Optimal. Leaf size=28 \[ -\frac {\sqrt {a+b x}}{2 x^2 \sqrt {-a-b x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {23, 30}
\begin {gather*} -\frac {\sqrt {a+b x}}{2 x^2 \sqrt {-a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 30
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{x^3 \sqrt {-a-b x}} \, dx &=\frac {\sqrt {a+b x} \int \frac {1}{x^3} \, dx}{\sqrt {-a-b x}}\\ &=-\frac {\sqrt {a+b x}}{2 x^2 \sqrt {-a-b x}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+b x}}{2 x^2 \sqrt {-a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 23, normalized size = 0.82
method | result | size |
gosper | \(-\frac {\sqrt {b x +a}}{2 x^{2} \sqrt {-b x -a}}\) | \(23\) |
default | \(\frac {\sqrt {-b x -a}}{2 \sqrt {b x +a}\, x^{2}}\) | \(23\) |
risch | \(\frac {i \sqrt {\frac {-b x -a}{b x +a}}\, \sqrt {b x +a}}{2 \sqrt {-b x -a}\, x^{2}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (22) = 44\).
time = 0.30, size = 60, normalized size = 2.14 \begin {gather*} -\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2}} b}{2 \, a^{2} x} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2}}}{2 \, a x^{2}} \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 = 1.11, size = 88, normalized size = 3.14 \begin {gather*} \frac {2 i a b^{3} \left (\frac {a}{b} + x\right )}{2 a^{4} - 4 a^{3} b \left (\frac {a}{b} + x\right ) + 2 a^{2} b^{2} \left (\frac {a}{b} + x\right )^{2}} - \frac {i b^{4} \left (\frac {a}{b} + x\right )^{2}}{2 a^{4} - 4 a^{3} b \left (\frac {a}{b} + x\right ) + 2 a^{2} b^{2} \left (\frac {a}{b} + x\right )^{2}} \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 = 1.85, size = 5, normalized size = 0.18 \begin {gather*} \frac {i}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 22, normalized size = 0.79 \begin {gather*} \frac {\sqrt {-a-b\,x}}{2\,x^2\,\sqrt {a+b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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