∫ 1_________√ a + ax√___

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

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

________________________________________________________________________________________

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 = {65, 223, 209} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+a}}{\sqrt {a} \sqrt {c-c x}}\right )}{\sqrt {a} \sqrt {c}} \end {gather*}

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

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,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;

FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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,

\begin {align*} \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+a x}\right )}{a}\\ &=\frac {2 \text {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 = 0.04, size = 47, normalized size = 1.09 \begin {gather*} \frac {2 \sqrt {1+x} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {1+x}}{\sqrt {c-c x}}\right )}{\sqrt {c} \sqrt {a (1+x)}} \end {gather*}

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

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

________________________________________________________________________________________

Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 15.53, size = 63, normalized size = 1.47 \begin {gather*} \frac {-I \text {meijerg}\left [\left \{\left \{\frac {1}{4},\frac {3}{4}\right \},\left \{\frac {1}{2},\frac {1}{2},1,1\right \}\right \},\left \{\left \{0,\frac {1}{4},\frac {1}{2},\frac {3}{4},1,0\right \},\left \{\right \}\right \},\frac {1}{x^2}\right ]+\text {meijerg}\left [\left \{\left \{-\frac {1}{2},-\frac {1}{4},0,\frac {1}{4},\frac {1}{2},1\right \},\left \{\right \}\right \},\left \{\left \{-\frac {1}{4},\frac {1}{4}\right \},\left \{-\frac {1}{2},0,0,0\right \}\right \},\frac {\text {exp\_polar}\left [-2 I \text {Pi}\right ]}{x^2}\right ]}{4 \text {Pi}^{\frac {3}{2}} \sqrt {a} \sqrt {c}} \end {gather*}

mathics('Integrate[1/((a + a*x)^(1/2)*(c - c*x)^(1/2)),x]')

(-I meijerg[{{1 / 4, 3 / 4}, {1 / 2, 1 / 2, 1, 1}}, {{0, 1 / 4, 1 / 2, 3 / 4, 1, 0}, {}}, 1 / x ^ 2] + 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]

________________________________________________________________________________________


method	result	size

default	\(\frac {\sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{\sqrt {a x +a}\, \sqrt {-c x +c}\, \sqrt {a c}}\)	\(57\)

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

((-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))

________________________________________________________________________________________

Maxima [A]
time = 0.36, size = 8, normalized size = 0.19 \begin {gather*} \frac {\arcsin \left (x\right )}{\sqrt {a c}} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.30, size = 101, normalized size = 2.35 \begin {gather*} \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 {gather*}

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

[-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)]

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 13.20, size = 85, normalized size = 1.98 \begin {gather*} - \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 {gather*}

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

-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*

________________________________________________________________________________________

Giac [A]
time = 0.01, size = 62, normalized size = 1.44 \begin {gather*} -\frac {2 a^{2} \ln \left |\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right |}{\left |a\right | a \sqrt {-a c}} \end {gather*}

-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))

________________________________________________________________________________________

Mupad [B]
time = 0.18, size = 44, normalized size = 1.02 \begin {gather*} -\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}} \end {gather*}

-(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)

________________________________________________________________________________________

3.12.40 \(\int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx\) [1140]