Integrand size = 20, antiderivative size = 37 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{9/2}} \, dx=-\frac {2 A b}{5 x^{5/2}}-\frac {2 (b B+A c)}{3 x^{3/2}}-\frac {2 B c}{\sqrt {x}} \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {779} \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{9/2}} \, dx=-\frac {2 (A c+b B)}{3 x^{3/2}}-\frac {2 A b}{5 x^{5/2}}-\frac {2 B c}{\sqrt {x}} \]
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Rule 779
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A b}{x^{7/2}}+\frac {b B+A c}{x^{5/2}}+\frac {B c}{x^{3/2}}\right ) \, dx \\ & = -\frac {2 A b}{5 x^{5/2}}-\frac {2 (b B+A c)}{3 x^{3/2}}-\frac {2 B c}{\sqrt {x}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{9/2}} \, dx=-\frac {2 \left (3 A b+5 b B x+5 A c x+15 B c x^2\right )}{15 x^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76
method | result | size |
gosper | \(-\frac {2 \left (15 B c \,x^{2}+5 A c x +5 B b x +3 A b \right )}{15 x^{\frac {5}{2}}}\) | \(28\) |
derivativedivides | \(-\frac {2 A b}{5 x^{\frac {5}{2}}}-\frac {2 \left (A c +B b \right )}{3 x^{\frac {3}{2}}}-\frac {2 B c}{\sqrt {x}}\) | \(28\) |
default | \(-\frac {2 A b}{5 x^{\frac {5}{2}}}-\frac {2 \left (A c +B b \right )}{3 x^{\frac {3}{2}}}-\frac {2 B c}{\sqrt {x}}\) | \(28\) |
trager | \(-\frac {2 \left (15 B c \,x^{2}+5 A c x +5 B b x +3 A b \right )}{15 x^{\frac {5}{2}}}\) | \(28\) |
risch | \(-\frac {2 \left (15 B c \,x^{2}+5 A c x +5 B b x +3 A b \right )}{15 x^{\frac {5}{2}}}\) | \(28\) |
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{9/2}} \, dx=-\frac {2 \, {\left (15 \, B c x^{2} + 3 \, A b + 5 \, {\left (B b + A c\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \]
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Time = 0.37 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.24 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{9/2}} \, dx=- \frac {2 A b}{5 x^{\frac {5}{2}}} - \frac {2 A c}{3 x^{\frac {3}{2}}} - \frac {2 B b}{3 x^{\frac {3}{2}}} - \frac {2 B c}{\sqrt {x}} \]
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Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{9/2}} \, dx=-\frac {2 \, {\left (15 \, B c x^{2} + 3 \, A b + 5 \, {\left (B b + A c\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \]
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{9/2}} \, dx=-\frac {2 \, {\left (15 \, B c x^{2} + 5 \, B b x + 5 \, A c x + 3 \, A b\right )}}{15 \, x^{\frac {5}{2}}} \]
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Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{x^{9/2}} \, dx=-\frac {2\,B\,c\,x^2+\left (\frac {2\,A\,c}{3}+\frac {2\,B\,b}{3}\right )\,x+\frac {2\,A\,b}{5}}{x^{5/2}} \]
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