3.1554 \(\int \frac{1}{\sqrt{\frac{b-b c}{d}+b x} \sqrt{c-d x}} \, dx\)

Optimal. Leaf size=42 \[ \frac{2 \sin ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{b (1-c)}{d}+b x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.0161989, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {63, 216} \[ \frac{2 \sin ^{-1}\left (\frac{\sqrt{d} \sqrt{\frac{b (1-c)}{d}+b x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[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 63

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 216

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*} \int \frac{1}{\sqrt{\frac{b-b c}{d}+b x} \sqrt{c-d x}} \, dx &=\frac{2 \operatorname{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]  time = 0.0501009, size = 67, normalized size = 1.6 \[ \frac{2 \sqrt{-d} \sqrt{-c+d x+1} \sinh ^{-1}\left (\frac{\sqrt{-d} \sqrt{c-d x}}{\sqrt{d}}\right )}{d^{3/2} \sqrt{\frac{b (-c+d x+1)}{d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-d]*Sqrt[1 - c + d*x]*ArcSinh[(Sqrt[-d]*Sqrt[c - d*x])/Sqrt[d]])/(d^(3/2)*Sqrt[(b*(1 - c + d*x))/d])

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Maple [B]  time = 0.023, size = 118, normalized size = 2.8 \begin{align*}{\sqrt{ \left ({\frac{b \left ( 1-c \right ) }{d}}+bx \right ) \left ( -dx+c \right ) }\arctan \left ({\sqrt{bd} \left ( x-{\frac{-b \left ( 1-c \right ) +bc}{2\,bd}} \right ){\frac{1}{\sqrt{-d{x}^{2}b+ \left ( -b \left ( 1-c \right ) +bc \right ) x+{\frac{b \left ( 1-c \right ) c}{d}}}}}} \right ){\frac{1}{\sqrt{{\frac{b \left ( 1-c \right ) }{d}}+bx}}}{\frac{1}{\sqrt{-dx+c}}}{\frac{1}{\sqrt{bd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.11584, size = 409, normalized size = 9.74 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \left (- \frac{c}{d} + x + \frac{1}{d}\right )} \sqrt{c - d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.0759, size = 90, normalized size = 2.14 \begin{align*} -\frac{2 \, b \log \left (-\sqrt{-b d} \sqrt{b x - \frac{b c - b}{d}} + \sqrt{-{\left (b x - \frac{b c - b}{d}\right )} b d + b^{2}}\right )}{\sqrt{-b d}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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