Integrand size = 19, antiderivative size = 136 \[ \int x^{7/2} \sqrt {b x+c x^2} \, dx=\frac {256 b^4 \left (b x+c x^2\right )^{3/2}}{3465 c^5 x^{3/2}}-\frac {128 b^3 \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt {x}}+\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac {16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]
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Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {670, 662} \[ \int x^{7/2} \sqrt {b x+c x^2} \, dx=\frac {256 b^4 \left (b x+c x^2\right )^{3/2}}{3465 c^5 x^{3/2}}-\frac {128 b^3 \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt {x}}+\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac {16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]
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Rule 662
Rule 670
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac {(8 b) \int x^{5/2} \sqrt {b x+c x^2} \, dx}{11 c} \\ & = -\frac {16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}+\frac {\left (16 b^2\right ) \int x^{3/2} \sqrt {b x+c x^2} \, dx}{33 c^2} \\ & = \frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac {16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac {\left (64 b^3\right ) \int \sqrt {x} \sqrt {b x+c x^2} \, dx}{231 c^3} \\ & = -\frac {128 b^3 \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt {x}}+\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac {16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}+\frac {\left (128 b^4\right ) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{1155 c^4} \\ & = \frac {256 b^4 \left (b x+c x^2\right )^{3/2}}{3465 c^5 x^{3/2}}-\frac {128 b^3 \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt {x}}+\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac {16 b x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.47 \[ \int x^{7/2} \sqrt {b x+c x^2} \, dx=\frac {2 (x (b+c x))^{3/2} \left (128 b^4-192 b^3 c x+240 b^2 c^2 x^2-280 b c^3 x^3+315 c^4 x^4\right )}{3465 c^5 x^{3/2}} \]
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Time = 2.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.47
method | result | size |
default | \(\frac {2 \left (c x +b \right ) \left (315 c^{4} x^{4}-280 b \,c^{3} x^{3}+240 b^{2} c^{2} x^{2}-192 b^{3} c x +128 b^{4}\right ) \sqrt {x \left (c x +b \right )}}{3465 c^{5} \sqrt {x}}\) | \(64\) |
gosper | \(\frac {2 \left (c x +b \right ) \left (315 c^{4} x^{4}-280 b \,c^{3} x^{3}+240 b^{2} c^{2} x^{2}-192 b^{3} c x +128 b^{4}\right ) \sqrt {c \,x^{2}+b x}}{3465 c^{5} \sqrt {x}}\) | \(66\) |
risch | \(\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (315 c^{5} x^{5}+35 b \,x^{4} c^{4}-40 b^{2} c^{3} x^{3}+48 x^{2} b^{3} c^{2}-64 c x \,b^{4}+128 b^{5}\right )}{3465 \sqrt {x \left (c x +b \right )}\, c^{5}}\) | \(75\) |
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Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.52 \[ \int x^{7/2} \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt {c x^{2} + b x}}{3465 \, c^{5} \sqrt {x}} \]
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\[ \int x^{7/2} \sqrt {b x+c x^2} \, dx=\int x^{\frac {7}{2}} \sqrt {x \left (b + c x\right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.47 \[ \int x^{7/2} \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt {c x + b}}{3465 \, c^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.51 \[ \int x^{7/2} \sqrt {b x+c x^2} \, dx=-\frac {256 \, b^{\frac {11}{2}}}{3465 \, c^{5}} + \frac {2 \, {\left (315 \, {\left (c x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (c x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{4}\right )}}{3465 \, c^{5}} \]
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Timed out. \[ \int x^{7/2} \sqrt {b x+c x^2} \, dx=\int x^{7/2}\,\sqrt {c\,x^2+b\,x} \,d x \]
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