Integrand size = 24, antiderivative size = 170 \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {32 b^3 (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{15015 c^5 x^{5/2}}-\frac {16 b^2 (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac {4 b (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt {x}}-\frac {2 (8 b B-13 A c) \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \]
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Time = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {808, 670, 662} \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {32 b^3 \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{15015 c^5 x^{5/2}}-\frac {16 b^2 \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{3003 c^4 x^{3/2}}+\frac {4 b \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{429 c^3 \sqrt {x}}-\frac {2 \sqrt {x} \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{143 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \]
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Rule 662
Rule 670
Rule 808
Rubi steps \begin{align*} \text {integral}& = \frac {2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}+\frac {\left (2 \left (\frac {3}{2} (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right )\right ) \int x^{3/2} \left (b x+c x^2\right )^{3/2} \, dx}{13 c} \\ & = -\frac {2 (8 b B-13 A c) \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}+\frac {(6 b (8 b B-13 A c)) \int \sqrt {x} \left (b x+c x^2\right )^{3/2} \, dx}{143 c^2} \\ & = \frac {4 b (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt {x}}-\frac {2 (8 b B-13 A c) \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}-\frac {\left (8 b^2 (8 b B-13 A c)\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{\sqrt {x}} \, dx}{429 c^3} \\ & = -\frac {16 b^2 (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac {4 b (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt {x}}-\frac {2 (8 b B-13 A c) \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}+\frac {\left (16 b^3 (8 b B-13 A c)\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{3003 c^4} \\ & = \frac {32 b^3 (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{15015 c^5 x^{5/2}}-\frac {16 b^2 (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac {4 b (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt {x}}-\frac {2 (8 b B-13 A c) \sqrt {x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.55 \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 (x (b+c x))^{5/2} \left (128 b^4 B+105 c^4 x^3 (13 A+11 B x)-70 b c^3 x^2 (13 A+12 B x)+40 b^2 c^2 x (13 A+14 B x)-16 b^3 c (13 A+20 B x)\right )}{15015 c^5 x^{5/2}} \]
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Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-1155 B \,x^{4} c^{4}-1365 A \,c^{4} x^{3}+840 B b \,c^{3} x^{3}+910 A b \,c^{3} x^{2}-560 B \,b^{2} c^{2} x^{2}-520 A \,b^{2} c^{2} x +320 B \,b^{3} c x +208 A \,b^{3} c -128 b^{4} B \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15015 c^{5} x^{\frac {3}{2}}}\) | \(107\) |
default | \(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (c x +b \right )^{2} \left (-1155 B \,x^{4} c^{4}-1365 A \,c^{4} x^{3}+840 B b \,c^{3} x^{3}+910 A b \,c^{3} x^{2}-560 B \,b^{2} c^{2} x^{2}-520 A \,b^{2} c^{2} x +320 B \,b^{3} c x +208 A \,b^{3} c -128 b^{4} B \right )}{15015 \sqrt {x}\, c^{5}}\) | \(107\) |
risch | \(-\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (-1155 B \,c^{6} x^{6}-1365 A \,c^{6} x^{5}-1470 B b \,c^{5} x^{5}-1820 A b \,c^{5} x^{4}-35 B \,b^{2} c^{4} x^{4}-65 A \,b^{2} c^{4} x^{3}+40 B \,b^{3} c^{3} x^{3}+78 A \,b^{3} c^{3} x^{2}-48 B \,b^{4} c^{2} x^{2}-104 A \,b^{4} c^{2} x +64 B \,b^{5} c x +208 A \,b^{5} c -128 B \,b^{6}\right )}{15015 \sqrt {x \left (c x +b \right )}\, c^{5}}\) | \(153\) |
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Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.88 \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left (1155 \, B c^{6} x^{6} + 128 \, B b^{6} - 208 \, A b^{5} c + 105 \, {\left (14 \, B b c^{5} + 13 \, A c^{6}\right )} x^{5} + 35 \, {\left (B b^{2} c^{4} + 52 \, A b c^{5}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} c^{3} - 13 \, A b^{2} c^{4}\right )} x^{3} + 6 \, {\left (8 \, B b^{4} c^{2} - 13 \, A b^{3} c^{3}\right )} x^{2} - 8 \, {\left (8 \, B b^{5} c - 13 \, A b^{4} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{15015 \, c^{5} \sqrt {x}} \]
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\[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\int x^{\frac {3}{2}} \left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )\, dx \]
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Time = 0.21 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.61 \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left ({\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} x^{4} + 11 \, {\left (35 \, b c^{4} x^{5} + 5 \, b^{2} c^{3} x^{4} - 6 \, b^{3} c^{2} x^{3} + 8 \, b^{4} c x^{2} - 16 \, b^{5} x\right )} x^{3}\right )} \sqrt {c x + b} A}{3465 \, c^{4} x^{4}} + \frac {2 \, {\left (5 \, {\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} x^{5} + 13 \, {\left (315 \, b c^{5} x^{6} + 35 \, b^{2} c^{4} x^{5} - 40 \, b^{3} c^{3} x^{4} + 48 \, b^{4} c^{2} x^{3} - 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )} x^{4}\right )} \sqrt {c x + b} B}{45045 \, c^{5} x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (140) = 280\).
Time = 0.29 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.74 \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\frac {2}{9009} \, B c {\left (\frac {256 \, b^{\frac {13}{2}}}{c^{6}} + \frac {693 \, {\left (c x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (c x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{5}}{c^{6}}\right )} - \frac {2}{3465} \, B b {\left (\frac {128 \, b^{\frac {11}{2}}}{c^{5}} - \frac {315 \, {\left (c x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (c x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{4}}{c^{5}}\right )} - \frac {2}{3465} \, A c {\left (\frac {128 \, b^{\frac {11}{2}}}{c^{5}} - \frac {315 \, {\left (c x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (c x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{4}}{c^{5}}\right )} + \frac {2}{315} \, A b {\left (\frac {16 \, b^{\frac {9}{2}}}{c^{4}} + \frac {35 \, {\left (c x + b\right )}^{\frac {9}{2}} - 135 \, {\left (c x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{3}}{c^{4}}\right )} \]
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Timed out. \[ \int x^{3/2} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx=\int x^{3/2}\,{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \]
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