Optimal. Leaf size=126 \[ \frac{5}{8} a^{5/2} c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )+\frac{5}{16} a^2 c^2 x \sqrt{a x+a} \sqrt{c-c x}+\frac{5}{24} a c x (a x+a)^{3/2} (c-c x)^{3/2}+\frac{1}{6} x (a x+a)^{5/2} (c-c x)^{5/2} \]
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Rubi [A] time = 0.127767, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{5}{8} a^{5/2} c^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )+\frac{5}{16} a^2 c^2 x \sqrt{a x+a} \sqrt{c-c x}+\frac{5}{24} a c x (a x+a)^{3/2} (c-c x)^{3/2}+\frac{1}{6} x (a x+a)^{5/2} (c-c x)^{5/2} \]
Antiderivative was successfully verified.
[In] Int[(a + a*x)^(5/2)*(c - c*x)^(5/2),x]
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Rubi in Sympy [A] time = 19.1528, size = 116, normalized size = 0.92 \[ - \frac{5 a^{\frac{5}{2}} c^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} \sqrt{- c x + c}}{\sqrt{c} \sqrt{a x + a}} \right )}}{8} + \frac{5 a^{2} c^{2} x \sqrt{a x + a} \sqrt{- c x + c}}{16} + \frac{5 a c x \left (a x + a\right )^{\frac{3}{2}} \left (- c x + c\right )^{\frac{3}{2}}}{24} + \frac{x \left (a x + a\right )^{\frac{5}{2}} \left (- c x + c\right )^{\frac{5}{2}}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*x+a)**(5/2)*(-c*x+c)**(5/2),x)
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Mathematica [A] time = 0.180811, size = 91, normalized size = 0.72 \[ \frac{c^2 (a (x+1))^{5/2} \left (x \sqrt{x+1} \left (8 x^4-26 x^2+33\right ) \sqrt{c-c x}-30 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{x+1} \sqrt{c-c x}}{\sqrt{c} (x-1)}\right )\right )}{48 (x+1)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + a*x)^(5/2)*(c - c*x)^(5/2),x]
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Maple [B] time = 0.02, size = 193, normalized size = 1.5 \[ -{\frac{1}{6\,c} \left ( ax+a \right ) ^{{\frac{5}{2}}} \left ( -cx+c \right ) ^{{\frac{7}{2}}}}-{\frac{a}{6\,c} \left ( ax+a \right ) ^{{\frac{3}{2}}} \left ( -cx+c \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{2}}{8\,c}\sqrt{ax+a} \left ( -cx+c \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}}{24} \left ( -cx+c \right ) ^{{\frac{5}{2}}}\sqrt{ax+a}}+{\frac{5\,{a}^{2}c}{48} \left ( -cx+c \right ) ^{{\frac{3}{2}}}\sqrt{ax+a}}+{\frac{5\,{a}^{2}{c}^{2}}{16}\sqrt{ax+a}\sqrt{-cx+c}}+{\frac{5\,{a}^{3}{c}^{3}}{16}\sqrt{ \left ( -cx+c \right ) \left ( ax+a \right ) }\arctan \left ({x\sqrt{ac}{\frac{1}{\sqrt{-ac{x}^{2}+ac}}}} \right ){\frac{1}{\sqrt{ax+a}}}{\frac{1}{\sqrt{-cx+c}}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*x+a)^(5/2)*(-c*x+c)^(5/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + a)^(5/2)*(-c*x + c)^(5/2),x, algorithm="maxima")
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Fricas [A] time = 0.220706, size = 1, normalized size = 0.01 \[ \left [\frac{5}{32} \, \sqrt{-a c} a^{2} c^{2} \log \left (2 \, a c x^{2} + 2 \, \sqrt{-a c} \sqrt{a x + a} \sqrt{-c x + c} x - a c\right ) + \frac{1}{48} \,{\left (8 \, a^{2} c^{2} x^{5} - 26 \, a^{2} c^{2} x^{3} + 33 \, a^{2} c^{2} x\right )} \sqrt{a x + a} \sqrt{-c x + c}, \frac{5}{16} \, \sqrt{a c} a^{2} c^{2} \arctan \left (\frac{a c x}{\sqrt{a c} \sqrt{a x + a} \sqrt{-c x + c}}\right ) + \frac{1}{48} \,{\left (8 \, a^{2} c^{2} x^{5} - 26 \, a^{2} c^{2} x^{3} + 33 \, a^{2} c^{2} x\right )} \sqrt{a x + a} \sqrt{-c x + c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + a)^(5/2)*(-c*x + c)^(5/2),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x+a)**(5/2)*(-c*x+c)**(5/2),x)
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GIAC/XCAS [A] time = 0.33746, size = 478, normalized size = 3.79 \[ -\frac{{\left (\frac{6 \, a^{3} c{\rm ln}\left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}{\left ({\left (2 \,{\left ({\left (a x + a\right )}{\left (4 \,{\left (a x + a\right )}{\left (\frac{a x + a}{a^{4}} - \frac{5}{a^{3}}\right )} + \frac{39}{a^{2}}\right )} - \frac{37}{a}\right )}{\left (a x + a\right )} + 31\right )}{\left (a x + a\right )} - 3 \, a\right )} \sqrt{a x + a}\right )} c^{2}{\left | a \right |}}{48 \, a} - \frac{{\left (\frac{2 \, a^{3} c{\rm ln}\left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt{a x + a} a x\right )} c^{2}{\left | a \right |}}{2 \, a} + \frac{{\left (\frac{2 \, a^{3} c{\rm ln}\left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}} - \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c}{\left ({\left (a x + a\right )}{\left (2 \,{\left (a x + a\right )}{\left (\frac{a x + a}{a^{2}} - \frac{3}{a}\right )} + 5\right )} - a\right )} \sqrt{a x + a}\right )} c^{2}{\left | a \right |}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x + a)^(5/2)*(-c*x + c)^(5/2),x, algorithm="giac")
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