Integrand size = 20, antiderivative size = 183 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{5/2}}{39 a^2 x^{13/2}}-\frac {16 b (2 A b-3 a B) (a+b x)^{5/2}}{429 a^3 x^{11/2}}+\frac {32 b^2 (2 A b-3 a B) (a+b x)^{5/2}}{1287 a^4 x^{9/2}}-\frac {128 b^3 (2 A b-3 a B) (a+b x)^{5/2}}{9009 a^5 x^{7/2}}+\frac {256 b^4 (2 A b-3 a B) (a+b x)^{5/2}}{45045 a^6 x^{5/2}} \]
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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} \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=\frac {256 b^4 (a+b x)^{5/2} (2 A b-3 a B)}{45045 a^6 x^{5/2}}-\frac {128 b^3 (a+b x)^{5/2} (2 A b-3 a B)}{9009 a^5 x^{7/2}}+\frac {32 b^2 (a+b x)^{5/2} (2 A b-3 a B)}{1287 a^4 x^{9/2}}-\frac {16 b (a+b x)^{5/2} (2 A b-3 a B)}{429 a^3 x^{11/2}}+\frac {2 (a+b x)^{5/2} (2 A b-3 a B)}{39 a^2 x^{13/2}}-\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}+\frac {\left (2 \left (-5 A b+\frac {15 a B}{2}\right )\right ) \int \frac {(a+b x)^{3/2}}{x^{15/2}} \, dx}{15 a} \\ & = -\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{5/2}}{39 a^2 x^{13/2}}+\frac {(8 b (2 A b-3 a B)) \int \frac {(a+b x)^{3/2}}{x^{13/2}} \, dx}{39 a^2} \\ & = -\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{5/2}}{39 a^2 x^{13/2}}-\frac {16 b (2 A b-3 a B) (a+b x)^{5/2}}{429 a^3 x^{11/2}}-\frac {\left (16 b^2 (2 A b-3 a B)\right ) \int \frac {(a+b x)^{3/2}}{x^{11/2}} \, dx}{143 a^3} \\ & = -\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{5/2}}{39 a^2 x^{13/2}}-\frac {16 b (2 A b-3 a B) (a+b x)^{5/2}}{429 a^3 x^{11/2}}+\frac {32 b^2 (2 A b-3 a B) (a+b x)^{5/2}}{1287 a^4 x^{9/2}}+\frac {\left (64 b^3 (2 A b-3 a B)\right ) \int \frac {(a+b x)^{3/2}}{x^{9/2}} \, dx}{1287 a^4} \\ & = -\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{5/2}}{39 a^2 x^{13/2}}-\frac {16 b (2 A b-3 a B) (a+b x)^{5/2}}{429 a^3 x^{11/2}}+\frac {32 b^2 (2 A b-3 a B) (a+b x)^{5/2}}{1287 a^4 x^{9/2}}-\frac {128 b^3 (2 A b-3 a B) (a+b x)^{5/2}}{9009 a^5 x^{7/2}}-\frac {\left (128 b^4 (2 A b-3 a B)\right ) \int \frac {(a+b x)^{3/2}}{x^{7/2}} \, dx}{9009 a^5} \\ & = -\frac {2 A (a+b x)^{5/2}}{15 a x^{15/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{5/2}}{39 a^2 x^{13/2}}-\frac {16 b (2 A b-3 a B) (a+b x)^{5/2}}{429 a^3 x^{11/2}}+\frac {32 b^2 (2 A b-3 a B) (a+b x)^{5/2}}{1287 a^4 x^{9/2}}-\frac {128 b^3 (2 A b-3 a B) (a+b x)^{5/2}}{9009 a^5 x^{7/2}}+\frac {256 b^4 (2 A b-3 a B) (a+b x)^{5/2}}{45045 a^6 x^{5/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.61 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 (a+b x)^{5/2} \left (-256 A b^5 x^5+1680 a^3 b^2 x^2 (A+B x)+128 a b^4 x^4 (5 A+3 B x)-160 a^2 b^3 x^3 (7 A+6 B x)-210 a^4 b x (11 A+12 B x)+231 a^5 (13 A+15 B x)\right )}{45045 a^6 x^{15/2}} \]
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Time = 0.51 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-256 A \,b^{5} x^{5}+384 B a \,b^{4} x^{5}+640 a A \,b^{4} x^{4}-960 B \,a^{2} b^{3} x^{4}-1120 a^{2} A \,b^{3} x^{3}+1680 B \,a^{3} b^{2} x^{3}+1680 a^{3} A \,b^{2} x^{2}-2520 B \,a^{4} b \,x^{2}-2310 a^{4} A b x +3465 a^{5} B x +3003 a^{5} A \right )}{45045 x^{\frac {15}{2}} a^{6}}\) | \(125\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-256 A \,b^{6} x^{6}+384 B a \,b^{5} x^{6}+384 A a \,b^{5} x^{5}-576 B \,a^{2} b^{4} x^{5}-480 A \,a^{2} b^{4} x^{4}+720 B \,a^{3} b^{3} x^{4}+560 A \,a^{3} b^{3} x^{3}-840 B \,a^{4} b^{2} x^{3}-630 A \,a^{4} b^{2} x^{2}+945 B \,a^{5} b \,x^{2}+693 A \,a^{5} b x +3465 B \,a^{6} x +3003 A \,a^{6}\right )}{45045 x^{\frac {15}{2}} a^{6}}\) | \(149\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-256 A \,b^{7} x^{7}+384 B a \,b^{6} x^{7}+128 A a \,b^{6} x^{6}-192 B \,a^{2} b^{5} x^{6}-96 A \,a^{2} b^{5} x^{5}+144 B \,a^{3} b^{4} x^{5}+80 A \,a^{3} b^{4} x^{4}-120 B \,a^{4} b^{3} x^{4}-70 A \,a^{4} b^{3} x^{3}+105 B \,a^{5} b^{2} x^{3}+63 A \,a^{5} b^{2} x^{2}+4410 B \,a^{6} b \,x^{2}+3696 A \,a^{6} b x +3465 B \,a^{7} x +3003 A \,a^{7}\right )}{45045 x^{\frac {15}{2}} a^{6}}\) | \(173\) |
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Time = 0.23 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 \, {\left (3003 \, A a^{7} + 128 \, {\left (3 \, B a b^{6} - 2 \, A b^{7}\right )} x^{7} - 64 \, {\left (3 \, B a^{2} b^{5} - 2 \, A a b^{6}\right )} x^{6} + 48 \, {\left (3 \, B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )} x^{5} - 40 \, {\left (3 \, B a^{4} b^{3} - 2 \, A a^{3} b^{4}\right )} x^{4} + 35 \, {\left (3 \, B a^{5} b^{2} - 2 \, A a^{4} b^{3}\right )} x^{3} + 63 \, {\left (70 \, B a^{6} b + A a^{5} b^{2}\right )} x^{2} + 231 \, {\left (15 \, B a^{7} + 16 \, A a^{6} b\right )} x\right )} \sqrt {b x + a}}{45045 \, a^{6} x^{\frac {15}{2}}} \]
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Timed out. \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (147) = 294\).
Time = 0.20 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {256 \, \sqrt {b x^{2} + a x} B b^{6}}{15015 \, a^{5} x} + \frac {512 \, \sqrt {b x^{2} + a x} A b^{7}}{45045 \, a^{6} x} + \frac {128 \, \sqrt {b x^{2} + a x} B b^{5}}{15015 \, a^{4} x^{2}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{6}}{45045 \, a^{5} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} B b^{4}}{5005 \, a^{3} x^{3}} + \frac {64 \, \sqrt {b x^{2} + a x} A b^{5}}{15015 \, a^{4} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} B b^{3}}{3003 \, a^{2} x^{4}} - \frac {32 \, \sqrt {b x^{2} + a x} A b^{4}}{9009 \, a^{3} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{429 \, a x^{5}} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{3}}{1287 \, a^{2} x^{5}} + \frac {3 \, \sqrt {b x^{2} + a x} B b}{715 \, x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{715 \, a x^{6}} + \frac {3 \, \sqrt {b x^{2} + a x} B a}{65 \, x^{7}} + \frac {\sqrt {b x^{2} + a x} A b}{390 \, x^{7}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{5 \, x^{8}} + \frac {\sqrt {b x^{2} + a x} A a}{30 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{6 \, x^{9}} \]
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Time = 0.35 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (3 \, B a^{2} b^{14} - 2 \, A a b^{15}\right )} {\left (b x + a\right )}}{a^{7}} - \frac {15 \, {\left (3 \, B a^{3} b^{14} - 2 \, A a^{2} b^{15}\right )}}{a^{7}}\right )} + \frac {195 \, {\left (3 \, B a^{4} b^{14} - 2 \, A a^{3} b^{15}\right )}}{a^{7}}\right )} - \frac {715 \, {\left (3 \, B a^{5} b^{14} - 2 \, A a^{4} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {6435 \, {\left (3 \, B a^{6} b^{14} - 2 \, A a^{5} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )} - \frac {9009 \, {\left (B a^{7} b^{14} - A a^{6} b^{15}\right )}}{a^{7}}\right )} {\left (b x + a\right )}^{\frac {5}{2}} b}{45045 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {15}{2}} {\left | b \right |}} \]
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Time = 1.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^{17/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{15}+x\,\left (\frac {32\,A\,b}{195}+\frac {2\,B\,a}{13}\right )-\frac {x^7\,\left (512\,A\,b^7-768\,B\,a\,b^6\right )}{45045\,a^6}-\frac {2\,b^2\,x^3\,\left (2\,A\,b-3\,B\,a\right )}{1287\,a^2}+\frac {16\,b^3\,x^4\,\left (2\,A\,b-3\,B\,a\right )}{9009\,a^3}-\frac {32\,b^4\,x^5\,\left (2\,A\,b-3\,B\,a\right )}{15015\,a^4}+\frac {128\,b^5\,x^6\,\left (2\,A\,b-3\,B\,a\right )}{45045\,a^5}+\frac {2\,b\,x^2\,\left (A\,b+70\,B\,a\right )}{715\,a}\right )}{x^{15/2}} \]
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