Optimal. Leaf size=43 \[ \frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-\frac {e \log \left (a-c x^2\right )}{2 c} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {635, 208, 260} \begin {gather*} \frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-\frac {e \log \left (a-c x^2\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 260
Rule 635
Rubi steps
\begin {align*} \int \frac {d+e x}{a-c x^2} \, dx &=d \int \frac {1}{a-c x^2} \, dx+e \int \frac {x}{a-c x^2} \, dx\\ &=\frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-\frac {e \log \left (a-c x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.00 \begin {gather*} \frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-\frac {e \log \left (a-c x^2\right )}{2 c} \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 {d+e x}{a-c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 102, normalized size = 2.37 \begin {gather*} \left [-\frac {a e \log \left (c x^{2} - a\right ) - \sqrt {a c} d \log \left (\frac {c x^{2} + 2 \, \sqrt {a c} x + a}{c x^{2} - a}\right )}{2 \, a c}, -\frac {a e \log \left (c x^{2} - a\right ) + 2 \, \sqrt {-a c} d \arctan \left (\frac {\sqrt {-a c} x}{a}\right )}{2 \, a c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 37, normalized size = 0.86 \begin {gather*} -\frac {d \arctan \left (\frac {c x}{\sqrt {-a c}}\right )}{\sqrt {-a c}} - \frac {e \log \left (c x^{2} - a\right )}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 34, normalized size = 0.79 \begin {gather*} \frac {d \arctanh \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}-\frac {e \ln \left (c \,x^{2}-a \right )}{2 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 49, normalized size = 1.14 \begin {gather*} -\frac {d \log \left (\frac {c x - \sqrt {a c}}{c x + \sqrt {a c}}\right )}{2 \, \sqrt {a c}} - \frac {e \log \left (c x^{2} - a\right )}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 103, normalized size = 2.40 \begin {gather*} \frac {d\,\ln \left (a\,c+x\,\sqrt {a\,c^3}\right )\,\sqrt {a\,c^3}}{2\,a\,c^2}-\frac {e\,\ln \left (x\,\sqrt {a\,c^3}-a\,c\right )}{2\,c}-\frac {e\,\ln \left (a\,c+x\,\sqrt {a\,c^3}\right )}{2\,c}-\frac {d\,\ln \left (x\,\sqrt {a\,c^3}-a\,c\right )\,\sqrt {a\,c^3}}{2\,a\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.28, size = 119, normalized size = 2.77 \begin {gather*} - \left (\frac {e}{2 c} - \frac {d \sqrt {a c^{3}}}{2 a c^{2}}\right ) \log {\left (x + \frac {- 2 a c \left (\frac {e}{2 c} - \frac {d \sqrt {a c^{3}}}{2 a c^{2}}\right ) + a e}{c d} \right )} - \left (\frac {e}{2 c} + \frac {d \sqrt {a c^{3}}}{2 a c^{2}}\right ) \log {\left (x + \frac {- 2 a c \left (\frac {e}{2 c} + \frac {d \sqrt {a c^{3}}}{2 a c^{2}}\right ) + a e}{c d} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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