Integrand size = 20, antiderivative size = 180 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{3/2}} \, dx=-\frac {2 A}{9 a x^{9/2} \sqrt {a+b x}}-\frac {2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt {a+b x}}+\frac {16 (10 A b-9 a B) \sqrt {a+b x}}{63 a^3 x^{7/2}}-\frac {32 b (10 A b-9 a B) \sqrt {a+b x}}{105 a^4 x^{5/2}}+\frac {128 b^2 (10 A b-9 a B) \sqrt {a+b x}}{315 a^5 x^{3/2}}-\frac {256 b^3 (10 A b-9 a B) \sqrt {a+b x}}{315 a^6 \sqrt {x}} \]
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Time = 0.05 (sec) , antiderivative size = 180, 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} \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{3/2}} \, dx=-\frac {256 b^3 \sqrt {a+b x} (10 A b-9 a B)}{315 a^6 \sqrt {x}}+\frac {128 b^2 \sqrt {a+b x} (10 A b-9 a B)}{315 a^5 x^{3/2}}-\frac {32 b \sqrt {a+b x} (10 A b-9 a B)}{105 a^4 x^{5/2}}+\frac {16 \sqrt {a+b x} (10 A b-9 a B)}{63 a^3 x^{7/2}}-\frac {2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt {a+b x}}-\frac {2 A}{9 a x^{9/2} \sqrt {a+b x}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{9 a x^{9/2} \sqrt {a+b x}}+\frac {\left (2 \left (-5 A b+\frac {9 a B}{2}\right )\right ) \int \frac {1}{x^{9/2} (a+b x)^{3/2}} \, dx}{9 a} \\ & = -\frac {2 A}{9 a x^{9/2} \sqrt {a+b x}}-\frac {2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt {a+b x}}-\frac {(8 (10 A b-9 a B)) \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx}{9 a^2} \\ & = -\frac {2 A}{9 a x^{9/2} \sqrt {a+b x}}-\frac {2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt {a+b x}}+\frac {16 (10 A b-9 a B) \sqrt {a+b x}}{63 a^3 x^{7/2}}+\frac {(16 b (10 A b-9 a B)) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{21 a^3} \\ & = -\frac {2 A}{9 a x^{9/2} \sqrt {a+b x}}-\frac {2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt {a+b x}}+\frac {16 (10 A b-9 a B) \sqrt {a+b x}}{63 a^3 x^{7/2}}-\frac {32 b (10 A b-9 a B) \sqrt {a+b x}}{105 a^4 x^{5/2}}-\frac {\left (64 b^2 (10 A b-9 a B)\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{105 a^4} \\ & = -\frac {2 A}{9 a x^{9/2} \sqrt {a+b x}}-\frac {2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt {a+b x}}+\frac {16 (10 A b-9 a B) \sqrt {a+b x}}{63 a^3 x^{7/2}}-\frac {32 b (10 A b-9 a B) \sqrt {a+b x}}{105 a^4 x^{5/2}}+\frac {128 b^2 (10 A b-9 a B) \sqrt {a+b x}}{315 a^5 x^{3/2}}+\frac {\left (128 b^3 (10 A b-9 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{315 a^5} \\ & = -\frac {2 A}{9 a x^{9/2} \sqrt {a+b x}}-\frac {2 (10 A b-9 a B)}{9 a^2 x^{7/2} \sqrt {a+b x}}+\frac {16 (10 A b-9 a B) \sqrt {a+b x}}{63 a^3 x^{7/2}}-\frac {32 b (10 A b-9 a B) \sqrt {a+b x}}{105 a^4 x^{5/2}}+\frac {128 b^2 (10 A b-9 a B) \sqrt {a+b x}}{315 a^5 x^{3/2}}-\frac {256 b^3 (10 A b-9 a B) \sqrt {a+b x}}{315 a^6 \sqrt {x}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{3/2}} \, dx=-\frac {2 \left (1280 A b^5 x^5+128 a b^4 x^4 (5 A-9 B x)+16 a^3 b^2 x^2 (5 A+9 B x)+5 a^5 (7 A+9 B x)-32 a^2 b^3 x^3 (5 A+18 B x)-2 a^4 b x (25 A+36 B x)\right )}{315 a^6 x^{9/2} \sqrt {a+b x}} \]
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Time = 0.52 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.69
method | result | size |
gosper | \(-\frac {2 \left (1280 A \,b^{5} x^{5}-1152 B a \,b^{4} x^{5}+640 a A \,b^{4} x^{4}-576 B \,a^{2} b^{3} x^{4}-160 a^{2} A \,b^{3} x^{3}+144 B \,a^{3} b^{2} x^{3}+80 a^{3} A \,b^{2} x^{2}-72 B \,a^{4} b \,x^{2}-50 a^{4} A b x +45 a^{5} B x +35 a^{5} A \right )}{315 x^{\frac {9}{2}} \sqrt {b x +a}\, a^{6}}\) | \(125\) |
default | \(-\frac {2 \left (1280 A \,b^{5} x^{5}-1152 B a \,b^{4} x^{5}+640 a A \,b^{4} x^{4}-576 B \,a^{2} b^{3} x^{4}-160 a^{2} A \,b^{3} x^{3}+144 B \,a^{3} b^{2} x^{3}+80 a^{3} A \,b^{2} x^{2}-72 B \,a^{4} b \,x^{2}-50 a^{4} A b x +45 a^{5} B x +35 a^{5} A \right )}{315 x^{\frac {9}{2}} \sqrt {b x +a}\, a^{6}}\) | \(125\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (965 A \,b^{4} x^{4}-837 B a \,b^{3} x^{4}-325 A a \,b^{3} x^{3}+261 B \,a^{2} b^{2} x^{3}+165 A \,a^{2} b^{2} x^{2}-117 B \,a^{3} b \,x^{2}-85 A \,a^{3} b x +45 B \,a^{4} x +35 A \,a^{4}\right )}{315 a^{6} x^{\frac {9}{2}}}-\frac {2 b^{4} \left (A b -B a \right ) \sqrt {x}}{a^{6} \sqrt {b x +a}}\) | \(128\) |
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Time = 0.23 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (35 \, A a^{5} - 128 \, {\left (9 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} - 64 \, {\left (9 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 16 \, {\left (9 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 8 \, {\left (9 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{315 \, {\left (a^{6} b x^{6} + a^{7} x^{5}\right )}} \]
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Timed out. \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{3/2}} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.30 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{3/2}} \, dx=\frac {256 \, B b^{4} x}{35 \, \sqrt {b x^{2} + a x} a^{5}} - \frac {512 \, A b^{5} x}{63 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {128 \, B b^{3}}{35 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {256 \, A b^{4}}{63 \, \sqrt {b x^{2} + a x} a^{5}} - \frac {32 \, B b^{2}}{35 \, \sqrt {b x^{2} + a x} a^{3} x} + \frac {64 \, A b^{3}}{63 \, \sqrt {b x^{2} + a x} a^{4} x} + \frac {16 \, B b}{35 \, \sqrt {b x^{2} + a x} a^{2} x^{2}} - \frac {32 \, A b^{2}}{63 \, \sqrt {b x^{2} + a x} a^{3} x^{2}} - \frac {2 \, B}{7 \, \sqrt {b x^{2} + a x} a x^{3}} + \frac {20 \, A b}{63 \, \sqrt {b x^{2} + a x} a^{2} x^{3}} - \frac {2 \, A}{9 \, \sqrt {b x^{2} + a x} a x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (144) = 288\).
Time = 0.37 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.83 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{3/2}} \, dx=\frac {2 \, {\left ({\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {{\left (837 \, B a^{15} b^{13} - 965 \, A a^{14} b^{14}\right )} {\left (b x + a\right )}}{a^{20} b^{4} {\left | b \right |}} - \frac {9 \, {\left (401 \, B a^{16} b^{13} - 465 \, A a^{15} b^{14}\right )}}{a^{20} b^{4} {\left | b \right |}}\right )} + \frac {126 \, {\left (47 \, B a^{17} b^{13} - 55 \, A a^{16} b^{14}\right )}}{a^{20} b^{4} {\left | b \right |}}\right )} - \frac {210 \, {\left (21 \, B a^{18} b^{13} - 25 \, A a^{17} b^{14}\right )}}{a^{20} b^{4} {\left | b \right |}}\right )} {\left (b x + a\right )} + \frac {315 \, {\left (4 \, B a^{19} b^{13} - 5 \, A a^{18} b^{14}\right )}}{a^{20} b^{4} {\left | b \right |}}\right )} \sqrt {b x + a}}{315 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {9}{2}}} + \frac {4 \, {\left (B^{2} a^{2} b^{11} - 2 \, A B a b^{12} + A^{2} b^{13}\right )}}{{\left (B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {11}{2}} + B a^{2} b^{\frac {13}{2}} - A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {13}{2}} - A a b^{\frac {15}{2}}\right )} a^{5} {\left | b \right |}} \]
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Time = 1.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x}{x^{11/2} (a+b x)^{3/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{9\,a\,b}+\frac {16\,x^2\,\left (10\,A\,b-9\,B\,a\right )}{315\,a^3}+\frac {128\,b^2\,x^4\,\left (10\,A\,b-9\,B\,a\right )}{315\,a^5}+\frac {256\,b^3\,x^5\,\left (10\,A\,b-9\,B\,a\right )}{315\,a^6}-\frac {32\,b\,x^3\,\left (10\,A\,b-9\,B\,a\right )}{315\,a^4}+\frac {x\,\left (90\,B\,a^5-100\,A\,a^4\,b\right )}{315\,a^6\,b}\right )}{x^{11/2}+\frac {a\,x^{9/2}}{b}} \]
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