Optimal. Leaf size=27 \[ \frac{x}{a c \sqrt{a x+a} \sqrt{c-c x}} \]
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Rubi [A] time = 0.0226106, 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 \[ \frac{x}{a c \sqrt{a x+a} \sqrt{c-c x}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + a*x)^(3/2)*(c - c*x)^(3/2)),x]
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Rubi in Sympy [A] time = 4.16744, size = 20, normalized size = 0.74 \[ \frac{x}{a c \sqrt{a x + a} \sqrt{- c x + c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a*x+a)**(3/2)/(-c*x+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0414199, size = 33, normalized size = 1.22 \[ -\frac{x (x+1) \sqrt{c-c x}}{c^2 (x-1) (a (x+1))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + a*x)^(3/2)*(c - c*x)^(3/2)),x]
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Maple [A] time = 0.005, size = 25, normalized size = 0.9 \[ -{ \left ( 1+x \right ) \left ( -1+x \right ) x \left ( ax+a \right ) ^{-{\frac{3}{2}}} \left ( -cx+c \right ) ^{-{\frac{3}{2}}}} \]
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]
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Maxima [A] time = 1.34945, size = 28, normalized size = 1.04 \[ \frac{x}{\sqrt{-a c x^{2} + a c} a c} \]
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]
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Fricas [A] time = 0.207142, size = 53, normalized size = 1.96 \[ -\frac{\sqrt{a x + a} \sqrt{-c x + c} x}{a^{2} c^{2} x^{2} - a^{2} c^{2}} \]
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]
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Sympy [A] time = 20.1819, size = 82, normalized size = 3.04 \[ - \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & \frac{1}{2}, \frac{3}{2}, 2 \\\frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 2 & 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} a^{\frac{3}{2}} c^{\frac{3}{2}}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 & \\\frac{1}{4}, \frac{3}{4} & - \frac{1}{2}, 0, 1, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} a^{\frac{3}{2}} c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a*x+a)**(3/2)/(-c*x+c)**(3/2),x)
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GIAC/XCAS [A] time = 0.214495, size = 157, normalized size = 5.81 \[ -\frac{2 \, \sqrt{-a c} a}{{\left (2 \, a^{2} c -{\left (\sqrt{-a c} \sqrt{a x + a} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )} c{\left | a \right |}} - \frac{\sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a}}{2 \,{\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )} c{\left | a \right |}} \]
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]