3.4.8 \(\int \frac {1}{\sqrt [4]{a+3 x^2} (2 a+3 x^2)} \, dx\) [308]

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

[Out]

-1/6*arctan(1/3*a^(3/4)*(1+(3*x^2+a)^(1/2)/a^(1/2))/x/(3*x^2+a)^(1/4)*3^(1/2))/a^(3/4)*3^(1/2)-1/6*arctanh(1/3
*a^(3/4)*(1-(3*x^2+a)^(1/2)/a^(1/2))/x/(3*x^2+a)^(1/4)*3^(1/2))/a^(3/4)*3^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 120, normalized size of antiderivative = 1.00, 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 = {406} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {a^{3/4} \left (\frac {\sqrt {a+3 x^2}}{\sqrt {a}}+1\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + 3*x^2)^(1/4)*(2*a + 3*x^2)),x]

[Out]

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

Rule 406

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b^2/a, 4]}, Simp[(-b/(2*a
*d*q))*ArcTan[(b + q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))], x] - Simp[(b/(2*a*d*q))*ArcTanh[(b - q^2*S
qrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))], 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+3 x^2} \left (2 a+3 x^2\right )} \, dx &=-\frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+3 x^2}}{\sqrt {a}}\right )}{\sqrt {3} x \sqrt [4]{a+3 x^2}}\right )}{2 \sqrt {3} a^{3/4}}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 121, normalized size = 1.01 \begin {gather*} \frac {-\tan ^{-1}\left (\frac {-3 x^2+2 \sqrt {a} \sqrt {a+3 x^2}}{2 \sqrt {3} \sqrt [4]{a} x \sqrt [4]{a+3 x^2}}\right )+\tanh ^{-1}\left (\frac {2 \sqrt {3} \sqrt [4]{a} x \sqrt [4]{a+3 x^2}}{3 x^2+2 \sqrt {a} \sqrt {a+3 x^2}}\right )}{4 \sqrt {3} a^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + 3*x^2)^(1/4)*(2*a + 3*x^2)),x]

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (3 x^{2}+a \right )^{\frac {1}{4}} \left (3 x^{2}+2 a \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x)

[Out]

int(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x, algorithm="maxima")

[Out]

integrate(1/((3*x^2 + 2*a)*(3*x^2 + a)^(1/4)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (89) = 178\).
time = 4.69, size = 286, normalized size = 2.38 \begin {gather*} \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (6 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} a^{3} \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{36}\right )^{\frac {1}{4}} \sqrt {3 \, x^{2} + a} a \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {-a \sqrt {-\frac {1}{a^{3}}}} - \left (\frac {1}{36}\right )^{\frac {1}{4}} {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}}\right )}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} + a} a^{2} x \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} + {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a^{2} \sqrt {-\frac {1}{a^{3}}} - 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a x \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} + {\left (3 \, x^{2} + a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) + \frac {1}{4} \, \left (\frac {1}{36}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {18 \, \left (\frac {1}{36}\right )^{\frac {3}{4}} \sqrt {3 \, x^{2} + a} a^{2} x \left (-\frac {1}{a^{3}}\right )^{\frac {3}{4}} - {\left (3 \, x^{2} + a\right )}^{\frac {1}{4}} a^{2} \sqrt {-\frac {1}{a^{3}}} - 3 \, \left (\frac {1}{36}\right )^{\frac {1}{4}} a x \left (-\frac {1}{a^{3}}\right )^{\frac {1}{4}} - {\left (3 \, x^{2} + a\right )}^{\frac {3}{4}}}{3 \, x^{2} + 2 \, a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x, algorithm="fricas")

[Out]

(1/36)^(1/4)*(-1/a^3)^(1/4)*arctan(2*(sqrt(1/2)*(6*(1/36)^(3/4)*a^3*(-1/a^3)^(3/4) + (1/36)^(1/4)*sqrt(3*x^2 +
a)*a*(-1/a^3)^(1/4))*sqrt(-a*sqrt(-1/a^3)) - (1/36)^(1/4)*(3*x^2 + a)^(1/4)*a*(-1/a^3)^(1/4))/x) - 1/4*(1/36)
^(1/4)*(-1/a^3)^(1/4)*log((18*(1/36)^(3/4)*sqrt(3*x^2 + a)*a^2*x*(-1/a^3)^(3/4) + (3*x^2 + a)^(1/4)*a^2*sqrt(-
1/a^3) - 3*(1/36)^(1/4)*a*x*(-1/a^3)^(1/4) + (3*x^2 + a)^(3/4))/(3*x^2 + 2*a)) + 1/4*(1/36)^(1/4)*(-1/a^3)^(1/
4)*log(-(18*(1/36)^(3/4)*sqrt(3*x^2 + a)*a^2*x*(-1/a^3)^(3/4) - (3*x^2 + a)^(1/4)*a^2*sqrt(-1/a^3) - 3*(1/36)^
(1/4)*a*x*(-1/a^3)^(1/4) - (3*x^2 + a)^(3/4))/(3*x^2 + 2*a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{a + 3 x^{2}} \cdot \left (2 a + 3 x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x**2+a)**(1/4)/(3*x**2+2*a),x)

[Out]

Integral(1/((a + 3*x**2)**(1/4)*(2*a + 3*x**2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+a)^(1/4)/(3*x^2+2*a),x, algorithm="giac")

[Out]

integrate(1/((3*x^2 + 2*a)*(3*x^2 + a)^(1/4)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (3\,x^2+2\,a\right )\,{\left (3\,x^2+a\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((2*a + 3*x^2)*(a + 3*x^2)^(1/4)),x)

[Out]

int(1/((2*a + 3*x^2)*(a + 3*x^2)^(1/4)), x)

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