Optimal. Leaf size=57 \[ -\frac {a e-c d x}{2 a c \left (a+c x^2\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}} \]
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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {653, 211}
\begin {gather*} \frac {d \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}-\frac {a e-c d x}{2 a c \left (a+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 653
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+c x^2\right )^2} \, dx &=-\frac {a e-c d x}{2 a c \left (a+c x^2\right )}+\frac {d \int \frac {1}{a+c x^2} \, dx}{2 a}\\ &=-\frac {a e-c d x}{2 a c \left (a+c x^2\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 57, normalized size = 1.00 \begin {gather*} \frac {-a e+c d x}{2 a c \left (a+c x^2\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 49, normalized size = 0.86
method | result | size |
default | \(\frac {2 c d x -2 a e}{4 a c \left (c \,x^{2}+a \right )}+\frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 a \sqrt {a c}}\) | \(49\) |
risch | \(\frac {\frac {x d}{2 a}-\frac {e}{2 c}}{c \,x^{2}+a}-\frac {d \ln \left (c x +\sqrt {-a c}\right )}{4 \sqrt {-a c}\, a}+\frac {d \ln \left (-c x +\sqrt {-a c}\right )}{4 \sqrt {-a c}\, a}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 49, normalized size = 0.86 \begin {gather*} \frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a} + \frac {c d x - a e}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.12, size = 142, normalized size = 2.49 \begin {gather*} \left [\frac {2 \, a c d x - 2 \, a^{2} e - {\left (c d x^{2} + a d\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{4 \, {\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}, \frac {a c d x - a^{2} e + {\left (c d x^{2} + a d\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{2 \, {\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 90, normalized size = 1.58 \begin {gather*} d \left (- \frac {\sqrt {- \frac {1}{a^{3} c}} \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} c}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} c}} \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} c}} + x \right )}}{4}\right ) + \frac {- a e + c d x}{2 a^{2} c + 2 a c^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.89, size = 48, normalized size = 0.84 \begin {gather*} \frac {d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a} + \frac {c d x - a e}{2 \, {\left (c x^{2} + a\right )} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 44, normalized size = 0.77 \begin {gather*} \frac {d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {c}}-\frac {\frac {e}{2\,c}-\frac {d\,x}{2\,a}}{c\,x^2+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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