3.4.19 \(\int \frac {1}{(-2 a-b x^2) \sqrt [4]{-a-b x^2}} \, dx\) [319]

Optimal. Leaf size=103 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {407} \begin {gather*} -\frac {\text {ArcTan}\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 407

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b^2/a, 4]}, Simp[(b/(2*S
qrt[2]*a*d*q))*ArcTan[q*(x/(Sqrt[2]*(a + b*x^2)^(1/4)))], x] + Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTanh[q*(x/(Sqrt[2
]*(a + b*x^2)^(1/4)))], 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*}

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Mathematica [A]
time = 0.13, size = 90, normalized size = 0.87 \begin {gather*} -\frac {-\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}{\sqrt {b} x}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{-a-b x^2}}{\sqrt {b} x}\right )}{2 \sqrt {2} a^{3/4} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-b \,x^{2}-2 a \right ) \left (-b \,x^{2}-a \right )^{\frac {1}{4}}}\, dx\]

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 = 0.00, 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/(-b*x^2-2*a)/(-b*x^2-a)^(1/4),x, algorithm="maxima")

[Out]

-integrate(1/((b*x^2 + 2*a)*(-b*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. 350 vs. \(2 (73) = 146\).
time = 25.81, size = 350, normalized size = 3.40 \begin {gather*} -\left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {-b x^{2} - a} a \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}}\right )} \sqrt {-a b \sqrt {\frac {1}{a^{3} b^{2}}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (-b x^{2} - a\right )}^{\frac {1}{4}} a \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}}\right )}}{x}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} - a} a^{2} b^{2} x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} - a\right )}^{\frac {1}{4}} a^{2} b \sqrt {\frac {1}{a^{3} b^{2}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} - {\left (-b x^{2} - a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} - a} a^{2} b^{2} x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (-b x^{2} - a\right )}^{\frac {1}{4}} a^{2} b \sqrt {\frac {1}{a^{3} b^{2}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} - a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) \end {gather*}

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]

-(1/4)^(1/4)*(1/(a^3*b^2))^(1/4)*arctan(2*(sqrt(1/2)*(2*(1/4)^(3/4)*a^3*b*(1/(a^3*b^2))^(3/4) - (1/4)^(1/4)*sq
rt(-b*x^2 - a)*a*(1/(a^3*b^2))^(1/4))*sqrt(-a*b*sqrt(1/(a^3*b^2))) - (1/4)^(1/4)*(-b*x^2 - a)^(1/4)*a*(1/(a^3*
b^2))^(1/4))/x) + 1/4*(1/4)^(1/4)*(1/(a^3*b^2))^(1/4)*log(-(2*(1/4)^(3/4)*sqrt(-b*x^2 - a)*a^2*b^2*x*(1/(a^3*b
^2))^(3/4) + (-b*x^2 - a)^(1/4)*a^2*b*sqrt(1/(a^3*b^2)) - (1/4)^(1/4)*a*b*x*(1/(a^3*b^2))^(1/4) - (-b*x^2 - a)
^(3/4))/(b*x^2 + 2*a)) - 1/4*(1/4)^(1/4)*(1/(a^3*b^2))^(1/4)*log((2*(1/4)^(3/4)*sqrt(-b*x^2 - a)*a^2*b^2*x*(1/
(a^3*b^2))^(3/4) - (-b*x^2 - a)^(1/4)*a^2*b*sqrt(1/(a^3*b^2)) - (1/4)^(1/4)*a*b*x*(1/(a^3*b^2))^(1/4) + (-b*x^
2 - a)^(3/4))/(b*x^2 + 2*a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{2 a \sqrt [4]{- a - b x^{2}} + b x^{2} \sqrt [4]{- a - b x^{2}}}\, dx \end {gather*}

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

[Out]

integrate(-1/((b*x^2 + 2*a)*(-b*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 (-b\,x^2-a\right )}^{1/4}\,\left (b\,x^2+2\,a\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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