3.1558 \(\int \frac{1}{\sqrt{a-b x} \sqrt{c+d x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0277071, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {63, 217, 203} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a-b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTan[(Sqrt[d]*Sqrt[a - b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(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 217

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]

Rule 203

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

Rubi steps

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

Mathematica [B]  time = 0.0787922, size = 103, normalized size = 2.4 \[ \frac{2 \sqrt{-b} \sqrt{-a d-b c} \sqrt{\frac{b (c+d x)}{a d+b c}} \sin ^{-1}\left (\frac{\sqrt{-b} \sqrt{d} \sqrt{a-b x}}{\sqrt{b} \sqrt{-a d-b c}}\right )}{b^{3/2} \sqrt{d} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.007, size = 84, normalized size = 2. \begin{align*}{\sqrt{ \left ( -bx+a \right ) \left ( dx+c \right ) }\arctan \left ({\sqrt{bd} \left ( x-{\frac{ad-bc}{2\,bd}} \right ){\frac{1}{\sqrt{-d{x}^{2}b+ \left ( ad-bc \right ) x+ac}}}} \right ){\frac{1}{\sqrt{-bx+a}}}{\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*x+a)^(1/2)/(d*x+c)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.12206, size = 417, normalized size = 9.7 \begin{align*} \left [-\frac{\sqrt{-b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2} - 4 \,{\left (2 \, b d x + b c - a d\right )} \sqrt{-b d} \sqrt{-b x + a} \sqrt{d x + c} + 8 \,{\left (b^{2} c d - a b d^{2}\right )} x\right )}{2 \, b d}, -\frac{\sqrt{b d} \arctan \left (\frac{{\left (2 \, b d x + b c - a d\right )} \sqrt{b d} \sqrt{-b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} d^{2} x^{2} - a b c d +{\left (b^{2} c d - a b d^{2}\right )} x\right )}}\right )}{b d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 - 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b*c - a*d)*sqrt(-b*d)*sqrt(-
b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d - a*b*d^2)*x)/(b*d), -sqrt(b*d)*arctan(1/2*(2*b*d*x + b*c - a*d)*sqrt(b*d)
*sqrt(-b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 - a*b*c*d + (b^2*c*d - a*b*d^2)*x))/(b*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.0829, size = 73, normalized size = 1.7 \begin{align*} \frac{2 \, b \log \left ({\left | -\sqrt{-b d} \sqrt{-b x + a} + \sqrt{b^{2} c +{\left (b x - a\right )} b d + a b d} \right |}\right )}{\sqrt{-b d}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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