3.6.92 \(\int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx\) [592]

Optimal. Leaf size=50 \[ -\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 50, 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 = {52, 65, 223, 209} \begin {gather*} \frac {a \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{3/2}}-\frac {\sqrt {x} \sqrt {a-b x}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/Sqrt[a - b*x],x]

[Out]

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

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,
b, c, d, m, n, x]

Rule 209

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])

Rule 223

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,
b}, x] &&  !GtQ[a, 0]

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 56, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {x} \sqrt {a-b x}}{b}+\frac {a \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{(-b)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/Sqrt[a - b*x],x]

[Out]

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

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Maple [A]
time = 0.12, size = 70, normalized size = 1.40

method	result	size
default	\(-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\)	\(70\)
risch	\(-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^(1/2)*(-b*x+a)^(1/2)/b+1/2*a/b^(3/2)*(x*(-b*x+a))^(1/2)/x^(1/2)/(-b*x+a)^(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 = 0.49, size = 56, normalized size = 1.12 \begin {gather*} -\frac {a \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {3}{2}}} - \frac {\sqrt {-b x + a} a}{{\left (b^{2} - \frac {{\left (b x - a\right )} b}{x}\right )} \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.69, size = 93, normalized size = 1.86 \begin {gather*} \left [-\frac {a \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, \sqrt {-b x + a} b \sqrt {x}}{2 \, b^{2}}, -\frac {a \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + \sqrt {-b x + a} b \sqrt {x}}{b^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(a*sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) + 2*sqrt(-b*x + a)*b*sqrt(x))/b^2, -(a*s
qrt(b)*arctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) + sqrt(-b*x + a)*b*sqrt(x))/b^2]

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Sympy [C] Result contains complex when optimal does not.
time = 1.23, size = 121, normalized size = 2.42 \begin {gather*} \begin {cases} \frac {i \sqrt {a} \sqrt {x}}{b \sqrt {-1 + \frac {b x}{a}}} - \frac {i a \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {i x^{\frac {3}{2}}}{\sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {\sqrt {a} \sqrt {x} \sqrt {1 - \frac {b x}{a}}}{b} + \frac {a \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*sqrt(a)*sqrt(x)/(b*sqrt(-1 + b*x/a)) - I*a*acosh(sqrt(b)*sqrt(x)/sqrt(a))/b**(3/2) - I*x**(3/2)/(
sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (-sqrt(a)*sqrt(x)*sqrt(1 - b*x/a)/b + a*asin(sqrt(b)*sqrt(x)/sqrt(
a))/b**(3/2), True))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{-4,[1
,0,0]%%%}+%%%{4,[0,1,1]%%%}+%%%{4,[0,1,0]%%%}+%%%{-4,[0,0,1]%%%},0,%%%{6,[2,0,0]%%%}+%%%{-12,[1,1,1]%%%}+%%%{-
4,[1,1,0]%%%}+%%%{4,[

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Mupad [B]
time = 0.52, size = 47, normalized size = 0.94 \begin {gather*} \frac {2\,a\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a-b\,x}-\sqrt {a}}\right )}{b^{3/2}}-\frac {\sqrt {x}\,\sqrt {a-b\,x}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a - b*x)^(1/2),x)

[Out]

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

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