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

Optimal. Leaf size=120 \[ -\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0214657, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {397} \[ -\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 397

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b^2/a, 4]}, -Simp[(b*ArcT
an[(b + q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))])/(2*a*d*q), x] - Simp[(b*ArcTanh[(b - q^2*Sqrt[a + b*x
^2])/(q^3*x*(a + b*x^2)^(1/4))])/(2*a*d*q), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a
]

Rubi steps

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

Mathematica [C]  time = 0.148803, size = 165, normalized size = 1.38 \[ \frac{6 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right ) \left (6 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )-b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*a*x*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -(b*x^2)/(2*a)])/((a + b*x^2)^(1/4)*(2*a + b*x^2)*(6*a*AppellF
1[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 = 0.044, 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+a)^(1/4)/(b*x^2+2*a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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