Integrand size = 15, antiderivative size = 75 \[ \int \frac {d+e x}{\left (a+c x^2\right )^3} \, dx=\frac {-a e+c d x}{4 a c \left (a+c x^2\right )^2}+\frac {3 d x}{8 a^2 \left (a+c x^2\right )}+\frac {3 d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}} \]
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Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {653, 205, 211} \[ \int \frac {d+e x}{\left (a+c x^2\right )^3} \, dx=\frac {3 d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}}+\frac {3 d x}{8 a^2 \left (a+c x^2\right )}-\frac {a e-c d x}{4 a c \left (a+c x^2\right )^2} \]
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Rule 205
Rule 211
Rule 653
Rubi steps \begin{align*} \text {integral}& = -\frac {a e-c d x}{4 a c \left (a+c x^2\right )^2}+\frac {(3 d) \int \frac {1}{\left (a+c x^2\right )^2} \, dx}{4 a} \\ & = -\frac {a e-c d x}{4 a c \left (a+c x^2\right )^2}+\frac {3 d x}{8 a^2 \left (a+c x^2\right )}+\frac {(3 d) \int \frac {1}{a+c x^2} \, dx}{8 a^2} \\ & = -\frac {a e-c d x}{4 a c \left (a+c x^2\right )^2}+\frac {3 d x}{8 a^2 \left (a+c x^2\right )}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int \frac {d+e x}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {\sqrt {a} \left (-2 a^2 e+5 a c d x+3 c^2 d x^3\right )}{\left (a+c x^2\right )^2}+3 \sqrt {c} d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c} \]
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Time = 2.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {2 c d x -2 a e}{8 a c \left (c \,x^{2}+a \right )^{2}}+\frac {3 d \left (\frac {x}{2 a \left (c \,x^{2}+a \right )}+\frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 a \sqrt {a c}}\right )}{4 a}\) | \(70\) |
risch | \(\frac {\frac {3 d c \,x^{3}}{8 a^{2}}+\frac {5 x d}{8 a}-\frac {e}{4 c}}{\left (c \,x^{2}+a \right )^{2}}-\frac {3 d \ln \left (c x +\sqrt {-a c}\right )}{16 \sqrt {-a c}\, a^{2}}+\frac {3 d \ln \left (-c x +\sqrt {-a c}\right )}{16 \sqrt {-a c}\, a^{2}}\) | \(83\) |
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Time = 0.46 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.83 \[ \int \frac {d+e x}{\left (a+c x^2\right )^3} \, dx=\left [\frac {6 \, a c^{2} d x^{3} + 10 \, a^{2} c d x - 4 \, a^{3} e - 3 \, {\left (c^{2} d x^{4} + 2 \, a c d x^{2} + a^{2} d\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{16 \, {\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}, \frac {3 \, a c^{2} d x^{3} + 5 \, a^{2} c d x - 2 \, a^{3} e + 3 \, {\left (c^{2} d x^{4} + 2 \, a c d x^{2} + a^{2} d\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{8 \, {\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}\right ] \]
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Time = 0.24 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.65 \[ \int \frac {d+e x}{\left (a+c x^2\right )^3} \, dx=d \left (- \frac {3 \sqrt {- \frac {1}{a^{5} c}} \log {\left (- a^{3} \sqrt {- \frac {1}{a^{5} c}} + x \right )}}{16} + \frac {3 \sqrt {- \frac {1}{a^{5} c}} \log {\left (a^{3} \sqrt {- \frac {1}{a^{5} c}} + x \right )}}{16}\right ) + \frac {- 2 a^{2} e + 5 a c d x + 3 c^{2} d x^{3}}{8 a^{4} c + 16 a^{3} c^{2} x^{2} + 8 a^{2} c^{3} x^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {d+e x}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, c^{2} d x^{3} + 5 \, a c d x - 2 \, a^{2} e}{8 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} + \frac {3 \, d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2}} \]
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Time = 0.32 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80 \[ \int \frac {d+e x}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2}} + \frac {3 \, c^{2} d x^{3} + 5 \, a c d x - 2 \, a^{2} e}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c} \]
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Time = 9.35 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {d+e x}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {5\,d\,x}{8\,a}-\frac {e}{4\,c}+\frac {3\,c\,d\,x^3}{8\,a^2}}{a^2+2\,a\,c\,x^2+c^2\,x^4}+\frac {3\,d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {c}} \]
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