Optimal. Leaf size=23 \[ \frac {x \sqrt {a+b x}}{\sqrt {-a-b x}} \]
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Rubi [A] time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {23, 8} \[ \frac {x \sqrt {a+b x}}{\sqrt {-a-b x}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 23
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{\sqrt {-a-b x}} \, dx &=\frac {\sqrt {a+b x} \int 1 \, dx}{\sqrt {-a-b x}}\\ &=\frac {x \sqrt {a+b x}}{\sqrt {-a-b x}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.00 \[ \frac {x \sqrt {a+b x}}{\sqrt {-a-b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 1, normalized size = 0.04 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.48, size = 10, normalized size = 0.43 \[ -\frac {i \, {\left (b x + a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {\sqrt {b x +a}\, \sqrt {-b x -a}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {a+b\,x}}{\sqrt {-a-b\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.91, size = 37, normalized size = 1.61 \[ \begin {cases} - i \left (\frac {a}{b} + x\right ) & \text {for}\: \left |{\frac {a}{b} + x}\right | > 1 \vee \left |{\frac {a}{b} + x}\right | < 1 \\- i {G_{2, 2}^{1, 1}\left (\begin {matrix} 1 & 2 \\1 & 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} - i {G_{2, 2}^{0, 2}\left (\begin {matrix} 2, 1 & \\ & 1, 0 \end {matrix} \middle | {\frac {a}{b} + x} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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