3.1141 \(\int \frac{1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac{x}{a c \sqrt{a x+a} \sqrt{c-c x}} \]

[Out]

x/(a*c*Sqrt[a + a*x]*Sqrt[c - c*x])

________________________________________________________________________________________

Rubi [A]  time = 0.0031864, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {39} \[ \frac{x}{a c \sqrt{a x+a} \sqrt{c-c x}} \]

Antiderivative was successfully verified.

[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*} \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}}\\ \end{align*}

Mathematica [A]  time = 0.0163892, size = 27, normalized size = 1. \[ \frac{x (x+1)}{c (a (x+1))^{3/2} \sqrt{c-c x}} \]

Antiderivative was successfully verified.

[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]  time = 0.001, size = 25, normalized size = 0.9 \begin{align*} -{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( ax+a \right ) ^{-{\frac{3}{2}}} \left ( -cx+c \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00216, size = 28, normalized size = 1.04 \begin{align*} \frac{x}{\sqrt{-a c x^{2} + a c} a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 1.57577, size = 77, normalized size = 2.85 \begin{align*} -\frac{\sqrt{a x + a} \sqrt{-c x + c} x}{a^{2} c^{2} x^{2} - a^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 5.44478, size = 82, normalized size = 3.04 \begin{align*} - \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [B]  time = 1.0892, size = 157, normalized size = 5.81 \begin{align*} -\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 |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))