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

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

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Rubi [A]  time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {39} \begin {gather*} \frac {x}{a c \sqrt {a x+a} \sqrt {c-c x}} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 1.00 \begin {gather*} \frac {x (x+1)}{c (a (x+1))^{3/2} \sqrt {c-c x}} \end {gather*}

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])

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IntegrateAlgebraic [A]  time = 0.10, size = 47, normalized size = 1.74 \begin {gather*} \frac {\sqrt {a x+a} \left (c-\frac {a (c-c x)}{a x+a}\right )}{2 a^2 c^2 \sqrt {c-c x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.48, size = 39, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {a x + a} \sqrt {-c x + c} x}{a^{2} c^{2} x^{2} - a^{2} c^{2}} \end {gather*}

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)

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giac [B]  time = 0.70, size = 116, normalized size = 4.30 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.00, size = 25, normalized size = 0.93 \begin {gather*} -\frac {\left (x +1\right ) \left (x -1\right ) x}{\left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {3}{2}}} \end {gather*}

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]

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

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maxima [A]  time = 1.31, size = 21, normalized size = 0.78 \begin {gather*} \frac {x}{\sqrt {-a c x^{2} + a c} a c} \end {gather*}

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)

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mupad [B]  time = 0.39, size = 23, normalized size = 0.85 \begin {gather*} \frac {x}{a\,c\,\sqrt {a+a\,x}\,\sqrt {c-c\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [C]  time = 4.44, size = 82, normalized size = 3.04 \begin {gather*} - \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 {gather*}

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))

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