\(\int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx\) [1554]

Optimal result

Integrand size = 29, antiderivative size = 42 \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\frac {2 \arcsin \left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]

[Out]

2*arcsin(d^(1/2)*(b*(1-c)/d+b*x)^(1/2)/b^(1/2))/b^(1/2)/d^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {65, 222} \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\frac {2 \arcsin \left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \]

[In]

Int[1/(Sqrt[(b - b*c)/d + b*x]*Sqrt[c - d*x]),x]

[Out]

(2*ArcSin[(Sqrt[d]*Sqrt[(b*(1 - c))/d + b*x])/Sqrt[b]])/(Sqrt[b]*Sqrt[d])

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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {b-b c}{b}-\frac {d x^2}{b}}} \, dx,x,\sqrt {\frac {b-b c}{d}+b x}\right )}{b} \\ & = \frac {2 \sin ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {b (1-c)}{d}+b x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {d}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\frac {2 \sqrt {1-c+d x} \arctan \left (\frac {\sqrt {1-c+d x}}{\sqrt {c-d x}}\right )}{d \sqrt {\frac {b (1-c+d x)}{d}}} \]

[In]

Integrate[1/(Sqrt[(b - b*c)/d + b*x]*Sqrt[c - d*x]),x]

[Out]

(2*Sqrt[1 - c + d*x]*ArcTan[Sqrt[1 - c + d*x]/Sqrt[c - d*x]])/(d*Sqrt[(b*(1 - c + d*x))/d])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(32)=64\).

Time = 0.60 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.60

method	result	size
default	\(\frac {\sqrt {\left (b x -\frac {b \left (c -1\right )}{d}\right ) \left (-d x +c \right )}\, \arctan \left (\frac {\sqrt {b d}\, \left (x -\frac {b \left (c -1\right )+b c}{2 b d}\right )}{\sqrt {-b d \,x^{2}+\left (b \left (c -1\right )+b c \right ) x -\frac {b \left (c -1\right ) c}{d}}}\right )}{\sqrt {b x -\frac {b \left (c -1\right )}{d}}\, \sqrt {-d x +c}\, \sqrt {b d}}\)	\(109\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (31) = 62\).

Time = 0.24 (sec) , antiderivative size = 176, normalized size of antiderivative = 4.19 \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\left [-\frac {\sqrt {-b d} \log \left (8 \, b d^{2} x^{2} + 8 \, b c^{2} - 8 \, {\left (2 \, b c - b\right )} d x - 4 \, \sqrt {-b d} {\left (2 \, d x - 2 \, c + 1\right )} \sqrt {-d x + c} \sqrt {\frac {b d x - b c + b}{d}} - 8 \, b c + b\right )}{2 \, b d}, -\frac {\sqrt {b d} \arctan \left (\frac {\sqrt {b d} {\left (2 \, d x - 2 \, c + 1\right )} \sqrt {-d x + c} \sqrt {\frac {b d x - b c + b}{d}}}{2 \, {\left (b d^{2} x^{2} + b c^{2} - {\left (2 \, b c - b\right )} d x - b c\right )}}\right )}{b d}\right ] \]

[In]

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

[Out]

[-1/2*sqrt(-b*d)*log(8*b*d^2*x^2 + 8*b*c^2 - 8*(2*b*c - b)*d*x - 4*sqrt(-b*d)*(2*d*x - 2*c + 1)*sqrt(-d*x + c)
*sqrt((b*d*x - b*c + b)/d) - 8*b*c + b)/(b*d), -sqrt(b*d)*arctan(1/2*sqrt(b*d)*(2*d*x - 2*c + 1)*sqrt(-d*x + c
)*sqrt((b*d*x - b*c + b)/d)/(b*d^2*x^2 + b*c^2 - (2*b*c - b)*d*x - b*c))/(b*d)]

Sympy [F]

\[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\int \frac {1}{\sqrt {b \left (- \frac {c}{d} + x + \frac {1}{d}\right )} \sqrt {c - d x}}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(b*(-c/d + x + 1/d))*sqrt(c - d*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(2*c-1>0)', see `assume?` for m
ore details)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (31) = 62\).

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=-\frac {2 \, {\left | b \right |} \log \left (-\sqrt {b d^{2} x - b c d + b d} \sqrt {-b d} + \sqrt {b^{2} d^{2} - {\left (b d^{2} x - b c d + b d\right )} b d}\right )}{\sqrt {-b d} b} \]

[In]

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

[Out]

-2*abs(b)*log(-sqrt(b*d^2*x - b*c*d + b*d)*sqrt(-b*d) + sqrt(b^2*d^2 - (b*d^2*x - b*c*d + b*d)*b*d))/(sqrt(-b*
d)*b)

Mupad [B] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\sqrt {\frac {b-b c}{d}+b x} \sqrt {c-d x}} \, dx=-\frac {4\,\mathrm {atan}\left (-\frac {d\,\left (\sqrt {\frac {b-b\,c}{d}+b\,x}-\sqrt {\frac {b-b\,c}{d}}\right )}{\sqrt {b\,d}\,\left (\sqrt {c-d\,x}-\sqrt {c}\right )}\right )}{\sqrt {b\,d}} \]

[In]

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

[Out]

-(4*atan(-(d*(((b - b*c)/d + b*x)^(1/2) - ((b - b*c)/d)^(1/2)))/((b*d)^(1/2)*((c - d*x)^(1/2) - c^(1/2)))))/(b
*d)^(1/2)