Optimal. Leaf size=55 \[ \frac {d \tanh ^{-1}\left (\frac {c x}{a}\right )}{2 a^3 c}+\frac {a^2 e+c^2 d x}{2 a^2 c^2 \left (a^2-c^2 x^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {639, 208} \begin {gather*} \frac {a^2 e+c^2 d x}{2 a^2 c^2 \left (a^2-c^2 x^2\right )}+\frac {d \tanh ^{-1}\left (\frac {c x}{a}\right )}{2 a^3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 639
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a^2-c^2 x^2\right )^2} \, dx &=\frac {a^2 e+c^2 d x}{2 a^2 c^2 \left (a^2-c^2 x^2\right )}+\frac {d \int \frac {1}{a^2-c^2 x^2} \, dx}{2 a^2}\\ &=\frac {a^2 e+c^2 d x}{2 a^2 c^2 \left (a^2-c^2 x^2\right )}+\frac {d \tanh ^{-1}\left (\frac {c x}{a}\right )}{2 a^3 c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 58, normalized size = 1.05 \begin {gather*} \frac {d \tanh ^{-1}\left (\frac {c x}{a}\right )}{2 a^3 c}+\frac {a^2 (-e)-c^2 d x}{2 a^2 c^2 \left (c^2 x^2-a^2\right )} \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}{\left (a^2-c^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.45, size = 87, normalized size = 1.58 \begin {gather*} -\frac {2 \, a c^{2} d x + 2 \, a^{3} e - {\left (c^{3} d x^{2} - a^{2} c d\right )} \log \left (c x + a\right ) + {\left (c^{3} d x^{2} - a^{2} c d\right )} \log \left (c x - a\right )}{4 \, {\left (a^{3} c^{4} x^{2} - a^{5} c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 71, normalized size = 1.29 \begin {gather*} \frac {d \log \left ({\left | c x + a \right |}\right )}{4 \, a^{3} c} - \frac {d \log \left ({\left | c x - a \right |}\right )}{4 \, a^{3} c} - \frac {c^{2} d x + a^{2} e}{2 \, {\left (c^{2} x^{2} - a^{2}\right )} a^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 102, normalized size = 1.85 \begin {gather*} \frac {e}{4 \left (c x +a \right ) a \,c^{2}}-\frac {e}{4 \left (c x -a \right ) a \,c^{2}}-\frac {d}{4 \left (c x +a \right ) a^{2} c}-\frac {d}{4 \left (c x -a \right ) a^{2} c}-\frac {d \ln \left (c x -a \right )}{4 a^{3} c}+\frac {d \ln \left (c x +a \right )}{4 a^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 68, normalized size = 1.24 \begin {gather*} -\frac {c^{2} d x + a^{2} e}{2 \, {\left (a^{2} c^{4} x^{2} - a^{4} c^{2}\right )}} + \frac {d \log \left (c x + a\right )}{4 \, a^{3} c} - \frac {d \log \left (c x - a\right )}{4 \, a^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 46, normalized size = 0.84 \begin {gather*} \frac {\frac {e}{2\,c^2}+\frac {d\,x}{2\,a^2}}{a^2-c^2\,x^2}+\frac {d\,\mathrm {atanh}\left (\frac {c\,x}{a}\right )}{2\,a^3\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 56, normalized size = 1.02 \begin {gather*} \frac {- a^{2} e - c^{2} d x}{- 2 a^{4} c^{2} + 2 a^{2} c^{4} x^{2}} + \frac {d \left (- \frac {\log {\left (- \frac {a}{c} + x \right )}}{4} + \frac {\log {\left (\frac {a}{c} + x \right )}}{4}\right )}{a^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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