Integrand size = 20, antiderivative size = 147 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{3/2}} \, dx=-\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}+\frac {12 (8 A b-7 a B) \sqrt {a+b x}}{35 a^3 x^{5/2}}-\frac {16 b (8 A b-7 a B) \sqrt {a+b x}}{35 a^4 x^{3/2}}+\frac {32 b^2 (8 A b-7 a B) \sqrt {a+b x}}{35 a^5 \sqrt {x}} \]
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Time = 0.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{3/2}} \, dx=\frac {32 b^2 \sqrt {a+b x} (8 A b-7 a B)}{35 a^5 \sqrt {x}}-\frac {16 b \sqrt {a+b x} (8 A b-7 a B)}{35 a^4 x^{3/2}}+\frac {12 \sqrt {a+b x} (8 A b-7 a B)}{35 a^3 x^{5/2}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}-\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}+\frac {\left (2 \left (-4 A b+\frac {7 a B}{2}\right )\right ) \int \frac {1}{x^{7/2} (a+b x)^{3/2}} \, dx}{7 a} \\ & = -\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}-\frac {(6 (8 A b-7 a B)) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{7 a^2} \\ & = -\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}+\frac {12 (8 A b-7 a B) \sqrt {a+b x}}{35 a^3 x^{5/2}}+\frac {(24 b (8 A b-7 a B)) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{35 a^3} \\ & = -\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}+\frac {12 (8 A b-7 a B) \sqrt {a+b x}}{35 a^3 x^{5/2}}-\frac {16 b (8 A b-7 a B) \sqrt {a+b x}}{35 a^4 x^{3/2}}-\frac {\left (16 b^2 (8 A b-7 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{35 a^4} \\ & = -\frac {2 A}{7 a x^{7/2} \sqrt {a+b x}}-\frac {2 (8 A b-7 a B)}{7 a^2 x^{5/2} \sqrt {a+b x}}+\frac {12 (8 A b-7 a B) \sqrt {a+b x}}{35 a^3 x^{5/2}}-\frac {16 b (8 A b-7 a B) \sqrt {a+b x}}{35 a^4 x^{3/2}}+\frac {32 b^2 (8 A b-7 a B) \sqrt {a+b x}}{35 a^5 \sqrt {x}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.64 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{3/2}} \, dx=-\frac {2 \left (-128 A b^4 x^4+16 a b^3 x^3 (-4 A+7 B x)+8 a^2 b^2 x^2 (2 A+7 B x)-2 a^3 b x (4 A+7 B x)+a^4 (5 A+7 B x)\right )}{35 a^5 x^{7/2} \sqrt {a+b x}} \]
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Time = 1.47 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.69
method | result | size |
gosper | \(-\frac {2 \left (-128 A \,b^{4} x^{4}+112 B a \,b^{3} x^{4}-64 A a \,b^{3} x^{3}+56 B \,a^{2} b^{2} x^{3}+16 A \,a^{2} b^{2} x^{2}-14 B \,a^{3} b \,x^{2}-8 A \,a^{3} b x +7 B \,a^{4} x +5 A \,a^{4}\right )}{35 x^{\frac {7}{2}} \sqrt {b x +a}\, a^{5}}\) | \(101\) |
default | \(-\frac {2 \left (-128 A \,b^{4} x^{4}+112 B a \,b^{3} x^{4}-64 A a \,b^{3} x^{3}+56 B \,a^{2} b^{2} x^{3}+16 A \,a^{2} b^{2} x^{2}-14 B \,a^{3} b \,x^{2}-8 A \,a^{3} b x +7 B \,a^{4} x +5 A \,a^{4}\right )}{35 x^{\frac {7}{2}} \sqrt {b x +a}\, a^{5}}\) | \(101\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-93 A \,b^{3} x^{3}+77 B a \,b^{2} x^{3}+29 a A \,b^{2} x^{2}-21 B \,a^{2} b \,x^{2}-13 a^{2} A b x +7 a^{3} B x +5 a^{3} A \right )}{35 a^{5} x^{\frac {7}{2}}}+\frac {2 b^{3} \left (A b -B a \right ) \sqrt {x}}{a^{5} \sqrt {b x +a}}\) | \(104\) |
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Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (5 \, A a^{4} + 16 \, {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} + 8 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 2 \, {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} + {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{35 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1008 vs. \(2 (144) = 288\).
Time = 80.04 (sec) , antiderivative size = 1008, normalized size of antiderivative = 6.86 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{3/2}} \, dx=A \left (- \frac {10 a^{7} b^{\frac {33}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} - \frac {14 a^{6} b^{\frac {35}{2}} x \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} - \frac {14 a^{5} b^{\frac {37}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} + \frac {70 a^{4} b^{\frac {39}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} + \frac {560 a^{3} b^{\frac {41}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} + \frac {1120 a^{2} b^{\frac {43}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} + \frac {896 a b^{\frac {45}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}} + \frac {256 b^{\frac {47}{2}} x^{7} \sqrt {\frac {a}{b x} + 1}}{35 a^{9} b^{16} x^{3} + 140 a^{8} b^{17} x^{4} + 210 a^{7} b^{18} x^{5} + 140 a^{6} b^{19} x^{6} + 35 a^{5} b^{20} x^{7}}\right ) + B \left (- \frac {2 a^{5} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {10 a^{3} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {60 a^{2} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {80 a b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac {32 b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.28 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{3/2}} \, dx=-\frac {32 \, B b^{3} x}{5 \, \sqrt {b x^{2} + a x} a^{4}} + \frac {256 \, A b^{4} x}{35 \, \sqrt {b x^{2} + a x} a^{5}} - \frac {16 \, B b^{2}}{5 \, \sqrt {b x^{2} + a x} a^{3}} + \frac {128 \, A b^{3}}{35 \, \sqrt {b x^{2} + a x} a^{4}} + \frac {4 \, B b}{5 \, \sqrt {b x^{2} + a x} a^{2} x} - \frac {32 \, A b^{2}}{35 \, \sqrt {b x^{2} + a x} a^{3} x} - \frac {2 \, B}{5 \, \sqrt {b x^{2} + a x} a x^{2}} + \frac {16 \, A b}{35 \, \sqrt {b x^{2} + a x} a^{2} x^{2}} - \frac {2 \, A}{7 \, \sqrt {b x^{2} + a x} a x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (117) = 234\).
Time = 0.35 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.99 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {{\left (77 \, B a^{10} b^{9} {\left | b \right |} - 93 \, A a^{9} b^{10} {\left | b \right |}\right )} {\left (b x + a\right )}}{a^{14} b^{4}} - \frac {28 \, {\left (9 \, B a^{11} b^{9} {\left | b \right |} - 11 \, A a^{10} b^{10} {\left | b \right |}\right )}}{a^{14} b^{4}}\right )} + \frac {70 \, {\left (4 \, B a^{12} b^{9} {\left | b \right |} - 5 \, A a^{11} b^{10} {\left | b \right |}\right )}}{a^{14} b^{4}}\right )} - \frac {35 \, {\left (3 \, B a^{13} b^{9} {\left | b \right |} - 4 \, A a^{12} b^{10} {\left | b \right |}\right )}}{a^{14} b^{4}}\right )} \sqrt {b x + a}}{35 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {7}{2}}} - \frac {4 \, {\left (B^{2} a^{2} b^{9} - 2 \, A B a b^{10} + A^{2} b^{11}\right )}}{{\left (B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {9}{2}} + B a^{2} b^{\frac {11}{2}} - A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {11}{2}} - A a b^{\frac {13}{2}}\right )} a^{4} {\left | b \right |}} \]
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Time = 0.92 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^{9/2} (a+b x)^{3/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{7\,a\,b}+\frac {4\,x^2\,\left (8\,A\,b-7\,B\,a\right )}{35\,a^3}-\frac {x^4\,\left (256\,A\,b^4-224\,B\,a\,b^3\right )}{35\,a^5\,b}-\frac {16\,b\,x^3\,\left (8\,A\,b-7\,B\,a\right )}{35\,a^4}+\frac {x\,\left (14\,B\,a^4-16\,A\,a^3\,b\right )}{35\,a^5\,b}\right )}{x^{9/2}+\frac {a\,x^{7/2}}{b}} \]
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