Optimal. Leaf size=49 \[ -\frac {d \log \left (a+c x^2\right )}{2 a}+\frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {d \log (x)}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {801, 635, 205, 260} \begin {gather*} -\frac {d \log \left (a+c x^2\right )}{2 a}+\frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {d \log (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 801
Rubi steps
\begin {align*} \int \frac {d+e x}{x \left (a+c x^2\right )} \, dx &=\int \left (\frac {d}{a x}+\frac {a e-c d x}{a \left (a+c x^2\right )}\right ) \, dx\\ &=\frac {d \log (x)}{a}+\frac {\int \frac {a e-c d x}{a+c x^2} \, dx}{a}\\ &=\frac {d \log (x)}{a}-\frac {(c d) \int \frac {x}{a+c x^2} \, dx}{a}+e \int \frac {1}{a+c x^2} \, dx\\ &=\frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {d \log (x)}{a}-\frac {d \log \left (a+c x^2\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 1.00 \begin {gather*} -\frac {d \log \left (a+c x^2\right )}{2 a}+\frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\frac {d \log (x)}{a} \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}{x \left (a+c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 109, normalized size = 2.22 \begin {gather*} \left [-\frac {c d \log \left (c x^{2} + a\right ) - 2 \, c d \log \relax (x) + \sqrt {-a c} e \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{2 \, a c}, -\frac {c d \log \left (c x^{2} + a\right ) - 2 \, c d \log \relax (x) - 2 \, \sqrt {a c} e \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 = 40, normalized size = 0.82 \begin {gather*} \frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right ) e}{\sqrt {a c}} - \frac {d \log \left (c x^{2} + a\right )}{2 \, a} + \frac {d \log \left ({\left | x \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 39, normalized size = 0.80 \begin {gather*} \frac {e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}+\frac {d \ln \relax (x )}{a}-\frac {d \ln \left (c \,x^{2}+a \right )}{2 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 38, normalized size = 0.78 \begin {gather*} \frac {e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}} - \frac {d \log \left (c x^{2} + a\right )}{2 \, a} + \frac {d \log \relax (x)}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 216, normalized size = 4.41 \begin {gather*} \frac {d\,\ln \relax (x)}{a}-\frac {d\,\ln \left (a\,e\,\sqrt {-a^3\,c}+3\,a^2\,c\,d-a^2\,c\,e\,x+3\,c\,d\,x\,\sqrt {-a^3\,c}\right )}{2\,a}-\frac {d\,\ln \left (a\,e\,\sqrt {-a^3\,c}-3\,a^2\,c\,d+a^2\,c\,e\,x+3\,c\,d\,x\,\sqrt {-a^3\,c}\right )}{2\,a}+\frac {e\,\ln \left (a\,e\,\sqrt {-a^3\,c}-3\,a^2\,c\,d+a^2\,c\,e\,x+3\,c\,d\,x\,\sqrt {-a^3\,c}\right )\,\sqrt {-a^3\,c}}{2\,a^2\,c}-\frac {e\,\ln \left (a\,e\,\sqrt {-a^3\,c}+3\,a^2\,c\,d-a^2\,c\,e\,x+3\,c\,d\,x\,\sqrt {-a^3\,c}\right )\,\sqrt {-a^3\,c}}{2\,a^2\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.47, size = 321, normalized size = 6.55 \begin {gather*} \left (- \frac {d}{2 a} - \frac {e \sqrt {- a^{3} c}}{2 a^{2} c}\right ) \log {\left (x + \frac {- 12 a^{2} c d \left (- \frac {d}{2 a} - \frac {e \sqrt {- a^{3} c}}{2 a^{2} c}\right )^{2} + 2 a^{2} e^{2} \left (- \frac {d}{2 a} - \frac {e \sqrt {- a^{3} c}}{2 a^{2} c}\right ) + 6 a c d^{2} \left (- \frac {d}{2 a} - \frac {e \sqrt {- a^{3} c}}{2 a^{2} c}\right ) - 2 a d e^{2} + 6 c d^{3}}{a e^{3} + 9 c d^{2} e} \right )} + \left (- \frac {d}{2 a} + \frac {e \sqrt {- a^{3} c}}{2 a^{2} c}\right ) \log {\left (x + \frac {- 12 a^{2} c d \left (- \frac {d}{2 a} + \frac {e \sqrt {- a^{3} c}}{2 a^{2} c}\right )^{2} + 2 a^{2} e^{2} \left (- \frac {d}{2 a} + \frac {e \sqrt {- a^{3} c}}{2 a^{2} c}\right ) + 6 a c d^{2} \left (- \frac {d}{2 a} + \frac {e \sqrt {- a^{3} c}}{2 a^{2} c}\right ) - 2 a d e^{2} + 6 c d^{3}}{a e^{3} + 9 c d^{2} e} \right )} + \frac {d \log {\relax (x )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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