Integrand size = 17, antiderivative size = 28 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {a e-c d x}{a c \sqrt {a+c x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {651} \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {a e-c d x}{a c \sqrt {a+c x^2}} \]
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Rule 651
Rubi steps \begin{align*} \text {integral}& = -\frac {a e-c d x}{a c \sqrt {a+c x^2}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {-a e+c d x}{a c \sqrt {a+c x^2}} \]
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Time = 1.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96
method | result | size |
gosper | \(-\frac {-c d x +a e}{\sqrt {c \,x^{2}+a}\, a c}\) | \(27\) |
trager | \(-\frac {-c d x +a e}{\sqrt {c \,x^{2}+a}\, a c}\) | \(27\) |
default | \(\frac {d x}{a \sqrt {c \,x^{2}+a}}-\frac {e}{c \sqrt {c \,x^{2}+a}}\) | \(32\) |
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none
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c^{2} x^{2} + a^{2} c} \]
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Time = 1.91 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=e \left (\begin {cases} - \frac {1}{c \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {d x}{a^{\frac {3}{2}} \sqrt {1 + \frac {c x^{2}}{a}}} \]
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none
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {d x}{\sqrt {c x^{2} + a} a} - \frac {e}{\sqrt {c x^{2} + a} c} \]
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none
Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\frac {d x}{a} - \frac {e}{c}}{\sqrt {c x^{2} + a}} \]
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Time = 9.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {\frac {e}{c}-\frac {d\,x}{a}}{\sqrt {c\,x^2+a}} \]
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