Optimal. Leaf size=31 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {247, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{a+b (c+d x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 109, normalized size = 3.52 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {b d^{2} x^{2} + 2 \, b c d x + b c^{2} - 2 \, \sqrt {-a b} {\left (d x + c\right )} - a}{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}\right )}{2 \, a b d}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} {\left (d x + c\right )}}{a}\right )}{a b d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 24, normalized size = 0.77 \[ \frac {\arctan \left (\frac {b d x + b c}{\sqrt {a b}}\right )}{\sqrt {a b} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 34, normalized size = 1.10 \[ \frac {\arctan \left (\frac {2 b \,d^{2} x +2 b d c}{2 \sqrt {a b}\, d}\right )}{\sqrt {a b}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.56, size = 30, normalized size = 0.97 \[ \frac {\arctan \left (\frac {b d^{2} x + b c d}{\sqrt {a b} d}\right )}{\sqrt {a b} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 27, normalized size = 0.87 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,c+\sqrt {b}\,d\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.21, size = 61, normalized size = 1.97 \[ \frac {- \frac {\sqrt {- \frac {1}{a b}} \log {\left (x + \frac {- a \sqrt {- \frac {1}{a b}} + c}{d} \right )}}{2} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (x + \frac {a \sqrt {- \frac {1}{a b}} + c}{d} \right )}}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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