Optimal. Leaf size=27 \[ \frac {x}{a c \sqrt {a+a x} \sqrt {c-c x}} \]
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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.
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Rule 39
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.
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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.
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Maxima [A]
time = 0.29, 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.
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Fricas [A]
time = 1.44, 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.
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Sympy [C] Result contains complex when optimal does not.
time = 2.19, 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.
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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 = 1.01, 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.
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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.
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