Integrand size = 20, antiderivative size = 183 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{19/2}} \, dx=-\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}-\frac {16 b (10 A b-17 a B) (a+b x)^{7/2}}{3315 a^3 x^{13/2}}+\frac {32 b^2 (10 A b-17 a B) (a+b x)^{7/2}}{12155 a^4 x^{11/2}}-\frac {128 b^3 (10 A b-17 a B) (a+b x)^{7/2}}{109395 a^5 x^{9/2}}+\frac {256 b^4 (10 A b-17 a B) (a+b x)^{7/2}}{765765 a^6 x^{7/2}} \]
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Time = 0.05 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{19/2}} \, dx=\frac {256 b^4 (a+b x)^{7/2} (10 A b-17 a B)}{765765 a^6 x^{7/2}}-\frac {128 b^3 (a+b x)^{7/2} (10 A b-17 a B)}{109395 a^5 x^{9/2}}+\frac {32 b^2 (a+b x)^{7/2} (10 A b-17 a B)}{12155 a^4 x^{11/2}}-\frac {16 b (a+b x)^{7/2} (10 A b-17 a B)}{3315 a^3 x^{13/2}}+\frac {2 (a+b x)^{7/2} (10 A b-17 a B)}{255 a^2 x^{15/2}}-\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {\left (2 \left (-5 A b+\frac {17 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{17/2}} \, dx}{17 a} \\ & = -\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}+\frac {(8 b (10 A b-17 a B)) \int \frac {(a+b x)^{5/2}}{x^{15/2}} \, dx}{255 a^2} \\ & = -\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}-\frac {16 b (10 A b-17 a B) (a+b x)^{7/2}}{3315 a^3 x^{13/2}}-\frac {\left (16 b^2 (10 A b-17 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx}{1105 a^3} \\ & = -\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}-\frac {16 b (10 A b-17 a B) (a+b x)^{7/2}}{3315 a^3 x^{13/2}}+\frac {32 b^2 (10 A b-17 a B) (a+b x)^{7/2}}{12155 a^4 x^{11/2}}+\frac {\left (64 b^3 (10 A b-17 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{11/2}} \, dx}{12155 a^4} \\ & = -\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}-\frac {16 b (10 A b-17 a B) (a+b x)^{7/2}}{3315 a^3 x^{13/2}}+\frac {32 b^2 (10 A b-17 a B) (a+b x)^{7/2}}{12155 a^4 x^{11/2}}-\frac {128 b^3 (10 A b-17 a B) (a+b x)^{7/2}}{109395 a^5 x^{9/2}}-\frac {\left (128 b^4 (10 A b-17 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{9/2}} \, dx}{109395 a^5} \\ & = -\frac {2 A (a+b x)^{7/2}}{17 a x^{17/2}}+\frac {2 (10 A b-17 a B) (a+b x)^{7/2}}{255 a^2 x^{15/2}}-\frac {16 b (10 A b-17 a B) (a+b x)^{7/2}}{3315 a^3 x^{13/2}}+\frac {32 b^2 (10 A b-17 a B) (a+b x)^{7/2}}{12155 a^4 x^{11/2}}-\frac {128 b^3 (10 A b-17 a B) (a+b x)^{7/2}}{109395 a^5 x^{9/2}}+\frac {256 b^4 (10 A b-17 a B) (a+b x)^{7/2}}{765765 a^6 x^{7/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.62 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{19/2}} \, dx=-\frac {2 (a+b x)^{7/2} \left (-1280 A b^5 x^5+3003 a^5 (15 A+17 B x)+128 a b^4 x^4 (35 A+17 B x)-224 a^2 b^3 x^3 (45 A+34 B x)+336 a^3 b^2 x^2 (55 A+51 B x)-462 a^4 b x (65 A+68 B x)\right )}{765765 a^6 x^{17/2}} \]
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Time = 0.52 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-1280 A \,b^{5} x^{5}+2176 B a \,b^{4} x^{5}+4480 a A \,b^{4} x^{4}-7616 B \,a^{2} b^{3} x^{4}-10080 a^{2} A \,b^{3} x^{3}+17136 B \,a^{3} b^{2} x^{3}+18480 a^{3} A \,b^{2} x^{2}-31416 B \,a^{4} b \,x^{2}-30030 a^{4} A b x +51051 a^{5} B x +45045 a^{5} A \right )}{765765 x^{\frac {17}{2}} a^{6}}\) | \(125\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-1280 A \,b^{7} x^{7}+2176 B a \,b^{6} x^{7}+1920 A a \,b^{6} x^{6}-3264 B \,a^{2} b^{5} x^{6}-2400 A \,a^{2} b^{5} x^{5}+4080 B \,a^{3} b^{4} x^{5}+2800 A \,a^{3} b^{4} x^{4}-4760 B \,a^{4} b^{3} x^{4}-3150 A \,a^{4} b^{3} x^{3}+5355 B \,a^{5} b^{2} x^{3}+3465 A \,a^{5} b^{2} x^{2}+70686 B \,a^{6} b \,x^{2}+60060 A \,a^{6} b x +51051 B \,a^{7} x +45045 A \,a^{7}\right )}{765765 x^{\frac {17}{2}} a^{6}}\) | \(173\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-1280 A \,b^{8} x^{8}+2176 B a \,b^{7} x^{8}+640 A a \,b^{7} x^{7}-1088 B \,a^{2} b^{6} x^{7}-480 A \,a^{2} b^{6} x^{6}+816 B \,a^{3} b^{5} x^{6}+400 A \,a^{3} b^{5} x^{5}-680 B \,a^{4} b^{4} x^{5}-350 A \,a^{4} b^{4} x^{4}+595 B \,a^{5} b^{3} x^{4}+315 A \,a^{5} b^{3} x^{3}+76041 B \,a^{6} b^{2} x^{3}+63525 A \,a^{6} b^{2} x^{2}+121737 B \,a^{7} b \,x^{2}+105105 A \,a^{7} b x +51051 B \,a^{8} x +45045 A \,a^{8}\right )}{765765 x^{\frac {17}{2}} a^{6}}\) | \(197\) |
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Time = 0.24 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{19/2}} \, dx=-\frac {2 \, {\left (45045 \, A a^{8} + 128 \, {\left (17 \, B a b^{7} - 10 \, A b^{8}\right )} x^{8} - 64 \, {\left (17 \, B a^{2} b^{6} - 10 \, A a b^{7}\right )} x^{7} + 48 \, {\left (17 \, B a^{3} b^{5} - 10 \, A a^{2} b^{6}\right )} x^{6} - 40 \, {\left (17 \, B a^{4} b^{4} - 10 \, A a^{3} b^{5}\right )} x^{5} + 35 \, {\left (17 \, B a^{5} b^{3} - 10 \, A a^{4} b^{4}\right )} x^{4} + 63 \, {\left (1207 \, B a^{6} b^{2} + 5 \, A a^{5} b^{3}\right )} x^{3} + 231 \, {\left (527 \, B a^{7} b + 275 \, A a^{6} b^{2}\right )} x^{2} + 3003 \, {\left (17 \, B a^{8} + 35 \, A a^{7} b\right )} x\right )} \sqrt {b x + a}}{765765 \, a^{6} x^{\frac {17}{2}}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{19/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (147) = 294\).
Time = 0.21 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.42 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{19/2}} \, dx=-\frac {256 \, \sqrt {b x^{2} + a x} B b^{7}}{45045 \, a^{5} x} + \frac {512 \, \sqrt {b x^{2} + a x} A b^{8}}{153153 \, a^{6} x} + \frac {128 \, \sqrt {b x^{2} + a x} B b^{6}}{45045 \, a^{4} x^{2}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{7}}{153153 \, a^{5} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} B b^{5}}{15015 \, a^{3} x^{3}} + \frac {64 \, \sqrt {b x^{2} + a x} A b^{6}}{51051 \, a^{4} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} B b^{4}}{9009 \, a^{2} x^{4}} - \frac {160 \, \sqrt {b x^{2} + a x} A b^{5}}{153153 \, a^{3} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{3}}{1287 \, a x^{5}} + \frac {20 \, \sqrt {b x^{2} + a x} A b^{4}}{21879 \, a^{2} x^{5}} + \frac {\sqrt {b x^{2} + a x} B b^{2}}{715 \, x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{3}}{2431 \, a x^{6}} - \frac {\sqrt {b x^{2} + a x} B a b}{780 \, x^{7}} + \frac {\sqrt {b x^{2} + a x} A b^{2}}{1326 \, x^{7}} - \frac {\sqrt {b x^{2} + a x} B a^{2}}{60 \, x^{8}} - \frac {\sqrt {b x^{2} + a x} A a b}{1428 \, x^{8}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{12 \, x^{9}} - \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{476 \, x^{9}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{5 \, x^{10}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{84 \, x^{10}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{6 \, x^{11}} \]
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Time = 0.40 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{19/2}} \, dx=-\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (17 \, B a^{3} b^{16} - 10 \, A a^{2} b^{17}\right )} {\left (b x + a\right )}}{a^{8}} - \frac {17 \, {\left (17 \, B a^{4} b^{16} - 10 \, A a^{3} b^{17}\right )}}{a^{8}}\right )} + \frac {255 \, {\left (17 \, B a^{5} b^{16} - 10 \, A a^{4} b^{17}\right )}}{a^{8}}\right )} - \frac {1105 \, {\left (17 \, B a^{6} b^{16} - 10 \, A a^{5} b^{17}\right )}}{a^{8}}\right )} {\left (b x + a\right )} + \frac {12155 \, {\left (17 \, B a^{7} b^{16} - 10 \, A a^{6} b^{17}\right )}}{a^{8}}\right )} {\left (b x + a\right )} - \frac {109395 \, {\left (B a^{8} b^{16} - A a^{7} b^{17}\right )}}{a^{8}}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b}{765765 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {17}{2}} {\left | b \right |}} \]
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Time = 1.20 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{19/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a^2}{17}+\frac {2\,a\,x\,\left (35\,A\,b+17\,B\,a\right )}{255}+\frac {2\,b\,x^2\,\left (275\,A\,b+527\,B\,a\right )}{3315}-\frac {2\,b^3\,x^4\,\left (10\,A\,b-17\,B\,a\right )}{21879\,a^2}+\frac {16\,b^4\,x^5\,\left (10\,A\,b-17\,B\,a\right )}{153153\,a^3}-\frac {32\,b^5\,x^6\,\left (10\,A\,b-17\,B\,a\right )}{255255\,a^4}+\frac {128\,b^6\,x^7\,\left (10\,A\,b-17\,B\,a\right )}{765765\,a^5}-\frac {256\,b^7\,x^8\,\left (10\,A\,b-17\,B\,a\right )}{765765\,a^6}+\frac {2\,b^2\,x^3\,\left (5\,A\,b+1207\,B\,a\right )}{12155\,a}\right )}{x^{17/2}} \]
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