Optimal. Leaf size=63 \[ \frac {(a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{3/2}}-\frac {x (b c-a d)}{2 c d \left (c+d x^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {385, 205} \begin {gather*} \frac {(a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{3/2}}-\frac {x (b c-a d)}{2 c d \left (c+d x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rubi steps
\begin {align*} \int \frac {a+b x^2}{\left (c+d x^2\right )^2} \, dx &=-\frac {(b c-a d) x}{2 c d \left (c+d x^2\right )}+\frac {(b c+a d) \int \frac {1}{c+d x^2} \, dx}{2 c d}\\ &=-\frac {(b c-a d) x}{2 c d \left (c+d x^2\right )}+\frac {(b c+a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 1.00 \begin {gather*} \frac {(a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{3/2}}-\frac {x (b c-a d)}{2 c d \left (c+d x^2\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 {a+b x^2}{\left (c+d x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.82, size = 182, normalized size = 2.89 \begin {gather*} \left [-\frac {{\left (b c^{2} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x}{4 \, {\left (c^{2} d^{3} x^{2} + c^{3} d^{2}\right )}}, \frac {{\left (b c^{2} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - {\left (b c^{2} d - a c d^{2}\right )} x}{2 \, {\left (c^{2} d^{3} x^{2} + c^{3} d^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 57, normalized size = 0.90 \begin {gather*} \frac {{\left (b c + a d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {c d} c d} - \frac {b c x - a d x}{2 \, {\left (d x^{2} + c\right )} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 1.08 \begin {gather*} \frac {a \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}\, c}+\frac {b \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}\, d}+\frac {\left (a d -b c \right ) x}{2 \left (d \,x^{2}+c \right ) c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.11, size = 57, normalized size = 0.90 \begin {gather*} -\frac {{\left (b c - a d\right )} x}{2 \, {\left (c d^{2} x^{2} + c^{2} d\right )}} + \frac {{\left (b c + a d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {c d} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.01, size = 51, normalized size = 0.81 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )\,\left (a\,d+b\,c\right )}{2\,c^{3/2}\,d^{3/2}}+\frac {x\,\left (a\,d-b\,c\right )}{2\,c\,d\,\left (d\,x^2+c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.39, size = 112, normalized size = 1.78 \begin {gather*} \frac {x \left (a d - b c\right )}{2 c^{2} d + 2 c d^{2} x^{2}} - \frac {\sqrt {- \frac {1}{c^{3} d^{3}}} \left (a d + b c\right ) \log {\left (- c^{2} d \sqrt {- \frac {1}{c^{3} d^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{c^{3} d^{3}}} \left (a d + b c\right ) \log {\left (c^{2} d \sqrt {- \frac {1}{c^{3} d^{3}}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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