3.318 \(\int \frac{1}{(-2 a+b x^2) \sqrt [4]{-a+b x^2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}} \]

[Out]

-ArcTan[(Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*(-a + b*x^2)^(1/4))]/(2*Sqrt[2]*a^(3/4)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/(S
qrt[2]*a^(1/4)*(-a + b*x^2)^(1/4))]/(2*Sqrt[2]*a^(3/4)*Sqrt[b])

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Rubi [A]  time = 0.0193968, antiderivative size = 101, 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, Rules used = {398} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[1/((-2*a + b*x^2)*(-a + b*x^2)^(1/4)),x]

[Out]

-ArcTan[(Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*(-a + b*x^2)^(1/4))]/(2*Sqrt[2]*a^(3/4)*Sqrt[b]) - ArcTanh[(Sqrt[b]*x)/(S
qrt[2]*a^(1/4)*(-a + b*x^2)^(1/4))]/(2*Sqrt[2]*a^(3/4)*Sqrt[b])

Rule 398

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b^2/a), 4]}, Simp[(b*Ar
cTan[(q*x)/(Sqrt[2]*(a + b*x^2)^(1/4))])/(2*Sqrt[2]*a*d*q), x] + Simp[(b*ArcTanh[(q*x)/(Sqrt[2]*(a + b*x^2)^(1
/4))])/(2*Sqrt[2]*a*d*q), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]

Rubi steps

\begin{align*} \int \frac{1}{\left (-2 a+b x^2\right ) \sqrt [4]{-a+b x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{-a+b x^2}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}\\ \end{align*}

Mathematica [C]  time = 0.156617, size = 163, normalized size = 1.61 \[ -\frac{6 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )}{\left (2 a-b x^2\right ) \sqrt [4]{b x^2-a} \left (b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )\right )+6 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((-2*a + b*x^2)*(-a + b*x^2)^(1/4)),x]

[Out]

(-6*a*x*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)])/((2*a - b*x^2)*(-a + b*x^2)^(1/4)*(6*a*AppellF1[
1/2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)] + b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (b*x^2)/a, (b*x^2)/(2*a)] +
AppellF1[3/2, 5/4, 1, 5/2, (b*x^2)/a, (b*x^2)/(2*a)])))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{2}-2\,a}{\frac{1}{\sqrt [4]{b{x}^{2}-a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2-2*a)/(b*x^2-a)^(1/4),x)

[Out]

int(1/(b*x^2-2*a)/(b*x^2-a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} - a\right )}^{\frac{1}{4}}{\left (b x^{2} - 2 \, a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2-2*a)/(b*x^2-a)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 - a)^(1/4)*(b*x^2 - 2*a)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2-2*a)/(b*x^2-a)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- 2 a + b x^{2}\right ) \sqrt [4]{- a + b x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2-2*a)/(b*x**2-a)**(1/4),x)

[Out]

Integral(1/((-2*a + b*x**2)*(-a + b*x**2)**(1/4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} - a\right )}^{\frac{1}{4}}{\left (b x^{2} - 2 \, a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2-2*a)/(b*x^2-a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 - a)^(1/4)*(b*x^2 - 2*a)), x)