Integrand size = 20, antiderivative size = 177 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{5/2}} \, dx=-\frac {2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac {2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac {16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt {a+b x}}+\frac {32 (10 A b-7 a B) \sqrt {a+b x}}{35 a^4 x^{5/2}}-\frac {128 b (10 A b-7 a B) \sqrt {a+b x}}{105 a^5 x^{3/2}}+\frac {256 b^2 (10 A b-7 a B) \sqrt {a+b x}}{105 a^6 \sqrt {x}} \]
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Time = 0.05 (sec) , antiderivative size = 177, 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} \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{5/2}} \, dx=\frac {256 b^2 \sqrt {a+b x} (10 A b-7 a B)}{105 a^6 \sqrt {x}}-\frac {128 b \sqrt {a+b x} (10 A b-7 a B)}{105 a^5 x^{3/2}}+\frac {32 \sqrt {a+b x} (10 A b-7 a B)}{35 a^4 x^{5/2}}-\frac {16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt {a+b x}}-\frac {2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac {2 A}{7 a x^{7/2} (a+b x)^{3/2}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{7 a x^{7/2} (a+b x)^{3/2}}+\frac {\left (2 \left (-5 A b+\frac {7 a B}{2}\right )\right ) \int \frac {1}{x^{7/2} (a+b x)^{5/2}} \, dx}{7 a} \\ & = -\frac {2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac {2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac {(8 (10 A b-7 a B)) \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx}{21 a^2} \\ & = -\frac {2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac {2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac {16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt {a+b x}}-\frac {(16 (10 A b-7 a B)) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{7 a^3} \\ & = -\frac {2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac {2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac {16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt {a+b x}}+\frac {32 (10 A b-7 a B) \sqrt {a+b x}}{35 a^4 x^{5/2}}+\frac {(64 b (10 A b-7 a B)) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{35 a^4} \\ & = -\frac {2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac {2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac {16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt {a+b x}}+\frac {32 (10 A b-7 a B) \sqrt {a+b x}}{35 a^4 x^{5/2}}-\frac {128 b (10 A b-7 a B) \sqrt {a+b x}}{105 a^5 x^{3/2}}-\frac {\left (128 b^2 (10 A b-7 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{105 a^5} \\ & = -\frac {2 A}{7 a x^{7/2} (a+b x)^{3/2}}-\frac {2 (10 A b-7 a B)}{21 a^2 x^{5/2} (a+b x)^{3/2}}-\frac {16 (10 A b-7 a B)}{21 a^3 x^{5/2} \sqrt {a+b x}}+\frac {32 (10 A b-7 a B) \sqrt {a+b x}}{35 a^4 x^{5/2}}-\frac {128 b (10 A b-7 a B) \sqrt {a+b x}}{105 a^5 x^{3/2}}+\frac {256 b^2 (10 A b-7 a B) \sqrt {a+b x}}{105 a^6 \sqrt {x}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.64 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{5/2}} \, dx=-\frac {2 \left (-1280 A b^5 x^5+128 a b^4 x^4 (-15 A+7 B x)+3 a^5 (5 A+7 B x)+96 a^2 b^3 x^3 (-5 A+14 B x)+16 a^3 b^2 x^2 (5 A+21 B x)-2 a^4 b x (15 A+28 B x)\right )}{105 a^6 x^{7/2} (a+b x)^{3/2}} \]
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Time = 1.49 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-790 A \,b^{3} x^{3}+511 B a \,b^{2} x^{3}+185 a A \,b^{2} x^{2}-98 B \,a^{2} b \,x^{2}-60 a^{2} A b x +21 a^{3} B x +15 a^{3} A \right )}{105 a^{6} x^{\frac {7}{2}}}+\frac {2 b^{3} \left (14 A \,b^{2} x -11 B a b x +15 a b A -12 a^{2} B \right ) \sqrt {x}}{3 \left (b x +a \right )^{\frac {3}{2}} a^{6}}\) | \(121\) |
gosper | \(-\frac {2 \left (-1280 A \,b^{5} x^{5}+896 B a \,b^{4} x^{5}-1920 a A \,b^{4} x^{4}+1344 B \,a^{2} b^{3} x^{4}-480 a^{2} A \,b^{3} x^{3}+336 B \,a^{3} b^{2} x^{3}+80 a^{3} A \,b^{2} x^{2}-56 B \,a^{4} b \,x^{2}-30 a^{4} A b x +21 a^{5} B x +15 a^{5} A \right )}{105 x^{\frac {7}{2}} \left (b x +a \right )^{\frac {3}{2}} a^{6}}\) | \(125\) |
default | \(-\frac {2 \left (-1280 A \,b^{5} x^{5}+896 B a \,b^{4} x^{5}-1920 a A \,b^{4} x^{4}+1344 B \,a^{2} b^{3} x^{4}-480 a^{2} A \,b^{3} x^{3}+336 B \,a^{3} b^{2} x^{3}+80 a^{3} A \,b^{2} x^{2}-56 B \,a^{4} b \,x^{2}-30 a^{4} A b x +21 a^{5} B x +15 a^{5} A \right )}{105 x^{\frac {7}{2}} \left (b x +a \right )^{\frac {3}{2}} a^{6}}\) | \(125\) |
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Time = 0.24 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{5/2}} \, dx=-\frac {2 \, {\left (15 \, A a^{5} + 128 \, {\left (7 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} + 192 \, {\left (7 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 48 \, {\left (7 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 8 \, {\left (7 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 3 \, {\left (7 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{105 \, {\left (a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{5} + a^{8} x^{4}\right )}} \]
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Timed out. \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.27 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{5/2}} \, dx=\frac {32 \, B b^{2} x}{15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} - \frac {256 \, B b^{3} x}{15 \, \sqrt {b x^{2} + a x} a^{5}} - \frac {64 \, A b^{3} x}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{4}} + \frac {512 \, A b^{4} x}{21 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {16 \, B b}{15 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2}} - \frac {128 \, B b^{2}}{15 \, \sqrt {b x^{2} + a x} a^{4}} - \frac {32 \, A b^{2}}{21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3}} + \frac {256 \, A b^{3}}{21 \, \sqrt {b x^{2} + a x} a^{5}} - \frac {2 \, B}{5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x} + \frac {4 \, A b}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} x} - \frac {2 \, A}{7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (141) = 282\).
Time = 0.42 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.15 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{5/2}} \, dx=-\frac {2 \, {\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {{\left (511 \, B a^{13} b^{9} {\left | b \right |} - 790 \, A a^{12} b^{10} {\left | b \right |}\right )} {\left (b x + a\right )}}{a^{18} b^{4}} - \frac {7 \, {\left (233 \, B a^{14} b^{9} {\left | b \right |} - 365 \, A a^{13} b^{10} {\left | b \right |}\right )}}{a^{18} b^{4}}\right )} + \frac {350 \, {\left (5 \, B a^{15} b^{9} {\left | b \right |} - 8 \, A a^{14} b^{10} {\left | b \right |}\right )}}{a^{18} b^{4}}\right )} - \frac {210 \, {\left (3 \, B a^{16} b^{9} {\left | b \right |} - 5 \, A a^{15} b^{10} {\left | b \right |}\right )}}{a^{18} b^{4}}\right )} \sqrt {b x + a}}{105 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {7}{2}}} - \frac {4 \, {\left (9 \, B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {9}{2}} + 24 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {11}{2}} - 12 \, A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {11}{2}} + 11 \, B a^{3} b^{\frac {13}{2}} - 30 \, A a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {13}{2}} - 14 \, A a^{2} b^{\frac {15}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{5} {\left | b \right |}} \]
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Time = 0.99 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.83 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{5/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{7\,a\,b^2}-\frac {32\,x^3\,\left (10\,A\,b-7\,B\,a\right )}{35\,a^4}+\frac {16\,x^2\,\left (10\,A\,b-7\,B\,a\right )}{105\,a^3\,b}-\frac {x^5\,\left (2560\,A\,b^5-1792\,B\,a\,b^4\right )}{105\,a^6\,b^2}-\frac {128\,b\,x^4\,\left (10\,A\,b-7\,B\,a\right )}{35\,a^5}+\frac {x\,\left (42\,B\,a^5-60\,A\,a^4\,b\right )}{105\,a^6\,b^2}\right )}{x^{11/2}+\frac {2\,a\,x^{9/2}}{b}+\frac {a^2\,x^{7/2}}{b^2}} \]
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