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}
\begin {gather*} \frac {x \sqrt {a+b x}}{\sqrt {-a-b x}} \end {gather*}
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 \begin {gather*} \frac {x \sqrt {a+b x}}{\sqrt {-a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 23, normalized size = 1.00
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {-b x -a}}{b}\) | \(23\) |
risch | \(-\frac {i \sqrt {\frac {-b x -a}{b x +a}}\, \sqrt {b x +a}\, x}{\sqrt {-b x -a}}\) | \(40\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.83, size = 53, normalized size = 2.30 \begin {gather*} \begin {cases} 0 & \text {for}\: \frac {1}{\left |{\frac {a}{b} + x}\right |} < 1 \wedge \left |{\frac {a}{b} + x}\right | < 1 \\- i \left (\frac {a}{b} + x\right ) & \text {for}\: \frac {1}{\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 2.57, size = 12, normalized size = 0.52 \begin {gather*} \frac {-i \, b x - i \, a}{b} \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 = -0.04 \begin {gather*} \int \frac {\sqrt {a+b\,x}}{\sqrt {-a-b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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