Optimal. Leaf size=43 \[ -\frac {c}{a x}-\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \]
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Rubi [A]
time = 0.01, antiderivative size = 43, 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 = {464, 211}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)}{a^{3/2} \sqrt {b}}-\frac {c}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 464
Rubi steps
\begin {align*} \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )} \, dx &=-\frac {c}{a x}-\frac {(b c-a d) \int \frac {1}{a+b x^2} \, dx}{a}\\ &=-\frac {c}{a x}-\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 42, normalized size = 0.98 \begin {gather*} -\frac {c}{a x}+\frac {(-b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 37, normalized size = 0.86
method | result | size |
default | \(\frac {\left (a d -b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \sqrt {a b}}-\frac {c}{a x}\) | \(37\) |
risch | \(-\frac {c}{a x}-\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) d}{2 \sqrt {-a b}}+\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) b c}{2 \sqrt {-a b}\, a}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) d}{2 \sqrt {-a b}}-\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) b c}{2 \sqrt {-a b}\, a}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 37, normalized size = 0.86 \begin {gather*} -\frac {{\left (b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {c}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.27, size = 105, normalized size = 2.44 \begin {gather*} \left [\frac {\sqrt {-a b} {\left (b c - a d\right )} x \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, a b c}{2 \, a^{2} b x}, -\frac {\sqrt {a b} {\left (b c - a d\right )} x \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + a b c}{a^{2} b x}\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.
\(2 (36) = 72\).
time = 0.18, size = 82, normalized size = 1.91 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{3} b}} \left (a d - b c\right ) \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a^{3} b}} \left (a d - b c\right ) \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{2} - \frac {c}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.37, size = 37, normalized size = 0.86 \begin {gather*} -\frac {{\left (b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {c}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 34, normalized size = 0.79 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )}{a^{3/2}\,\sqrt {b}}-\frac {c}{a\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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