Integrand size = 19, antiderivative size = 136 \[ \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {256 b^4 \left (b x+c x^2\right )^{5/2}}{15015 c^5 x^{5/2}}-\frac {128 b^3 \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac {32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt {x}}-\frac {16 b \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \]
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Time = 0.04 (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} \[ \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {256 b^4 \left (b x+c x^2\right )^{5/2}}{15015 c^5 x^{5/2}}-\frac {128 b^3 \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac {32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt {x}}-\frac {16 b \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \]
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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 )^{5/2}}{13 c}-\frac {(8 b) \int x^{3/2} \left (b x+c x^2\right )^{3/2} \, dx}{13 c} \\ & = -\frac {16 b \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}+\frac {\left (48 b^2\right ) \int \sqrt {x} \left (b x+c x^2\right )^{3/2} \, dx}{143 c^2} \\ & = \frac {32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt {x}}-\frac {16 b \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}-\frac {\left (64 b^3\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{\sqrt {x}} \, dx}{429 c^3} \\ & = -\frac {128 b^3 \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac {32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt {x}}-\frac {16 b \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}+\frac {\left (128 b^4\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{3003 c^4} \\ & = \frac {256 b^4 \left (b x+c x^2\right )^{5/2}}{15015 c^5 x^{5/2}}-\frac {128 b^3 \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac {32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt {x}}-\frac {16 b \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.47 \[ \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 (x (b+c x))^{5/2} \left (128 b^4-320 b^3 c x+560 b^2 c^2 x^2-840 b c^3 x^3+1155 c^4 x^4\right )}{15015 c^5 x^{5/2}} \]
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Time = 2.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.49
method | result | size |
gosper | \(\frac {2 \left (c x +b \right ) \left (1155 c^{4} x^{4}-840 b \,c^{3} x^{3}+560 b^{2} c^{2} x^{2}-320 b^{3} c x +128 b^{4}\right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15015 c^{5} x^{\frac {3}{2}}}\) | \(66\) |
default | \(\frac {2 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right )^{2} \left (1155 c^{4} x^{4}-840 b \,c^{3} x^{3}+560 b^{2} c^{2} x^{2}-320 b^{3} c x +128 b^{4}\right )}{15015 \sqrt {x}\, c^{5}}\) | \(66\) |
risch | \(\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (1155 x^{6} c^{6}+1470 x^{5} b \,c^{5}+35 b^{2} x^{4} c^{4}-40 x^{3} b^{3} c^{3}+48 b^{4} x^{2} c^{2}-64 b^{5} x c +128 b^{6}\right )}{15015 \sqrt {x \left (c x +b \right )}\, c^{5}}\) | \(86\) |
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Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.60 \[ \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left (1155 \, c^{6} x^{6} + 1470 \, b c^{5} x^{5} + 35 \, b^{2} c^{4} x^{4} - 40 \, b^{3} c^{3} x^{3} + 48 \, b^{4} c^{2} x^{2} - 64 \, b^{5} c x + 128 \, b^{6}\right )} \sqrt {c x^{2} + b x}}{15015 \, c^{5} \sqrt {x}} \]
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\[ \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx=\int x^{\frac {5}{2}} \left (x \left (b + c x\right )\right )^{\frac {3}{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.08 \[ \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left (5 \, {\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} x^{5} + 13 \, {\left (315 \, b c^{5} x^{6} + 35 \, b^{2} c^{4} x^{5} - 40 \, b^{3} c^{3} x^{4} + 48 \, b^{4} c^{2} x^{3} - 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )} x^{4}\right )} \sqrt {c x + b}}{45045 \, c^{5} x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.16 \[ \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2}{9009} \, c {\left (\frac {256 \, b^{\frac {13}{2}}}{c^{6}} + \frac {693 \, {\left (c x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (c x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{5}}{c^{6}}\right )} - \frac {2}{3465} \, b {\left (\frac {128 \, b^{\frac {11}{2}}}{c^{5}} - \frac {315 \, {\left (c x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (c x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{4}}{c^{5}}\right )} \]
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Timed out. \[ \int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx=\int x^{5/2}\,{\left (c\,x^2+b\,x\right )}^{3/2} \,d x \]
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