Optimal. Leaf size=25 \[ \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 25, 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 = {21, 211}
\begin {gather*} \frac {c \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 211
Rubi steps
\begin {align*} \int \frac {a c+b c x^2}{\left (a+b x^2\right )^2} \, dx &=c \int \frac {1}{a+b x^2} \, dx\\ &=\frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 17, normalized size = 0.68
method | result | size |
default | \(\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\) | \(17\) |
risch | \(-\frac {c \ln \left (b x +\sqrt {-a b}\right )}{2 \sqrt {-a b}}+\frac {c \ln \left (-b x +\sqrt {-a b}\right )}{2 \sqrt {-a b}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 16, normalized size = 0.64 \begin {gather*} \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 69, normalized size = 2.76 \begin {gather*} \left [-\frac {\sqrt {-a b} c \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b}, \frac {\sqrt {a b} c \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a b}\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. 54 vs.
\(2 (24) = 48\).
time = 0.09, size = 54, normalized size = 2.16 \begin {gather*} c \left (- \frac {\sqrt {- \frac {1}{a b}} \log {\left (- a \sqrt {- \frac {1}{a b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (a \sqrt {- \frac {1}{a b}} + x \right )}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.30, size = 16, normalized size = 0.64 \begin {gather*} \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 17, normalized size = 0.68 \begin {gather*} \frac {c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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