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

Optimal. Leaf size=89 \[ \frac{\sqrt [4]{a} c \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{b x^4-a}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{b x^4-a}}\right )}{2 \sqrt{b}} \]

[Out]

(d*ArcTanh[(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 = 0.137626, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{\sqrt [4]{a} c \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt{b x^4-a}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{b x^4-a}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

(d*ArcTanh[(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 = 15.5759, size = 78, normalized size = 0.88 \[ \frac{\sqrt [4]{a} c \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{\sqrt [4]{b} \sqrt{- a + b x^{4}}} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{- a + b x^{4}}} \right )}}{2 \sqrt{b}} \]

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

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Mathematica [C]  time = 0.246353, size = 108, normalized size = 1.21 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{b x^4-a}}\right )}{2 \sqrt{b}}-\frac{i c \sqrt{1-\frac{b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \sqrt{b x^4-a}} \]

Antiderivative was successfully verified.

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

[Out]

(d*ArcTanh[(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 = 0.023, size = 95, normalized size = 1.1 \[{c\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{-{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{-{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}-a}}}}+{\frac{d}{2}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}-a} \right ){\frac{1}{\sqrt{b}}}} \]

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*ln(b^(1
/2)*x^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 = 3.01345, size = 90, normalized size = 1.01 \[ d \left (\begin{cases} \frac{\operatorname{acosh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{for}\: \left |{\frac{b x^{4}}{a}}\right | > 1 \\- \frac{i \operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{otherwise} \end{cases}\right ) - \frac{i c x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4}}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} \]

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