Integrand size = 19, antiderivative size = 80 \[ \int x^{3/2} \sqrt {b x+c x^2} \, dx=\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac {8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c} \]
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Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {670, 662} \[ \int x^{3/2} \sqrt {b x+c x^2} \, dx=\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac {8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c} \]
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Rule 662
Rule 670
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac {(4 b) \int \sqrt {x} \sqrt {b x+c x^2} \, dx}{7 c} \\ & = -\frac {8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (8 b^2\right ) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{35 c^2} \\ & = \frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac {8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.52 \[ \int x^{3/2} \sqrt {b x+c x^2} \, dx=\frac {2 (x (b+c x))^{3/2} \left (8 b^2-12 b c x+15 c^2 x^2\right )}{105 c^3 x^{3/2}} \]
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Time = 2.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.52
method | result | size |
default | \(\frac {2 \left (c x +b \right ) \left (15 c^{2} x^{2}-12 b c x +8 b^{2}\right ) \sqrt {x \left (c x +b \right )}}{105 c^{3} \sqrt {x}}\) | \(42\) |
gosper | \(\frac {2 \left (c x +b \right ) \left (15 c^{2} x^{2}-12 b c x +8 b^{2}\right ) \sqrt {c \,x^{2}+b x}}{105 c^{3} \sqrt {x}}\) | \(44\) |
risch | \(\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (15 c^{3} x^{3}+3 b \,c^{2} x^{2}-4 b^{2} c x +8 b^{3}\right )}{105 \sqrt {x \left (c x +b \right )}\, c^{3}}\) | \(53\) |
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Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.61 \[ \int x^{3/2} \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt {c x^{2} + b x}}{105 \, c^{3} \sqrt {x}} \]
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\[ \int x^{3/2} \sqrt {b x+c x^2} \, dx=\int x^{\frac {3}{2}} \sqrt {x \left (b + c x\right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.52 \[ \int x^{3/2} \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt {c x + b}}{105 \, c^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.58 \[ \int x^{3/2} \sqrt {b x+c x^2} \, dx=-\frac {16 \, b^{\frac {7}{2}}}{105 \, c^{3}} + \frac {2 \, {\left (15 \, {\left (c x + b\right )}^{\frac {7}{2}} - 42 \, {\left (c x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{2}\right )}}{105 \, c^{3}} \]
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Timed out. \[ \int x^{3/2} \sqrt {b x+c x^2} \, dx=\int x^{3/2}\,\sqrt {c\,x^2+b\,x} \,d x \]
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