Optimal. Leaf size=19 \[ \frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {a+b x}\right )}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {65, 221}
\begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {a+b x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 221
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x} \sqrt {4+a+b x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {4+x^2}} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 26, normalized size = 1.37 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {4+a+b x}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs.
\(2(15)=30\).
time = 0.17, size = 86, normalized size = 4.53
method | result | size |
default | \(\frac {\sqrt {\left (b x +a \right ) \left (b x +a +4\right )}\, \ln \left (\frac {\frac {a b}{2}+\frac {b \left (a +4\right )}{2}+b^{2} x}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}+\left (a b +b \left (a +4\right )\right ) x +a \left (a +4\right )}\right )}{\sqrt {b x +a}\, \sqrt {b x +a +4}\, \sqrt {b^{2}}}\) | \(86\) |
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. 48 vs.
\(2 (15) = 30\).
time = 0.27, size = 48, normalized size = 2.53 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + a^{2} + 2 \, {\left (a b + 2 \, b\right )} x + 4 \, a} b + 4 \, b\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (15) = 30\).
time = 0.29, size = 31, normalized size = 1.63 \begin {gather*} -\frac {\log \left (-b x + \sqrt {b x + a + 4} \sqrt {b x + a} - a - 2\right )}{b} \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 \frac {1}{\sqrt {a + b x} \sqrt {a + b x + 4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 27, normalized size = 1.42 \begin {gather*} -\frac {2 \ln \left (\sqrt {a+b x+4}-\sqrt {a+b x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 50, normalized size = 2.63 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {a+4}-\sqrt {a+b\,x+4}\right )}{\sqrt {-b^2}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{\sqrt {-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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