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

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 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])

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.45, size = 101, normalized size = 2.35 \[ \left [-\frac {\sqrt {-a c} \log \left (2 \, a c x^{2} - 2 \, \sqrt {-a c} \sqrt {a x + a} \sqrt {-c x + c} x - a c\right )}{2 \, a c}, -\frac {\sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {a x + a} \sqrt {-c x + c} x}{a c x^{2} - a c}\right )}{a c}\right ] \]

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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giac [A]  time = 0.76, size = 49, normalized size = 1.14 \[ -\frac {2 \, a \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c} {\left | a \right |}} \]

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))

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maple [A]  time = 0.00, size = 57, normalized size = 1.33 \[ \frac {\sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{\sqrt {a x +a}\, \sqrt {-c x +c}\, \sqrt {a c}} \]

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

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maxima [A]  time = 2.99, size = 8, normalized size = 0.19 \[ \frac {\arcsin \relax (x)}{\sqrt {a c}} \]

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]

arcsin(x)/sqrt(a*c)

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mupad [B]  time = 0.18, size = 44, normalized size = 1.02 \[ -\frac {4\,\mathrm {atan}\left (\frac {a\,\left (\sqrt {c-c\,x}-\sqrt {c}\right )}{\sqrt {a\,c}\,\left (\sqrt {a+a\,x}-\sqrt {a}\right )}\right )}{\sqrt {a\,c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 3.95, size = 85, normalized size = 1.98 \[ - \frac {i {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {c}} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {c}} \]

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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