Optimal. Leaf size=96 \[ \frac {3}{4} a^{3/2} c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+a}}{\sqrt {a} \sqrt {c-c x}}\right )+\frac {3}{8} a c x \sqrt {a x+a} \sqrt {c-c x}+\frac {1}{4} x (a x+a)^{3/2} (c-c x)^{3/2} \]
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Rubi [A] time = 0.04, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {38, 63, 217, 203} \[ \frac {3}{4} a^{3/2} c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+a}}{\sqrt {a} \sqrt {c-c x}}\right )+\frac {3}{8} a c x \sqrt {a x+a} \sqrt {c-c x}+\frac {1}{4} x (a x+a)^{3/2} (c-c x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 38
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int (a+a x)^{3/2} (c-c x)^{3/2} \, dx &=\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{4} (3 a c) \int \sqrt {a+a x} \sqrt {c-c x} \, dx\\ &=\frac {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{8} \left (3 a^2 c^2\right ) \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx\\ &=\frac {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{4} \left (3 a c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+a x}\right )\\ &=\frac {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {1}{4} \left (3 a c^2\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 {3}{8} a c x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{4} x (a+a x)^{3/2} (c-c x)^{3/2}+\frac {3}{4} a^{3/2} c^{3/2} \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 = 0.08, size = 104, normalized size = 1.08 \[ \frac {\sqrt {c} (a (x+1))^{3/2} \sqrt {c-c x} \left (\sqrt {c} x \sqrt {x+1} \left (-2 x^3+2 x^2+5 x-5\right )+6 \sqrt {c-c x} \sin ^{-1}\left (\frac {\sqrt {c-c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{8 (x-1) (x+1)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 155, normalized size = 1.61 \[ \left [\frac {3}{16} \, \sqrt {-a c} a c \log \left (2 \, a c x^{2} + 2 \, \sqrt {-a c} \sqrt {a x + a} \sqrt {-c x + c} x - a c\right ) - \frac {1}{8} \, {\left (2 \, a c x^{3} - 5 \, a c x\right )} \sqrt {a x + a} \sqrt {-c x + c}, -\frac {3}{8} \, \sqrt {a c} a c \arctan \left (\frac {\sqrt {a c} \sqrt {a x + a} \sqrt {-c x + c} x}{a c x^{2} - a c}\right ) - \frac {1}{8} \, {\left (2 \, a c x^{3} - 5 \, a c x\right )} \sqrt {a x + a} \sqrt {-c x + c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.25, size = 403, normalized size = 4.20 \[ -\frac {{\left (\frac {18 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} {\left ({\left (a x + a\right )} {\left (2 \, {\left (a x + a\right )} {\left (\frac {3 \, {\left (a x + a\right )}}{a^{3}} - \frac {13}{a^{2}}\right )} + \frac {43}{a}\right )} - 39\right )} \sqrt {a x + a}\right )} c {\left | a \right |}}{24 \, a} + \frac {{\left (\frac {6 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left ({\left (a x + a\right )} {\left (\frac {2 \, {\left (a x + a\right )}}{a^{2}} - \frac {7}{a}\right )} + 9\right )}\right )} c {\left | a \right |}}{6 \, a} - \frac {{\left (\frac {2 \, a^{2} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a}\right )} c {\left | a \right |}}{a} + \frac {{\left (\frac {2 \, a^{3} c \log \left ({\left | -\sqrt {-a c} \sqrt {a x + a} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt {-a c}} + \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left (a x - 2 \, a\right )}\right )} c {\left | a \right |}}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 143, normalized size = 1.49 \[ \frac {3 \sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, a^{2} c^{2} \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{8 \sqrt {-c x +c}\, \sqrt {a x +a}\, \sqrt {a c}}+\frac {3 \sqrt {-c x +c}\, \sqrt {a x +a}\, a c}{8}+\frac {\sqrt {a x +a}\, \left (-c x +c \right )^{\frac {3}{2}} a}{8}-\frac {\sqrt {a x +a}\, \left (-c x +c \right )^{\frac {5}{2}} a}{4 c}-\frac {\left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {5}{2}}}{4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.09, size = 50, normalized size = 0.52 \[ \frac {3 \, a^{2} c^{2} \arcsin \relax (x)}{8 \, \sqrt {a c}} + \frac {3}{8} \, \sqrt {-a c x^{2} + a c} a c x + \frac {1}{4} \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} x \]
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 \[ \int {\left (a+a\,x\right )}^{3/2}\,{\left (c-c\,x\right )}^{3/2} \,d x \]
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 \[ \int \left (a \left (x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (x - 1\right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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