∫ 1______√ x√____ a − bx dx [593]

Optimal. Leaf size=29 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}} \]

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

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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {65, 223, 209} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}} \end {gather*}

(2*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*x]])/Sqrt[b]

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

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

FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,

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

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Mathematica [A]
time = 0.04, size = 35, normalized size = 1.21 \begin {gather*} -\frac {2 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{\sqrt {-b}} \end {gather*}

(-2*Log[-(Sqrt[-b]*Sqrt[x]) + Sqrt[a - b*x]])/Sqrt[-b]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.20, size = 44, normalized size = 1.52 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {-2 I \text {ArcCosh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]}{\sqrt {b}},\text {Abs}\left [\frac {b x}{a}\right ]>1\right \}\right \},\frac {2 \text {ArcSin}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]}{\sqrt {b}}\right ] \end {gather*}

mathics('Integrate[1/(Sqrt[x]*Sqrt[a - b*x]),x]')

Piecewise[{{-2 I ArcCosh[Sqrt[b] Sqrt[x] / Sqrt[a]] / Sqrt[b], Abs[b x / a] > 1}}, 2 ArcSin[Sqrt[b] Sqrt[x] /

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(21)=42\).
time = 0.11, size = 51, normalized size = 1.76


method	result	size

default	\(\frac {\sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{\sqrt {x}\, \sqrt {-b x +a}\, \sqrt {b}}\)	\(51\)

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

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

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Maxima [A]
time = 0.37, size = 21, normalized size = 0.72 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \end {gather*}

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

-2*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)))/sqrt(b)

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Fricas [A]
time = 0.31, size = 57, normalized size = 1.97 \begin {gather*} \left [-\frac {\sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right )}{b}, -\frac {2 \, \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}}\right ] \end {gather*}

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

[-sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a)/b, -2*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x)))/

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Sympy [A]
time = 0.54, size = 54, normalized size = 1.86 \begin {gather*} \begin {cases} - \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {2 \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \end {gather*}

Piecewise((-2*I*acosh(sqrt(b)*sqrt(x)/sqrt(a))/sqrt(b), Abs(b*x/a) > 1), (2*asin(sqrt(b)*sqrt(x)/sqrt(a))/sqrt

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Giac [A]
time = 0.00, size = 36, normalized size = 1.24 \begin {gather*} -\frac {2 \ln \left |\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right |}{\sqrt {-b}} \end {gather*}

-2*log(abs(-sqrt(-b)*sqrt(x) + sqrt(-b*x + a)))/sqrt(-b)

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Mupad [B]
time = 0.03, size = 27, normalized size = 0.93 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {\sqrt {a-b\,x}-\sqrt {a}}{\sqrt {b}\,\sqrt {x}}\right )}{\sqrt {b}} \end {gather*}

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

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3.6.93 \(\int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx\) [593]