Optimal. Leaf size=39 \[ \frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {d x}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 39, 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 = {388, 205} \begin {gather*} \frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {d x}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rubi steps
\begin {align*} \int \frac {c+d x^2}{a+b x^2} \, dx &=\frac {d x}{b}-\frac {(-b c+a d) \int \frac {1}{a+b x^2} \, dx}{b}\\ &=\frac {d x}{b}+\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 1.03 \begin {gather*} \frac {d x}{b}-\frac {(a d-b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \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 {c+d x^2}{a+b x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.05, size = 98, normalized size = 2.51 \begin {gather*} \left [\frac {2 \, a b d x + \sqrt {-a b} {\left (b c - a d\right )} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b^{2}}, \frac {a b d x + \sqrt {a b} {\left (b c - a d\right )} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 33, normalized size = 0.85 \begin {gather*} \frac {d x}{b} + \frac {{\left (b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 45, normalized size = 1.15 \begin {gather*} -\frac {a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}+\frac {d x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 33, normalized size = 0.85 \begin {gather*} \frac {d x}{b} + \frac {{\left (b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 32, normalized size = 0.82 \begin {gather*} \frac {d\,x}{b}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )}{\sqrt {a}\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.28, size = 82, normalized size = 2.10 \begin {gather*} \frac {\sqrt {- \frac {1}{a b^{3}}} \left (a d - b c\right ) \log {\left (- a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{a b^{3}}} \left (a d - b c\right ) \log {\left (a b \sqrt {- \frac {1}{a b^{3}}} + x \right )}}{2} + \frac {d x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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