Optimal. Leaf size=68 \[ \frac {a^2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b}+\frac {1}{2} x \sqrt {a+b x} \sqrt {a c-b c x} \]
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Rubi [A] time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {38, 63, 217, 203} \[ \frac {a^2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b}+\frac {1}{2} x \sqrt {a+b x} \sqrt {a c-b c x} \]
Antiderivative was successfully verified.
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Rule 38
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \sqrt {a+b x} \sqrt {a c-b c x} \, dx &=\frac {1}{2} x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {1}{2} \left (a^2 c\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\\ &=\frac {1}{2} x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {\left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 a c-c x^2}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {1}{2} x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {\left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{1+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b}\\ &=\frac {1}{2} x \sqrt {a+b x} \sqrt {a c-b c x}+\frac {a^2 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {c (a-b x)}}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 95, normalized size = 1.40 \[ \frac {c \left (-2 a^{5/2} \sqrt {a-b x} \sqrt {\frac {b x}{a}+1} \sin ^{-1}\left (\frac {\sqrt {a-b x}}{\sqrt {2} \sqrt {a}}\right )+a^2 b x-b^3 x^3\right )}{2 b \sqrt {a+b x} \sqrt {c (a-b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 159, normalized size = 2.34 \[ \left [\frac {a^{2} \sqrt {-c} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b x}{4 \, b}, -\frac {a^{2} \sqrt {c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) - \sqrt {-b c x + a c} \sqrt {b x + a} b x}{2 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 127, normalized size = 1.87 \[ \frac {\sqrt {\left (b x +a \right ) \left (-b c x +a c \right )}\, a^{2} c \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{2 \sqrt {-b c x +a c}\, \sqrt {b x +a}\, \sqrt {b^{2} c}}+\frac {\sqrt {-b c x +a c}\, \sqrt {b x +a}\, a}{2 b}-\frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {3}{2}}}{2 b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.09, size = 39, normalized size = 0.57 \[ \frac {a^{2} \sqrt {c} \arcsin \left (\frac {b x}{a}\right )}{2 \, b} + \frac {1}{2} \, \sqrt {-b^{2} c x^{2} + a^{2} c} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 72, normalized size = 1.06 \[ \frac {x\,\sqrt {a\,c-b\,c\,x}\,\sqrt {a+b\,x}}{2}-\frac {a^2\,\sqrt {b}\,c^2\,\ln \left (\sqrt {-b\,c}\,\sqrt {c\,\left (a-b\,x\right )}\,\sqrt {a+b\,x}-b^{3/2}\,c\,x\right )}{2\,{\left (-b\,c\right )}^{3/2}} \]
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 \[ \int \sqrt {- c \left (- a + b x\right )} \sqrt {a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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