Integrand size = 19, antiderivative size = 108 \[ \int x^{5/2} \sqrt {b x+c x^2} \, dx=-\frac {32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]
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Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {670, 662} \[ \int x^{5/2} \sqrt {b x+c x^2} \, dx=-\frac {32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]
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Rule 662
Rule 670
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {(2 b) \int x^{3/2} \sqrt {b x+c x^2} \, dx}{3 c} \\ & = -\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}+\frac {\left (8 b^2\right ) \int \sqrt {x} \sqrt {b x+c x^2} \, dx}{21 c^2} \\ & = \frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {\left (16 b^3\right ) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{105 c^3} \\ & = -\frac {32 b^3 \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {4 b \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.49 \[ \int x^{5/2} \sqrt {b x+c x^2} \, dx=\frac {2 (x (b+c x))^{3/2} \left (-16 b^3+24 b^2 c x-30 b c^2 x^2+35 c^3 x^3\right )}{315 c^4 x^{3/2}} \]
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Time = 1.99 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.49
method | result | size |
default | \(-\frac {2 \left (c x +b \right ) \left (-35 c^{3} x^{3}+30 b \,c^{2} x^{2}-24 b^{2} c x +16 b^{3}\right ) \sqrt {x \left (c x +b \right )}}{315 c^{4} \sqrt {x}}\) | \(53\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-35 c^{3} x^{3}+30 b \,c^{2} x^{2}-24 b^{2} c x +16 b^{3}\right ) \sqrt {c \,x^{2}+b x}}{315 c^{4} \sqrt {x}}\) | \(55\) |
risch | \(-\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (-35 c^{4} x^{4}-5 b \,c^{3} x^{3}+6 b^{2} c^{2} x^{2}-8 b^{3} c x +16 b^{4}\right )}{315 \sqrt {x \left (c x +b \right )}\, c^{4}}\) | \(64\) |
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Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.56 \[ \int x^{5/2} \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt {c x^{2} + b x}}{315 \, c^{4} \sqrt {x}} \]
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\[ \int x^{5/2} \sqrt {b x+c x^2} \, dx=\int x^{\frac {5}{2}} \sqrt {x \left (b + c x\right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.49 \[ \int x^{5/2} \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt {c x + b}}{315 \, c^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.54 \[ \int x^{5/2} \sqrt {b x+c x^2} \, dx=\frac {32 \, b^{\frac {9}{2}}}{315 \, c^{4}} + \frac {2 \, {\left (35 \, {\left (c x + b\right )}^{\frac {9}{2}} - 135 \, {\left (c x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{3}\right )}}{315 \, c^{4}} \]
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Timed out. \[ \int x^{5/2} \sqrt {b x+c x^2} \, dx=\int x^{5/2}\,\sqrt {c\,x^2+b\,x} \,d x \]
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