Integrand size = 22, antiderivative size = 63 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{5} A b^2 x^{5/2}+\frac {2}{7} b (b B+2 A c) x^{7/2}+\frac {2}{9} c (2 b B+A c) x^{9/2}+\frac {2}{11} B c^2 x^{11/2} \]
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Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {779} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{5} A b^2 x^{5/2}+\frac {2}{9} c x^{9/2} (A c+2 b B)+\frac {2}{7} b x^{7/2} (2 A c+b B)+\frac {2}{11} B c^2 x^{11/2} \]
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Rule 779
Rubi steps \begin{align*} \text {integral}& = \int \left (A b^2 x^{3/2}+b (b B+2 A c) x^{5/2}+c (2 b B+A c) x^{7/2}+B c^2 x^{9/2}\right ) \, dx \\ & = \frac {2}{5} A b^2 x^{5/2}+\frac {2}{7} b (b B+2 A c) x^{7/2}+\frac {2}{9} c (2 b B+A c) x^{9/2}+\frac {2}{11} B c^2 x^{11/2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2 x^{5/2} \left (11 A \left (63 b^2+90 b c x+35 c^2 x^2\right )+5 B x \left (99 b^2+154 b c x+63 c^2 x^2\right )\right )}{3465} \]
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Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (315 B \,c^{2} x^{3}+385 A \,c^{2} x^{2}+770 B b c \,x^{2}+990 A b c x +495 b^{2} B x +693 A \,b^{2}\right )}{3465}\) | \(52\) |
derivativedivides | \(\frac {2 B \,c^{2} x^{\frac {11}{2}}}{11}+\frac {2 \left (A \,c^{2}+2 B b c \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 A b c +B \,b^{2}\right ) x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{2} x^{\frac {5}{2}}}{5}\) | \(52\) |
default | \(\frac {2 B \,c^{2} x^{\frac {11}{2}}}{11}+\frac {2 \left (A \,c^{2}+2 B b c \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (2 A b c +B \,b^{2}\right ) x^{\frac {7}{2}}}{7}+\frac {2 A \,b^{2} x^{\frac {5}{2}}}{5}\) | \(52\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (315 B \,c^{2} x^{3}+385 A \,c^{2} x^{2}+770 B b c \,x^{2}+990 A b c x +495 b^{2} B x +693 A \,b^{2}\right )}{3465}\) | \(52\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (315 B \,c^{2} x^{3}+385 A \,c^{2} x^{2}+770 B b c \,x^{2}+990 A b c x +495 b^{2} B x +693 A \,b^{2}\right )}{3465}\) | \(52\) |
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Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{3465} \, {\left (315 \, B c^{2} x^{5} + 693 \, A b^{2} x^{2} + 385 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 495 \, {\left (B b^{2} + 2 \, A b c\right )} x^{3}\right )} \sqrt {x} \]
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Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.27 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2 A b^{2} x^{\frac {5}{2}}}{5} + \frac {4 A b c x^{\frac {7}{2}}}{7} + \frac {2 A c^{2} x^{\frac {9}{2}}}{9} + \frac {2 B b^{2} x^{\frac {7}{2}}}{7} + \frac {4 B b c x^{\frac {9}{2}}}{9} + \frac {2 B c^{2} x^{\frac {11}{2}}}{11} \]
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Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{11} \, B c^{2} x^{\frac {11}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} + \frac {2}{9} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (B b^{2} + 2 \, A b c\right )} x^{\frac {7}{2}} \]
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Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {x}} \, dx=\frac {2}{11} \, B c^{2} x^{\frac {11}{2}} + \frac {4}{9} \, B b c x^{\frac {9}{2}} + \frac {2}{9} \, A c^{2} x^{\frac {9}{2}} + \frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + \frac {4}{7} \, A b c x^{\frac {7}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{\sqrt {x}} \, dx=x^{7/2}\,\left (\frac {2\,B\,b^2}{7}+\frac {4\,A\,c\,b}{7}\right )+x^{9/2}\,\left (\frac {2\,A\,c^2}{9}+\frac {4\,B\,b\,c}{9}\right )+\frac {2\,A\,b^2\,x^{5/2}}{5}+\frac {2\,B\,c^2\,x^{11/2}}{11} \]
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