3.1140 \(\int \frac{1}{\sqrt{a+a x} \sqrt{c-c x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0239438, 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} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )}{\sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0163171, size = 47, normalized size = 1.09 \[ \frac{2 \sqrt{x+1} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x+1}}{\sqrt{c-c x}}\right )}{\sqrt{c} \sqrt{a (x+1)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 57, normalized size = 1.3 \begin{align*}{\sqrt{ \left ( -cx+c \right ) \left ( ax+a \right ) }\arctan \left ({x\sqrt{ac}{\frac{1}{\sqrt{-ac{x}^{2}+ac}}}} \right ){\frac{1}{\sqrt{ax+a}}}{\frac{1}{\sqrt{-cx+c}}}{\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.66179, size = 239, normalized size = 5.56 \begin{align*} \left [-\frac{\sqrt{-a c} \log \left (2 \, a c x^{2} - 2 \, \sqrt{-a c} \sqrt{a x + a} \sqrt{-c x + c} x - a c\right )}{2 \, a c}, -\frac{\sqrt{a c} \arctan \left (\frac{\sqrt{a c} \sqrt{a x + a} \sqrt{-c x + c} x}{a c x^{2} - a c}\right )}{a c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 3.1786, size = 85, normalized size = 1.98 \begin{align*} - \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{a} \sqrt{c}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{a} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), x**(-2))/(4*pi**(3/2)*sqrt(a)*sqrt(
c)) + meijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ()), ((-1/4, 1/4), (-1/2, 0, 0, 0)), exp_polar(-2*I*pi)/x**2)/(4*
pi**(3/2)*sqrt(a)*sqrt(c))

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Giac [A]  time = 1.14295, size = 66, normalized size = 1.53 \begin{align*} -\frac{2 \, a \log \left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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