Integrand size = 20, antiderivative size = 117 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx=-\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac {2 (6 A b-13 a B) (a+b x)^{7/2}}{143 a^2 x^{11/2}}-\frac {8 b (6 A b-13 a B) (a+b x)^{7/2}}{1287 a^3 x^{9/2}}+\frac {16 b^2 (6 A b-13 a B) (a+b x)^{7/2}}{9009 a^4 x^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx=\frac {16 b^2 (a+b x)^{7/2} (6 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac {8 b (a+b x)^{7/2} (6 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac {2 (a+b x)^{7/2} (6 A b-13 a B)}{143 a^2 x^{11/2}}-\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac {\left (2 \left (-3 A b+\frac {13 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{13/2}} \, dx}{13 a} \\ & = -\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac {2 (6 A b-13 a B) (a+b x)^{7/2}}{143 a^2 x^{11/2}}+\frac {(4 b (6 A b-13 a B)) \int \frac {(a+b x)^{5/2}}{x^{11/2}} \, dx}{143 a^2} \\ & = -\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac {2 (6 A b-13 a B) (a+b x)^{7/2}}{143 a^2 x^{11/2}}-\frac {8 b (6 A b-13 a B) (a+b x)^{7/2}}{1287 a^3 x^{9/2}}-\frac {\left (8 b^2 (6 A b-13 a B)\right ) \int \frac {(a+b x)^{5/2}}{x^{9/2}} \, dx}{1287 a^3} \\ & = -\frac {2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac {2 (6 A b-13 a B) (a+b x)^{7/2}}{143 a^2 x^{11/2}}-\frac {8 b (6 A b-13 a B) (a+b x)^{7/2}}{1287 a^3 x^{9/2}}+\frac {16 b^2 (6 A b-13 a B) (a+b x)^{7/2}}{9009 a^4 x^{7/2}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.65 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx=-\frac {2 (a+b x)^{7/2} \left (-48 A b^3 x^3+63 a^3 (11 A+13 B x)+8 a b^2 x^2 (21 A+13 B x)-14 a^2 b x (27 A+26 B x)\right )}{9009 a^4 x^{13/2}} \]
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Time = 0.52 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-48 A \,b^{3} x^{3}+104 B a \,b^{2} x^{3}+168 a A \,b^{2} x^{2}-364 B \,a^{2} b \,x^{2}-378 a^{2} A b x +819 a^{3} B x +693 a^{3} A \right )}{9009 x^{\frac {13}{2}} a^{4}}\) | \(77\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-48 A \,b^{5} x^{5}+104 B a \,b^{4} x^{5}+72 a A \,b^{4} x^{4}-156 B \,a^{2} b^{3} x^{4}-90 a^{2} A \,b^{3} x^{3}+195 B \,a^{3} b^{2} x^{3}+105 a^{3} A \,b^{2} x^{2}+1274 B \,a^{4} b \,x^{2}+1008 a^{4} A b x +819 a^{5} B x +693 a^{5} A \right )}{9009 x^{\frac {13}{2}} a^{4}}\) | \(125\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{6} x^{6}+104 B a \,b^{5} x^{6}+24 A a \,b^{5} x^{5}-52 B \,a^{2} b^{4} x^{5}-18 A \,a^{2} b^{4} x^{4}+39 B \,a^{3} b^{3} x^{4}+15 A \,a^{3} b^{3} x^{3}+1469 B \,a^{4} b^{2} x^{3}+1113 A \,a^{4} b^{2} x^{2}+2093 B \,a^{5} b \,x^{2}+1701 A \,a^{5} b x +819 B \,a^{6} x +693 A \,a^{6}\right )}{9009 x^{\frac {13}{2}} a^{4}}\) | \(149\) |
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Time = 0.23 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx=-\frac {2 \, {\left (693 \, A a^{6} + 8 \, {\left (13 \, B a b^{5} - 6 \, A b^{6}\right )} x^{6} - 4 \, {\left (13 \, B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} + 3 \, {\left (13 \, B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} + {\left (1469 \, B a^{4} b^{2} + 15 \, A a^{3} b^{3}\right )} x^{3} + 7 \, {\left (299 \, B a^{5} b + 159 \, A a^{4} b^{2}\right )} x^{2} + 63 \, {\left (13 \, B a^{6} + 27 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{9009 \, a^{4} x^{\frac {13}{2}}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (93) = 186\).
Time = 0.22 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.99 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx=-\frac {16 \, \sqrt {b x^{2} + a x} B b^{5}}{693 \, a^{3} x} + \frac {32 \, \sqrt {b x^{2} + a x} A b^{6}}{3003 \, a^{4} x} + \frac {8 \, \sqrt {b x^{2} + a x} B b^{4}}{693 \, a^{2} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{5}}{3003 \, a^{3} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{3}}{231 \, a x^{3}} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{4}}{1001 \, a^{2} x^{3}} + \frac {5 \, \sqrt {b x^{2} + a x} B b^{2}}{693 \, x^{4}} - \frac {10 \, \sqrt {b x^{2} + a x} A b^{3}}{3003 \, a x^{4}} - \frac {5 \, \sqrt {b x^{2} + a x} B a b}{792 \, x^{5}} + \frac {5 \, \sqrt {b x^{2} + a x} A b^{2}}{1716 \, x^{5}} - \frac {5 \, \sqrt {b x^{2} + a x} B a^{2}}{88 \, x^{6}} - \frac {3 \, \sqrt {b x^{2} + a x} A a b}{1144 \, x^{6}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{24 \, x^{7}} - \frac {3 \, \sqrt {b x^{2} + a x} A a^{2}}{104 \, x^{7}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{3 \, x^{8}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{8 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{4 \, x^{9}} \]
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Time = 0.37 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx=-\frac {2 \, {\left ({\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (13 \, B a^{3} b^{12} - 6 \, A a^{2} b^{13}\right )} {\left (b x + a\right )}}{a^{6}} - \frac {13 \, {\left (13 \, B a^{4} b^{12} - 6 \, A a^{3} b^{13}\right )}}{a^{6}}\right )} + \frac {143 \, {\left (13 \, B a^{5} b^{12} - 6 \, A a^{4} b^{13}\right )}}{a^{6}}\right )} - \frac {1287 \, {\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )}^{\frac {7}{2}} b}{9009 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {13}{2}} {\left | b \right |}} \]
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Time = 0.90 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a^2}{13}+\frac {x\,\left (1638\,B\,a^6+3402\,A\,b\,a^5\right )}{9009\,a^4}-\frac {x^6\,\left (96\,A\,b^6-208\,B\,a\,b^5\right )}{9009\,a^4}+\frac {2\,b\,x^2\,\left (159\,A\,b+299\,B\,a\right )}{1287}-\frac {2\,b^3\,x^4\,\left (6\,A\,b-13\,B\,a\right )}{3003\,a^2}+\frac {8\,b^4\,x^5\,\left (6\,A\,b-13\,B\,a\right )}{9009\,a^3}+\frac {2\,b^2\,x^3\,\left (15\,A\,b+1469\,B\,a\right )}{9009\,a}\right )}{x^{13/2}} \]
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