Optimal. Leaf size=72 \[ -\frac {(a e-c d x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {737, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a e^2+c d^2\right )}{2 a^{3/2} c^{3/2}}-\frac {(d+e x) (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 737
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx &=-\frac {(a e-c d x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \int \frac {1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {(a e-c d x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 77, normalized size = 1.07 \begin {gather*} \frac {-2 a d e+c d^2 x-a e^2 x}{2 a c \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 74, normalized size = 1.03
method | result | size |
default | \(\frac {-\frac {\left (e^{2} a -c \,d^{2}\right ) x}{2 a c}-\frac {d e}{c}}{c \,x^{2}+a}+\frac {\left (e^{2} a +c \,d^{2}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 a c \sqrt {a c}}\) | \(74\) |
risch | \(\frac {-\frac {\left (e^{2} a -c \,d^{2}\right ) x}{2 a c}-\frac {d e}{c}}{c \,x^{2}+a}-\frac {\ln \left (c x +\sqrt {-a c}\right ) e^{2}}{4 \sqrt {-a c}\, c}-\frac {\ln \left (c x +\sqrt {-a c}\right ) d^{2}}{4 \sqrt {-a c}\, a}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) e^{2}}{4 \sqrt {-a c}\, c}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) d^{2}}{4 \sqrt {-a c}\, a}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 72, normalized size = 1.00 \begin {gather*} -\frac {2 \, a d e - {\left (c d^{2} - a e^{2}\right )} x}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} + \frac {{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.19, size = 216, normalized size = 3.00 \begin {gather*} \left [\frac {2 \, a c^{2} d^{2} x - 2 \, a^{2} c x e^{2} - 4 \, a^{2} c d e - {\left (c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}\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^{3} x^{2} + a^{3} c^{2}\right )}}, \frac {a c^{2} d^{2} x - a^{2} c x e^{2} - 2 \, a^{2} c d e + {\left (c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{2 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (63) = 126\).
time = 0.28, size = 129, normalized size = 1.79 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log {\left (- a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log {\left (a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x \right )}}{4} + \frac {- 2 a d e + x \left (- a e^{2} + c d^{2}\right )}{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 = 1.16, size = 69, normalized size = 0.96 \begin {gather*} \frac {{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} + \frac {c d^{2} x - a x e^{2} - 2 \, a d 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 = 0.32, size = 68, normalized size = 0.94 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (c\,d^2+a\,e^2\right )}{2\,a^{3/2}\,c^{3/2}}-\frac {\frac {d\,e}{c}+\frac {x\,\left (a\,e^2-c\,d^2\right )}{2\,a\,c}}{c\,x^2+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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