Optimal. Leaf size=47 \[ \frac {c x}{2 a \left (a+b x^2\right )}+\frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {21, 205, 211}
\begin {gather*} \frac {c \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {c x}{2 a \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 205
Rule 211
Rubi steps
\begin {align*} \int \frac {a c+b c x^2}{\left (a+b x^2\right )^3} \, dx &=c \int \frac {1}{\left (a+b x^2\right )^2} \, dx\\ &=\frac {c x}{2 a \left (a+b x^2\right )}+\frac {c \int \frac {1}{a+b x^2} \, dx}{2 a}\\ &=\frac {c x}{2 a \left (a+b x^2\right )}+\frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 1.00 \begin {gather*} c \left (\frac {x}{2 a \left (a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 38, normalized size = 0.81
method | result | size |
default | \(c \left (\frac {x}{2 a \left (b \,x^{2}+a \right )}+\frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )\) | \(38\) |
risch | \(\frac {c x}{2 a \left (b \,x^{2}+a \right )}-\frac {\ln \left (b x +\sqrt {-a b}\right ) c}{4 \sqrt {-a b}\, a}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) c}{4 \sqrt {-a b}\, a}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 37, normalized size = 0.79 \begin {gather*} \frac {c x}{2 \, {\left (a b x^{2} + a^{2}\right )}} + \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 128, normalized size = 2.72 \begin {gather*} \left [\frac {2 \, a b c x - {\left (b c x^{2} + a c\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{4 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}, \frac {a b c x + {\left (b c x^{2} + a c\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{2 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}\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. 80 vs.
\(2 (39) = 78\).
time = 0.13, size = 80, normalized size = 1.70 \begin {gather*} c \left (\frac {x}{2 a^{2} + 2 a b x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b}} \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.97, size = 37, normalized size = 0.79 \begin {gather*} \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} + \frac {c x}{2 \, {\left (b x^{2} + a\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 35, normalized size = 0.74 \begin {gather*} \frac {c\,x}{2\,a\,\left (b\,x^2+a\right )}+\frac {c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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