3.4.12 \(\int \frac {\sqrt {x}}{a-b x^2} \, dx\) [312]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {335, 304, 211, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}} \end {gather*}

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 211

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

Rule 214

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

Rule 304

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 335

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {x}}{a-b x^2} \, dx &=2 \text {Subst}\left (\int \frac {x^2}{a-b x^4} \, dx,x,\sqrt {x}\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a}-\sqrt {b} x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {b}}-\frac {\text {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*}

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Mathematica [A]
time = 0.04, size = 48, normalized size = 0.83 \begin {gather*} \frac {-\tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}} \end {gather*}

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 = 0.03, size = 58, normalized size = 1.00

method	result	size
derivativedivides	\(-\frac {2 \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sqrt {x}-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(58\)
default	\(-\frac {2 \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sqrt {x}-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (38) = 76\).
time = 0.50, size = 86, normalized size = 1.48 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\log \left (\frac {\sqrt {b} \sqrt {x} - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} \sqrt {x} + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (38) = 76\).
time = 1.54, size = 117, normalized size = 2.02 \begin {gather*} 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 {gather*}

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 = 1.00, size = 92, normalized size = 1.59 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2}{b \sqrt {x}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a} & \text {for}\: b = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt [4]{\frac {a}{b}} \right )}}{2 b \sqrt [4]{\frac {a}{b}}} + \frac {\log {\left (\sqrt {x} + \sqrt [4]{\frac {a}{b}} \right )}}{2 b \sqrt [4]{\frac {a}{b}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{\frac {a}{b}}} \right )}}{b \sqrt [4]{\frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

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/(b*sqrt(x)), Eq(a, 0)), (2*x**(3/2)/(3*a), Eq(b, 0)), (-log(s
qrt(x) - (a/b)**(1/4))/(2*b*(a/b)**(1/4)) + log(sqrt(x) + (a/b)**(1/4))/(2*b*(a/b)**(1/4)) - atan(sqrt(x)/(a/b
)**(1/4))/(b*(a/b)**(1/4)), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (38) = 76\).
time = 0.96, size = 194, normalized size = 3.34 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.08, size = 33, normalized size = 0.57 \begin {gather*} -\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{a^{1/4}}\right )-\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{a^{1/4}\,b^{3/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a - b*x^2),x)

[Out]

-(atan((b^(1/4)*x^(1/2))/a^(1/4)) - atanh((b^(1/4)*x^(1/2))/a^(1/4)))/(a^(1/4)*b^(3/4))

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