Optimal. Leaf size=39 \[ \frac {d x}{b}+\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {396, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)}{\sqrt {a} b^{3/2}}+\frac {d x}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 396
Rubi steps
\begin {align*} \int \frac {c+d x^2}{a+b x^2} \, dx &=\frac {d x}{b}-\frac {(-b c+a d) \int \frac {1}{a+b x^2} \, dx}{b}\\ &=\frac {d x}{b}+\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 40, normalized size = 1.03 \begin {gather*} \frac {d x}{b}-\frac {(-b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 34, normalized size = 0.87
method | result | size |
default | \(\frac {d x}{b}+\frac {\left (-a d +b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b \sqrt {a b}}\) | \(34\) |
risch | \(\frac {d x}{b}-\frac {\ln \left (b x -\sqrt {-a b}\right ) a d}{2 b \sqrt {-a b}}+\frac {\ln \left (b x -\sqrt {-a b}\right ) c}{2 \sqrt {-a b}}+\frac {\ln \left (-b x -\sqrt {-a b}\right ) a d}{2 b \sqrt {-a b}}-\frac {\ln \left (-b x -\sqrt {-a b}\right ) c}{2 \sqrt {-a b}}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 33, normalized size = 0.85 \begin {gather*} \frac {d x}{b} + \frac {{\left (b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.19, size = 98, normalized size = 2.51 \begin {gather*} \left [\frac {2 \, a b d x + \sqrt {-a b} {\left (b c - a d\right )} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b^{2}}, \frac {a b d x + \sqrt {a b} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (34) = 68\).
time = 0.17, size = 82, normalized size = 2.10 \begin {gather*} \frac {\sqrt {- \frac {1}{a b^{3}}} \left (a d - b c\right ) \log {\left (- a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{a b^{3}}} \left (a d - b c\right ) \log {\left (a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{2} + \frac {d x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.48, size = 33, normalized size = 0.85 \begin {gather*} \frac {d x}{b} + \frac {{\left (b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 32, normalized size = 0.82 \begin {gather*} \frac {d\,x}{b}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )}{\sqrt {a}\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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