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

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

[Out]

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

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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 = 0.06, size = 27, normalized size = 1.00 \begin {gather*} \frac {x (1+x)}{c (a (1+x))^{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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 4.45, size = 63, normalized size = 2.33 \begin {gather*} \frac {-I \text {meijerg}\left [\left \{\left \{\frac {3}{4},\frac {5}{4},1\right \},\left \{\frac {1}{2},\frac {3}{2},2\right \}\right \},\left \{\left \{\frac {3}{4},1,\frac {5}{4},\frac {3}{2},2\right \},\left \{0\right \}\right \},\frac {1}{x^2}\right ]+\text {meijerg}\left [\left \{\left \{-\frac {1}{2},0,\frac {1}{4},\frac {1}{2},\frac {3}{4},1\right \},\left \{\right \}\right \},\left \{\left \{\frac {1}{4},\frac {3}{4}\right \},\left \{-\frac {1}{2},0,1,0\right \}\right \},\frac {\text {exp\_polar}\left [-2 I \text {Pi}\right ]}{x^2}\right ]}{2 \text {Pi}^{\frac {3}{2}} a^{\frac {3}{2}} c^{\frac {3}{2}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/((a + a*x)^(3/2)*(c - c*x)^(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}}, 1 / x ^ 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 Pi ^ (3 / 2) a ^ (3 / 2) c ^ (3 / 2))

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Maple [A]
time = 0.14, size = 47, normalized size = 1.74

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	\(-\frac {1}{a c \sqrt {a x +a}\, \sqrt {-c x +c}}+\frac {\sqrt {a x +a}}{c \,a^{2} \sqrt {-c x +c}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, 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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Fricas [A]
time = 0.29, 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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Sympy [C] Result contains complex when optimal does not.
time = 2.25, 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (23) = 46\).
time = 0.01, size = 131, normalized size = 4.85 \begin {gather*} -2 \left (-\frac {\frac {1}{8}\cdot 2 \sqrt {a x+a} \sqrt {2 a^{2} c-a c \left (a x+a\right )}}{c \left |a\right | \left (2 a^{2} c-a c \left (a x+a\right )\right )}-\frac {2 a \sqrt {-a c}}{2 c \left |a\right | \left (\left (\sqrt {2 a^{2} c-a c \left (a x+a\right )}-\sqrt {-a c} \sqrt {a x+a}\right )^{2}-2 c a^{2}\right )}\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)

[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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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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