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 = 0.0340864, 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, Rules used = {329, 298, 205, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}} \]

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))

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0158488, size = 48, normalized size = 0.83 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-\tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a - b*x^2),x]

[Out]

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

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Maple [A]  time = 0.007, size = 66, normalized size = 1.1 \begin{align*} -{\frac{1}{b}\arctan \left ({\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{1}{2\,b}\ln \left ({ \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}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.38314, size = 319, normalized size = 5.5 \begin{align*} 2 \, \left (\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{a b \sqrt{\frac{1}{a b^{3}}} + x} b \left (\frac{1}{a b^{3}}\right )^{\frac{1}{4}} - b \sqrt{x} \left (\frac{1}{a b^{3}}\right )^{\frac{1}{4}}\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(1/(a*b^3))^(1/4)*arctan(sqrt(a*b*sqrt(1/(a*b^3)) + x)*b*(1/(a*b^3))^(1/4) - b*sqrt(x)*(1/(a*b^3))^(1/4)) +
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 = 4.39652, size = 128, normalized size = 2.21 \begin{align*} \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^{2} \left (\frac{1}{b}\right )^{\frac{5}{4}}} + \frac{\log{\left (\sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 \sqrt [4]{a} b^{2} \left (\frac{1}{b}\right )^{\frac{5}{4}}} - \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{\sqrt [4]{a} b^{2} \left (\frac{1}{b}\right )^{\frac{5}{4}}} & \text{otherwise} \end{cases} \end{align*}

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**2*(1/b)**(5/4)) + log(a**(1/4)*(1/b)**(1/4) + sqrt(x))/(2*a**(
1/4)*b**2*(1/b)**(5/4)) - atan(sqrt(x)/(a**(1/4)*(1/b)**(1/4)))/(a**(1/4)*b**2*(1/b)**(5/4)), True))

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Giac [B]  time = 1.88666, size = 262, normalized size = 4.52 \begin{align*} \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}} \log \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}} \log \left (-\sqrt{2} \sqrt{x} \left (-\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{-\frac{a}{b}}\right )}{4 \, a b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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