Optimal. Leaf size=34 \[ \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {d x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, 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 = {15, 211}
\begin {gather*} \frac {x \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 211
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {d x^2} \left (a+b x^2\right )} \, dx &=\frac {x \int \frac {1}{a+b x^2} \, dx}{\sqrt {d x^2}}\\ &=\frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} \frac {x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 24, normalized size = 0.71
method | result | size |
default | \(\frac {x \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {d \,x^{2}}\, \sqrt {a b}}\) | \(24\) |
risch | \(-\frac {x \ln \left (b x +\sqrt {-a b}\right )}{2 \sqrt {d \,x^{2}}\, \sqrt {-a b}}+\frac {x \ln \left (-b x +\sqrt {-a b}\right )}{2 \sqrt {d \,x^{2}}\, \sqrt {-a b}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 23, normalized size = 0.68 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {d x^{2}} b}{\sqrt {a b d}}\right )}{\sqrt {a b d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.83, size = 94, normalized size = 2.76 \begin {gather*} \left [-\frac {\sqrt {-a b d} \log \left (\frac {b d x^{2} - a d - 2 \, \sqrt {-a b d} \sqrt {d x^{2}}}{b x^{2} + a}\right )}{2 \, a b d}, \frac {\sqrt {a b d} \arctan \left (\frac {\sqrt {a b d} \sqrt {d x^{2}}}{a d}\right )}{a b d}\right ] \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 {x}{\sqrt {d x^{2}} \left (a + b x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.88, size = 22, normalized size = 0.65 \begin {gather*} \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} \sqrt {d} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 23, normalized size = 0.68 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x^2}}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}\,\sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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