Optimal. Leaf size=36 \[ -\frac {\sqrt {b} c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {c}{a x} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {21, 325, 205} \[ -\frac {\sqrt {b} c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {c}{a x} \]
Antiderivative was successfully verified.
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Rule 21
Rule 205
Rule 325
Rubi steps
\begin {align*} \int \frac {a c+b c x^2}{x^2 \left (a+b x^2\right )^2} \, dx &=c \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx\\ &=-\frac {c}{a x}-\frac {(b c) \int \frac {1}{a+b x^2} \, dx}{a}\\ &=-\frac {c}{a x}-\frac {\sqrt {b} c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 1.00 \[ c \left (-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {1}{a x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 86, normalized size = 2.39 \[ \left [\frac {c x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 2 \, c}{2 \, a x}, -\frac {c x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + c}{a x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 31, normalized size = 0.86 \[ -\frac {b c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {c}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.89 \[ -\frac {b c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}-\frac {c}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.35, size = 31, normalized size = 0.86 \[ -\frac {b c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {c}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 28, normalized size = 0.78 \[ -\frac {c}{a\,x}-\frac {\sqrt {b}\,c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 66, normalized size = 1.83 \[ c \left (\frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (- \frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{3}}} \log {\left (\frac {a^{2} \sqrt {- \frac {b}{a^{3}}}}{b} + x \right )}}{2} - \frac {1}{a x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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