Optimal. Leaf size=67 \[ \frac {1}{2} x \sqrt {a x+a} \sqrt {c-c x}+\sqrt {a} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+a}}{\sqrt {a} \sqrt {c-c x}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {1}{2} x \sqrt {a x+a} \sqrt {c-c x}+\sqrt {a} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+a}}{\sqrt {a} \sqrt {c-c x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \sqrt {a+a x} \sqrt {c-c x} \, dx &=\frac {1}{2} x \sqrt {a+a x} \sqrt {c-c x}+\frac {1}{2} (a c) \int \frac {1}{\sqrt {a+a x} \sqrt {c-c x}} \, dx\\ &=\frac {1}{2} x \sqrt {a+a x} \sqrt {c-c x}+c \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+a x}\right )\\ &=\frac {1}{2} x \sqrt {a+a x} \sqrt {c-c x}+c \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 {1}{2} x \sqrt {a+a x} \sqrt {c-c x}+\sqrt {a} \sqrt {c} \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 = 0.06, size = 69, normalized size = 1.03 \begin {gather*} \frac {\sqrt {a (x+1)} \left (x \sqrt {x+1} \sqrt {c-c x}-2 \sqrt {c} \sin ^{-1}\left (\frac {\sqrt {c-c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{2 \sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 105, normalized size = 1.57 \begin {gather*} -\frac {a c \sqrt {c-c x} \left (\frac {a (c-c x)}{a x+a}-c\right )}{\sqrt {a x+a} \left (\frac {a (c-c x)}{a x+a}+c\right )^2}-\sqrt {a} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c-c x}}{\sqrt {c} \sqrt {a x+a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.60, size = 127, normalized size = 1.90 \begin {gather*} \left [\frac {1}{2} \, \sqrt {a x + a} \sqrt {-c x + c} x + \frac {1}{4} \, \sqrt {-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}{2} \, \sqrt {a x + a} \sqrt {-c x + c} x - \frac {1}{2} \, \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {a x + a} \sqrt {-c x + c} x}{a c x^{2} - a c}\right )\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.90, size = 173, normalized size = 2.58 \begin {gather*} -\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 )} {\left | a \right |}}{a^{2}} + \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 )} {\left | a \right |}}{2 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 98, normalized size = 1.46 \begin {gather*} \frac {\sqrt {\left (-c x +c \right ) \left (a x +a \right )}\, a c \arctan \left (\frac {\sqrt {a c}\, x}{\sqrt {-a c \,x^{2}+a c}}\right )}{2 \sqrt {-c x +c}\, \sqrt {a x +a}\, \sqrt {a c}}-\frac {\sqrt {a x +a}\, \left (-c x +c \right )^{\frac {3}{2}}}{2 c}+\frac {\sqrt {a x +a}\, \sqrt {-c x +c}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.08, size = 28, normalized size = 0.42 \begin {gather*} \frac {a c \arcsin \relax (x)}{2 \, \sqrt {a c}} + \frac {1}{2} \, \sqrt {-a c x^{2} + a c} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 59, normalized size = 0.88 \begin {gather*} \frac {x\,\sqrt {a+a\,x}\,\sqrt {c-c\,x}}{2}-\frac {\sqrt {a}\,\sqrt {-c}\,\ln \left (\sqrt {-c}\,\sqrt {a\,\left (x+1\right )}\,\sqrt {-c\,\left (x-1\right )}-\sqrt {a}\,c\,x\right )}{2} \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 \sqrt {a \left (x + 1\right )} \sqrt {- c \left (x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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