Optimal. Leaf size=49 \[ -\frac {\sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}+\frac {e \log \left (a+c x^2\right )}{2 c}+\frac {d x}{c} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1394, 774, 635, 205, 260} \[ -\frac {\sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}+\frac {e \log \left (a+c x^2\right )}{2 c}+\frac {d x}{c} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 774
Rule 1394
Rubi steps
\begin {align*} \int \frac {d+\frac {e}{x}}{c+\frac {a}{x^2}} \, dx &=\int \frac {x (e+d x)}{a+c x^2} \, dx\\ &=\frac {d x}{c}+\frac {\int \frac {-a d+c e x}{a+c x^2} \, dx}{c}\\ &=\frac {d x}{c}-\frac {(a d) \int \frac {1}{a+c x^2} \, dx}{c}+e \int \frac {x}{a+c x^2} \, dx\\ &=\frac {d x}{c}-\frac {\sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}+\frac {e \log \left (a+c x^2\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 1.00 \[ -\frac {\sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}+\frac {e \log \left (a+c x^2\right )}{2 c}+\frac {d x}{c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 108, normalized size = 2.20 \[ \left [\frac {d \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {a}{c}} - a}{c x^{2} + a}\right ) + 2 \, d x + e \log \left (c x^{2} + a\right )}{2 \, c}, -\frac {2 \, d \sqrt {\frac {a}{c}} \arctan \left (\frac {c x \sqrt {\frac {a}{c}}}{a}\right ) - 2 \, d x - e \log \left (c x^{2} + a\right )}{2 \, c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 43, normalized size = 0.88 \[ -\frac {a d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} + \frac {d x}{c} + \frac {e \log \left (c x^{2} + a\right )}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.88 \[ -\frac {a d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}+\frac {d x}{c}+\frac {e \ln \left (c \,x^{2}+a \right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.62, size = 42, normalized size = 0.86 \[ -\frac {a d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} + \frac {d x}{c} + \frac {e \log \left (c x^{2} + a\right )}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.59, size = 39, normalized size = 0.80 \[ \frac {e\,\ln \left (c\,x^2+a\right )}{2\,c}+\frac {d\,x}{c}-\frac {\sqrt {a}\,d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.28, size = 112, normalized size = 2.29 \[ \left (\frac {e}{2 c} - \frac {d \sqrt {- a c^{3}}}{2 c^{3}}\right ) \log {\left (x + \frac {- 2 c \left (\frac {e}{2 c} - \frac {d \sqrt {- a c^{3}}}{2 c^{3}}\right ) + e}{d} \right )} + \left (\frac {e}{2 c} + \frac {d \sqrt {- a c^{3}}}{2 c^{3}}\right ) \log {\left (x + \frac {- 2 c \left (\frac {e}{2 c} + \frac {d \sqrt {- a c^{3}}}{2 c^{3}}\right ) + e}{d} \right )} + \frac {d x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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