Integrand size = 17, antiderivative size = 72 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx=-\frac {(a e-c d x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {737, 211} \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a e^2+c d^2\right )}{2 a^{3/2} c^{3/2}}-\frac {(d+e x) (a e-c d x)}{2 a c \left (a+c x^2\right )} \]
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Rule 211
Rule 737
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx=\frac {-2 a d e+c d^2 x-a e^2 x}{2 a c \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}} \]
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Time = 2.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 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 | \(\frac {-\frac {\left (e^{2} a -c \,d^{2}\right ) x}{2 a c}-\frac {d e}{c}}{c \,x^{2}+a}-\frac {\ln \left (c x +\sqrt {-a c}\right ) e^{2}}{4 \sqrt {-a c}\, c}-\frac {\ln \left (c x +\sqrt {-a c}\right ) d^{2}}{4 \sqrt {-a c}\, a}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) e^{2}}{4 \sqrt {-a c}\, c}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) d^{2}}{4 \sqrt {-a c}\, a}\) | \(143\) |
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Time = 0.43 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.10 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx=\left [-\frac {4 \, a^{2} c d e + {\left (a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} x}{4 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, -\frac {2 \, a^{2} c d e - {\left (a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} x}{2 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (63) = 126\).
Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.79 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx=- \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log {\left (- a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log {\left (a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x \right )}}{4} + \frac {- 2 a d e + x \left (- a e^{2} + c d^{2}\right )}{2 a^{2} c + 2 a c^{2} x^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx=-\frac {2 \, a d e - {\left (c d^{2} - a e^{2}\right )} x}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} + \frac {{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} \]
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Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx=\frac {{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} + \frac {c d^{2} x - a e^{2} x - 2 \, a d e}{2 \, {\left (c x^{2} + a\right )} a c} \]
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Time = 9.40 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (c\,d^2+a\,e^2\right )}{2\,a^{3/2}\,c^{3/2}}-\frac {\frac {d\,e}{c}+\frac {x\,\left (a\,e^2-c\,d^2\right )}{2\,a\,c}}{c\,x^2+a} \]
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