} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} \right )}}{2 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.269044, size = 106, normalized size = 1.22 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}}-\frac{i c \sqrt{1-\frac{b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \sqrt{a-b x^4}} \]

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[-(Sqrt[b]/Sqrt[a])]*x], -1])/(Sqrt[-(Sqrt[b]/Sqrt[a])]
*Sqrt[a - b*x^4])

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Maple [A]  time = 0.006, size = 90, normalized size = 1. \[{c\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}}+{\frac{d}{2}\arctan \left ({{x}^{2}\sqrt{b}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)/(-b*x^4+a)^(1/2),x)

[Out]

c/(1/a^(1/2)*b^(1/2))^(1/2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2)
)^(1/2)/(-b*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)+1/2*d*arctan(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 = 2.99458, size = 95, normalized size = 1.09 \[ d \left (\begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{for}\: \left |{\frac{b x^{4}}{a}}\right | > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{otherwise} \end{cases}\right ) + \frac{c x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*Piecewise((-I*acosh(sqrt(b)*x**2/sqrt(a))/(2*sqrt(b)), Abs(b*x**4/a) > 1), (as
in(sqrt(b)*x**2/sqrt(a))/(2*sqrt(b)), True)) + c*x*gamma(1/4)*hyper((1/4, 1/2),
(5/4,), b*x**4*exp_polar(2*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)