Optimal. Leaf size=47 \[ \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {c x}{2 b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 47, 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, 288, 205} \begin {gather*} \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {c x}{2 b \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 205
Rule 288
Rubi steps
\begin {align*} \int \frac {x^2 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx &=c \int \frac {x^2}{\left (a+b x^2\right )^2} \, dx\\ &=-\frac {c x}{2 b \left (a+b x^2\right )}+\frac {c \int \frac {1}{a+b x^2} \, dx}{2 b}\\ &=-\frac {c x}{2 b \left (a+b x^2\right )}+\frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.00 \begin {gather*} c \left (\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {x}{2 b \left (a+b x^2\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.66, 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 b^{3} x^{2} + a^{2} b^{2}\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 b^{3} x^{2} + a^{2} b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 37, normalized size = 0.79 \begin {gather*} \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b} - \frac {c x}{2 \, {\left (b x^{2} + a\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.81 \begin {gather*} -\frac {c x}{2 \left (b \,x^{2}+a \right ) b}+\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.39, size = 38, normalized size = 0.81 \begin {gather*} -\frac {c x}{2 \, {\left (b^{2} x^{2} + a b\right )}} + \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 35, normalized size = 0.74 \begin {gather*} \frac {c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,b^{3/2}}-\frac {c\,x}{2\,b\,\left (b\,x^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.22, size = 80, normalized size = 1.70 \begin {gather*} c \left (- \frac {x}{2 a b + 2 b^{2} x^{2}} - \frac {\sqrt {- \frac {1}{a b^{3}}} \log {\left (- a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a b^{3}}} \log {\left (a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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