\(\int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx\) [1149]

Optimal result

Integrand size = 23, antiderivative size = 30 \[ \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {x}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {39} \[ \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {x}{a^2 c \sqrt {a+b x} \sqrt {a c-b c x}} \]

[In]

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

[Out]

x/(a^2*c*Sqrt[a + b*x]*Sqrt[a*c - b*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^2 c \sqrt {a+b x} \sqrt {a c-b c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {x}{a^2 c \sqrt {c (a-b x)} \sqrt {a+b x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00

method	result	size
gosper	\(\frac {\left (-b x +a \right ) x}{\sqrt {b x +a}\, a^{2} \left (-b c x +a c \right )^{\frac {3}{2}}}\)	\(30\)
default	\(-\frac {1}{b a c \sqrt {b x +a}\, \sqrt {-b c x +a c}}+\frac {\sqrt {b x +a}}{b c \,a^{2} \sqrt {-b c x +a c}}\)	\(59\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=-\frac {\sqrt {-b c x + a c} \sqrt {b x + a} x}{a^{2} b^{2} c^{2} x^{2} - a^{4} c^{2}} \]

[In]

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

[Out]

-sqrt(-b*c*x + a*c)*sqrt(b*x + a)*x/(a^2*b^2*c^2*x^2 - a^4*c^2)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.53 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.13 \[ \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=- \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & \frac {1}{2}, \frac {3}{2}, 2 \\\frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 2 & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} a^{2} b c^{\frac {3}{2}}} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1 & \\\frac {1}{4}, \frac {3}{4} & - \frac {1}{2}, 0, 1, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} a^{2} b c^{\frac {3}{2}}} \]

[In]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {x}{\sqrt {-b^{2} c x^{2} + a^{2} c} a^{2} c} \]

[In]

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

[Out]

x/(sqrt(-b^2*c*x^2 + a^2*c)*a^2*c)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (26) = 52\).

Time = 0.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.43 \[ \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {\frac {4 \, \sqrt {-c}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} - 2 \, a c\right )} a c} - \frac {\sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}}{{\left ({\left (b x + a\right )} c - 2 \, a c\right )} a^{2} c}}{2 \, b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx=\frac {x}{a^2\,c\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}} \]

[In]

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

[Out]

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