Optimal. Leaf size=49 \[ -\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}+\frac {d \log \left (a+c x^2\right )}{2 c}+\frac {e x}{c} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {774, 635, 205, 260} \begin {gather*} -\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}+\frac {d \log \left (a+c x^2\right )}{2 c}+\frac {e x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 774
Rubi steps
\begin {align*} \int \frac {x (d+e x)}{a+c x^2} \, dx &=\frac {e x}{c}+\frac {\int \frac {-a e+c d x}{a+c x^2} \, dx}{c}\\ &=\frac {e x}{c}+d \int \frac {x}{a+c x^2} \, dx-\frac {(a e) \int \frac {1}{a+c x^2} \, dx}{c}\\ &=\frac {e x}{c}-\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}+\frac {d \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 \begin {gather*} -\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{c^{3/2}}+\frac {d \log \left (a+c x^2\right )}{2 c}+\frac {e x}{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 {x (d+e x)}{a+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 108, normalized size = 2.20 \begin {gather*} \left [\frac {e \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {a}{c}} - a}{c x^{2} + a}\right ) + 2 \, e x + d \log \left (c x^{2} + a\right )}{2 \, c}, -\frac {2 \, e \sqrt {\frac {a}{c}} \arctan \left (\frac {c x \sqrt {\frac {a}{c}}}{a}\right ) - 2 \, e x - d \log \left (c x^{2} + a\right )}{2 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 44, normalized size = 0.90 \begin {gather*} -\frac {a \arctan \left (\frac {c x}{\sqrt {a c}}\right ) e}{\sqrt {a c} c} + \frac {x e}{c} + \frac {d \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 = 43, normalized size = 0.88 \begin {gather*} -\frac {a e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}+\frac {d \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {e x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 42, normalized size = 0.86 \begin {gather*} -\frac {a e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} + \frac {e x}{c} + \frac {d \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 = 1.07, size = 39, normalized size = 0.80 \begin {gather*} \frac {d\,\ln \left (c\,x^2+a\right )}{2\,c}+\frac {e\,x}{c}-\frac {\sqrt {a}\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.30, size = 112, normalized size = 2.29 \begin {gather*} \left (\frac {d}{2 c} - \frac {e \sqrt {- a c^{3}}}{2 c^{3}}\right ) \log {\left (x + \frac {- 2 c \left (\frac {d}{2 c} - \frac {e \sqrt {- a c^{3}}}{2 c^{3}}\right ) + d}{e} \right )} + \left (\frac {d}{2 c} + \frac {e \sqrt {- a c^{3}}}{2 c^{3}}\right ) \log {\left (x + \frac {- 2 c \left (\frac {d}{2 c} + \frac {e \sqrt {- a c^{3}}}{2 c^{3}}\right ) + d}{e} \right )} + \frac {e x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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