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 = 0.05, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {38, 63, 217, 203} \begin {gather*} \frac {5}{16} a^2 c^2 x \sqrt {a x+a} \sqrt {c-c x}+\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}{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} \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int (a+a x)^{5/2} (c-c x)^{5/2} \, dx &=\frac {1}{6} x (a+a x)^{5/2} (c-c x)^{5/2}+\frac {1}{6} (5 a c) \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx\\ &=\frac {5}{24} a c x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{6} x (a+a x)^{5/2} (c-c x)^{5/2}+\frac {1}{8} \left (5 a^2 c^2\right ) \int \sqrt {a+a x} \sqrt {c-c x} \, dx\\ &=\frac {5}{16} a^2 c^2 x \sqrt {a+a x} \sqrt {c-c x}+\frac {5}{24} a c x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{6} x (a+a x)^{5/2} (c-c x)^{5/2}+\frac {1}{16} \left (5 a^3 c^3\right ) \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx\\ &=\frac {5}{16} a^2 c^2 x \sqrt {a+a x} \sqrt {c-c x}+\frac {5}{24} a c x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{6} x (a+a x)^{5/2} (c-c x)^{5/2}+\frac {1}{8} \left (5 a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+a x}\right )\\ &=\frac {5}{16} a^2 c^2 x \sqrt {a+a x} \sqrt {c-c x}+\frac {5}{24} a c x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{6} x (a+a x)^{5/2} (c-c x)^{5/2}+\frac {1}{8} \left (5 a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+a x}}{\sqrt {c-c x}}\right )\\ &=\frac {5}{16} a^2 c^2 x \sqrt {a+a x} \sqrt {c-c x}+\frac {5}{24} a c x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{6} x (a+a x)^{5/2} (c-c x)^{5/2}+\frac {5}{8} a^{5/2} c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+a x}}{\sqrt {a} \sqrt {c-c x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 114, normalized size = 0.90 \begin {gather*} \frac {c^{3/2} (a (x+1))^{5/2} \sqrt {c-c x} \left (\sqrt {c} x \sqrt {x+1} \left (8 x^5-8 x^4-26 x^3+26 x^2+33 x-33\right )+30 \sqrt {c-c x} \sin ^{-1}\left (\frac {\sqrt {c-c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{48 (x-1) (x+1)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.38, size = 206, normalized size = 1.63 \begin {gather*} -\frac {5}{8} a^{5/2} c^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c-c x}}{\sqrt {c} \sqrt {a x+a}}\right )-\frac {a^3 c^3 \sqrt {c-c x} \left (\frac {15 a^5 (c-c x)^5}{(a x+a)^5}+\frac {85 a^4 c (c-c x)^4}{(a x+a)^4}+\frac {198 a^3 c^2 (c-c x)^3}{(a x+a)^3}-\frac {198 a^2 c^3 (c-c x)^2}{(a x+a)^2}-\frac {85 a c^4 (c-c x)}{a x+a}-15 c^5\right )}{24 \sqrt {a x+a} \left (\frac {a (c-c x)}{a x+a}+c\right )^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.61, size = 201, normalized size = 1.60 \begin {gather*} \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 {\sqrt {a c} \sqrt {a x + a} \sqrt {-c x + c} x}{a c x^{2} - 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}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.57, size = 679, normalized size = 5.39 \begin {gather*} \frac {1}{240} \, {\left (\frac {150 \, a^{2} c \log \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 {5 \, {\left (a x + a\right )}}{a^{5}} - \frac {31}{a^{4}}\right )} + \frac {321}{a^{3}}\right )} - \frac {451}{a^{2}}\right )} {\left (a x + a\right )} + \frac {745}{a}\right )} {\left (a x + a\right )} - 405\right )} \sqrt {a x + a}\right )} c^{2} {\left | a \right |} - \frac {1}{120} \, {\left (\frac {90 \, a^{2} c \log \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 (a x + a\right )} {\left (3 \, {\left (a x + a\right )} {\left (\frac {4 \, {\left (a x + a\right )}}{a^{4}} - \frac {21}{a^{3}}\right )} + \frac {133}{a^{2}}\right )} - \frac {295}{a}\right )} {\left (a x + a\right )} + 195\right )} \sqrt {a x + a}\right )} c^{2} {\left | a \right |} - \frac {1}{12} \, {\left (\frac {18 \, a^{2} c \log \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 {3 \, {\left (a x + a\right )}}{a^{3}} - \frac {13}{a^{2}}\right )} + \frac {43}{a}\right )} - 39\right )} \sqrt {a x + a}\right )} c^{2} {\left | a \right |} + \frac {1}{3} \, {\left (\frac {6 \, a^{2} c \log \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} {\left ({\left (a x + a\right )} {\left (\frac {2 \, {\left (a x + a\right )}}{a^{2}} - \frac {7}{a}\right )} + 9\right )}\right )} c^{2} {\left | a \right |} - {\left (\frac {2 \, a^{2} c \log \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}\right )} c^{2} {\left | a \right |} + \frac {{\left (\frac {2 \, a^{3} c \log \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} {\left (a x - 2 \, a\right )}\right )} c^{2} {\left | a \right |}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 193, normalized size = 1.53 \begin {gather*} \frac {5 \sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, a^{3} c^{3} \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{16 \sqrt {-c x +c}\, \sqrt {a x +a}\, \sqrt {a c}}+\frac {5 \sqrt {-c x +c}\, \sqrt {a x +a}\, a^{2} c^{2}}{16}+\frac {5 \left (-c x +c \right )^{\frac {3}{2}} \sqrt {a x +a}\, a^{2} c}{48}+\frac {\left (-c x +c \right )^{\frac {5}{2}} \sqrt {a x +a}\, a^{2}}{24}-\frac {\sqrt {a x +a}\, \left (-c x +c \right )^{\frac {7}{2}} a^{2}}{8 c}-\frac {\left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {7}{2}} a}{6 c}-\frac {\left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {7}{2}}}{6 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 72, normalized size = 0.57 \begin {gather*} \frac {5 \, a^{3} c^{3} \arcsin \relax (x)}{16 \, \sqrt {a c}} + \frac {5}{16} \, \sqrt {-a c x^{2} + a c} a^{2} c^{2} x + \frac {5}{24} \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} a c x + \frac {1}{6} \, {\left (-a c x^{2} + a c\right )}^{\frac {5}{2}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+a\,x\right )}^{5/2}\,{\left (c-c\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (x + 1\right )\right )^{\frac {5}{2}} \left (- c \left (x - 1\right )\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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