Integrand size = 20, antiderivative size = 39 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{\sqrt {x}} \, dx=\frac {2}{3} A b x^{3/2}+\frac {2}{5} (b B+A c) x^{5/2}+\frac {2}{7} B c x^{7/2} \]
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Time = 0.01 (sec) , antiderivative size = 39, 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} \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{\sqrt {x}} \, dx=\frac {2}{5} x^{5/2} (A c+b B)+\frac {2}{3} A b x^{3/2}+\frac {2}{7} B c x^{7/2} \]
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Rule 779
Rubi steps \begin{align*} \text {integral}& = \int \left (A b \sqrt {x}+(b B+A c) x^{3/2}+B c x^{5/2}\right ) \, dx \\ & = \frac {2}{3} A b x^{3/2}+\frac {2}{5} (b B+A c) x^{5/2}+\frac {2}{7} B c x^{7/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{\sqrt {x}} \, dx=\frac {2}{105} x^{3/2} (7 A (5 b+3 c x)+3 B x (7 b+5 c x)) \]
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Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (15 B c \,x^{2}+21 A c x +21 B b x +35 A b \right )}{105}\) | \(28\) |
derivativedivides | \(\frac {2 A b \,x^{\frac {3}{2}}}{3}+\frac {2 \left (A c +B b \right ) x^{\frac {5}{2}}}{5}+\frac {2 B c \,x^{\frac {7}{2}}}{7}\) | \(28\) |
default | \(\frac {2 A b \,x^{\frac {3}{2}}}{3}+\frac {2 \left (A c +B b \right ) x^{\frac {5}{2}}}{5}+\frac {2 B c \,x^{\frac {7}{2}}}{7}\) | \(28\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (15 B c \,x^{2}+21 A c x +21 B b x +35 A b \right )}{105}\) | \(28\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (15 B c \,x^{2}+21 A c x +21 B b x +35 A b \right )}{105}\) | \(28\) |
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Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.77 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{\sqrt {x}} \, dx=\frac {2}{105} \, {\left (15 \, B c x^{3} + 35 \, A b x + 21 \, {\left (B b + A c\right )} x^{2}\right )} \sqrt {x} \]
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Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{\sqrt {x}} \, dx=\frac {2 A b x^{\frac {3}{2}}}{3} + \frac {2 A c x^{\frac {5}{2}}}{5} + \frac {2 B b x^{\frac {5}{2}}}{5} + \frac {2 B c x^{\frac {7}{2}}}{7} \]
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Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{\sqrt {x}} \, dx=\frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} + \frac {2}{5} \, {\left (B b + A c\right )} x^{\frac {5}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{\sqrt {x}} \, dx=\frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{5} \, B b x^{\frac {5}{2}} + \frac {2}{5} \, A c x^{\frac {5}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} \]
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Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{\sqrt {x}} \, dx=\frac {2\,x^{3/2}\,\left (35\,A\,b+21\,A\,c\,x+21\,B\,b\,x+15\,B\,c\,x^2\right )}{105} \]
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