3.1551 \(\int \frac{1}{\sqrt{\frac{-b+b c}{d}+b x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=43 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{b x-\frac{b (1-c)}{d}}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.0153134, antiderivative size = 43, 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, 215} \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{b x-\frac{b (1-c)}{d}}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcSinh[(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 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[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 \sinh ^{-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 = 0.0250398, size = 41, normalized size = 0.95 \[ \frac{2 \sqrt{c+d x-1} \sinh ^{-1}\left (\sqrt{c+d x-1}\right )}{d \sqrt{\frac{b (c+d x-1)}{d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.014, size = 100, normalized size = 2.3 \begin{align*}{\sqrt{ \left ( bx+{\frac{b \left ( c-1 \right ) }{d}} \right ) \left ( dx+c \right ) }\ln \left ({ \left ({\frac{b \left ( c-1 \right ) }{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( b \left ( c-1 \right ) +bc \right ) x+{\frac{b \left ( c-1 \right ) c}{d}}} \right ){\frac{1}{\sqrt{bx+{\frac{b \left ( c-1 \right ) }{d}}}}}{\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*x+b*(c-1)/d)*(d*x+c))^(1/2)/(b*x+b*(c-1)/d)^(1/2)/(d*x+c)^(1/2)*ln((1/2*b*(c-1)+1/2*b*c+b*d*x)/(b*d)^(1/2)
+(d*x^2*b+(b*(c-1)+b*c)*x+b*(c-1)/d*c)^(1/2))/(b*d)^(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 = 2.10981, size = 405, normalized size = 9.42 \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)*sqr
t((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)*s
qrt((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 = 1.06311, size = 84, normalized size = 1.95 \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))