3.312 \(\int \frac{\sqrt{x}}{a-b x^2} \, dx\)

Optimal. Leaf size=58 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.0895296, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a - b*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.6884, size = 53, normalized size = 0.91 \[ - \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{\sqrt [4]{a} b^{\frac{3}{4}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{\sqrt [4]{a} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(-b*x**2+a),x)

[Out]

-atan(b**(1/4)*sqrt(x)/a**(1/4))/(a**(1/4)*b**(3/4)) + atanh(b**(1/4)*sqrt(x)/a*
*(1/4))/(a**(1/4)*b**(3/4))

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Mathematica [A]  time = 0.0323244, size = 73, normalized size = 1.26 \[ -\frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} \sqrt{x}\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} \sqrt{x}\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(a - b*x^2),x]

[Out]

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

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Maple [A]  time = 0.012, size = 66, normalized size = 1.1 \[ -{\frac{1}{b}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{1}{2\,b}\ln \left ({1 \left ( \sqrt{x}+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( \sqrt{x}-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(-b*x^2+a),x)

[Out]

-1/b/(a/b)^(1/4)*arctan(x^(1/2)/(a/b)^(1/4))+1/2/b/(a/b)^(1/4)*ln((x^(1/2)+(a/b)
^(1/4))/(x^(1/2)-(a/b)^(1/4)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x)/(b*x^2 - a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255073, size = 149, normalized size = 2.57 \[ 2 \, \left (\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{a b^{2} \left (\frac{1}{a b^{3}}\right )^{\frac{3}{4}}}{\sqrt{a b \sqrt{\frac{1}{a b^{3}}} + x} + \sqrt{x}}\right ) + \frac{1}{2} \, \left (\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \log \left (a b^{2} \left (\frac{1}{a b^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) - \frac{1}{2} \, \left (\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \log \left (-a b^{2} \left (\frac{1}{a b^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x)/(b*x^2 - a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 17.5813, size = 128, normalized size = 2.21 \[ \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{3}{2}}}{3 a} & \text{for}\: b = 0 \\\frac{2}{b \sqrt{x}} & \text{for}\: a = 0 \\- \frac{\log{\left (- \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 \sqrt [4]{a} b^{3} \left (\frac{1}{b}\right )^{\frac{9}{4}}} + \frac{\log{\left (\sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 \sqrt [4]{a} b^{3} \left (\frac{1}{b}\right )^{\frac{9}{4}}} - \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{\sqrt [4]{a} b^{3} \left (\frac{1}{b}\right )^{\frac{9}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(-b*x**2+a),x)

[Out]

Piecewise((zoo/sqrt(x), Eq(a, 0) & Eq(b, 0)), (2*x**(3/2)/(3*a), Eq(b, 0)), (2/(
b*sqrt(x)), Eq(a, 0)), (-log(-a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(1/4)*b**3*
(1/b)**(9/4)) + log(a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(1/4)*b**3*(1/b)**(9/
4)) - atan(sqrt(x)/(a**(1/4)*(1/b)**(1/4)))/(a**(1/4)*b**3*(1/b)**(9/4)), True))

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GIAC/XCAS [A]  time = 0.218923, size = 262, normalized size = 4.52 \[ \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{3}} + \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{3}} - \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (-\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{-\frac{a}{b}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (-\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{-\frac{a}{b}}\right )}{4 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(x)/(b*x^2 - a),x, algorithm="giac")

[Out]

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