\(\int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx\) [1141]

Optimal result

Integrand size = 20, antiderivative size = 27 \[ \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx=\frac {x}{a c \sqrt {a+a x} \sqrt {c-c x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {39} \[ \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx=\frac {x}{a c \sqrt {a x+a} \sqrt {c-c x}} \]

[In]

Int[1/((a + a*x)^(3/2)*(c - c*x)^(3/2)),x]

[Out]

x/(a*c*Sqrt[a + a*x]*Sqrt[c - c*x])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {x}{a c \sqrt {a+a x} \sqrt {c-c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx=\frac {x (1+x)}{c (a (1+x))^{3/2} \sqrt {c-c x}} \]

[In]

Integrate[1/((a + a*x)^(3/2)*(c - c*x)^(3/2)),x]

[Out]

(x*(1 + x))/(c*(a*(1 + x))^(3/2)*Sqrt[c - c*x])

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89

method	result	size
risch	\(\frac {x}{a c \sqrt {a \left (1+x \right )}\, \sqrt {-c \left (-1+x \right )}}\)	\(24\)
gosper	\(-\frac {\left (-1+x \right ) \left (1+x \right ) x}{\left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {3}{2}}}\)	\(25\)
default	\(-\frac {1}{a c \sqrt {a x +a}\, \sqrt {-c x +c}}+\frac {\sqrt {a x +a}}{c \,a^{2} \sqrt {-c x +c}}\)	\(47\)

[In]

int(1/(a*x+a)^(3/2)/(-c*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx=-\frac {\sqrt {a x + a} \sqrt {-c x + c} x}{a^{2} c^{2} x^{2} - a^{2} c^{2}} \]

[In]

integrate(1/(a*x+a)^(3/2)/(-c*x+c)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.75 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.04 \[ \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx=- \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & \frac {1}{2}, \frac {3}{2}, 2 \\\frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 2 & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} a^{\frac {3}{2}} c^{\frac {3}{2}}} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1 & \\\frac {1}{4}, \frac {3}{4} & - \frac {1}{2}, 0, 1, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} a^{\frac {3}{2}} c^{\frac {3}{2}}} \]

[In]

integrate(1/(a*x+a)**(3/2)/(-c*x+c)**(3/2),x)

[Out]

-I*meijerg(((3/4, 5/4, 1), (1/2, 3/2, 2)), ((3/4, 1, 5/4, 3/2, 2), (0,)), x**(-2))/(2*pi**(3/2)*a**(3/2)*c**(3
/2)) + meijerg(((-1/2, 0, 1/4, 1/2, 3/4, 1), ()), ((1/4, 3/4), (-1/2, 0, 1, 0)), exp_polar(-2*I*pi)/x**2)/(2*p
i**(3/2)*a**(3/2)*c**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx=\frac {x}{\sqrt {-a c x^{2} + a c} a c} \]

[In]

integrate(1/(a*x+a)^(3/2)/(-c*x+c)^(3/2),x, algorithm="maxima")

[Out]

x/(sqrt(-a*c*x^2 + a*c)*a*c)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (23) = 46\).

Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.30 \[ \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx=-\frac {2 \, \sqrt {-a c} a}{{\left (2 \, a^{2} c - {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )} c {\left | a \right |}} - \frac {\sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a}}{2 \, {\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )} c {\left | a \right |}} \]

[In]

integrate(1/(a*x+a)^(3/2)/(-c*x+c)^(3/2),x, algorithm="giac")

[Out]

-2*sqrt(-a*c)*a/((2*a^2*c - (sqrt(-a*c)*sqrt(a*x + a) - sqrt(-(a*x + a)*a*c + 2*a^2*c))^2)*c*abs(a)) - 1/2*sqr
t(-(a*x + a)*a*c + 2*a^2*c)*sqrt(a*x + a)/(((a*x + a)*a*c - 2*a^2*c)*c*abs(a))

Mupad [B] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx=\frac {x}{a\,c\,\sqrt {a+a\,x}\,\sqrt {c-c\,x}} \]

[In]

int(1/((a + a*x)^(3/2)*(c - c*x)^(3/2)),x)

[Out]

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