Integrand size = 20, antiderivative size = 150 \[ \int \frac {A+B x}{x^{11/2} \sqrt {a+b x}} \, dx=-\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {2 (8 A b-9 a B) \sqrt {a+b x}}{63 a^2 x^{7/2}}-\frac {4 b (8 A b-9 a B) \sqrt {a+b x}}{105 a^3 x^{5/2}}+\frac {16 b^2 (8 A b-9 a B) \sqrt {a+b x}}{315 a^4 x^{3/2}}-\frac {32 b^3 (8 A b-9 a B) \sqrt {a+b x}}{315 a^5 \sqrt {x}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 150, 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} \[ \int \frac {A+B x}{x^{11/2} \sqrt {a+b x}} \, dx=-\frac {32 b^3 \sqrt {a+b x} (8 A b-9 a B)}{315 a^5 \sqrt {x}}+\frac {16 b^2 \sqrt {a+b x} (8 A b-9 a B)}{315 a^4 x^{3/2}}-\frac {4 b \sqrt {a+b x} (8 A b-9 a B)}{105 a^3 x^{5/2}}+\frac {2 \sqrt {a+b x} (8 A b-9 a B)}{63 a^2 x^{7/2}}-\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}} \]
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {\left (2 \left (-4 A b+\frac {9 a B}{2}\right )\right ) \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx}{9 a} \\ & = -\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {2 (8 A b-9 a B) \sqrt {a+b x}}{63 a^2 x^{7/2}}+\frac {(2 b (8 A b-9 a B)) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{21 a^2} \\ & = -\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {2 (8 A b-9 a B) \sqrt {a+b x}}{63 a^2 x^{7/2}}-\frac {4 b (8 A b-9 a B) \sqrt {a+b x}}{105 a^3 x^{5/2}}-\frac {\left (8 b^2 (8 A b-9 a B)\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{105 a^3} \\ & = -\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {2 (8 A b-9 a B) \sqrt {a+b x}}{63 a^2 x^{7/2}}-\frac {4 b (8 A b-9 a B) \sqrt {a+b x}}{105 a^3 x^{5/2}}+\frac {16 b^2 (8 A b-9 a B) \sqrt {a+b x}}{315 a^4 x^{3/2}}+\frac {\left (16 b^3 (8 A b-9 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{315 a^4} \\ & = -\frac {2 A \sqrt {a+b x}}{9 a x^{9/2}}+\frac {2 (8 A b-9 a B) \sqrt {a+b x}}{63 a^2 x^{7/2}}-\frac {4 b (8 A b-9 a B) \sqrt {a+b x}}{105 a^3 x^{5/2}}+\frac {16 b^2 (8 A b-9 a B) \sqrt {a+b x}}{315 a^4 x^{3/2}}-\frac {32 b^3 (8 A b-9 a B) \sqrt {a+b x}}{315 a^5 \sqrt {x}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x}{x^{11/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x} \left (128 A b^4 x^4+24 a^2 b^2 x^2 (2 A+3 B x)-16 a b^3 x^3 (4 A+9 B x)+5 a^4 (7 A+9 B x)-2 a^3 b x (20 A+27 B x)\right )}{315 a^5 x^{9/2}} \]
[In]
[Out]
Time = 0.51 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{4} x^{4}-144 B a \,b^{3} x^{4}-64 A a \,b^{3} x^{3}+72 B \,a^{2} b^{2} x^{3}+48 A \,a^{2} b^{2} x^{2}-54 B \,a^{3} b \,x^{2}-40 A \,a^{3} b x +45 B \,a^{4} x +35 A \,a^{4}\right )}{315 x^{\frac {9}{2}} a^{5}}\) | \(101\) |
default | \(-\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{4} x^{4}-144 B a \,b^{3} x^{4}-64 A a \,b^{3} x^{3}+72 B \,a^{2} b^{2} x^{3}+48 A \,a^{2} b^{2} x^{2}-54 B \,a^{3} b \,x^{2}-40 A \,a^{3} b x +45 B \,a^{4} x +35 A \,a^{4}\right )}{315 x^{\frac {9}{2}} a^{5}}\) | \(101\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (128 A \,b^{4} x^{4}-144 B a \,b^{3} x^{4}-64 A a \,b^{3} x^{3}+72 B \,a^{2} b^{2} x^{3}+48 A \,a^{2} b^{2} x^{2}-54 B \,a^{3} b \,x^{2}-40 A \,a^{3} b x +45 B \,a^{4} x +35 A \,a^{4}\right )}{315 x^{\frac {9}{2}} a^{5}}\) | \(101\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.68 \[ \int \frac {A+B x}{x^{11/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (35 \, A a^{4} - 16 \, {\left (9 \, B a b^{3} - 8 \, A b^{4}\right )} x^{4} + 8 \, {\left (9 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{3} - 6 \, {\left (9 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{4} - 8 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{315 \, a^{5} x^{\frac {9}{2}}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1255 vs. \(2 (150) = 300\).
Time = 36.17 (sec) , antiderivative size = 1255, normalized size of antiderivative = 8.37 \[ \int \frac {A+B x}{x^{11/2} \sqrt {a+b x}} \, dx=- \frac {70 A a^{8} b^{\frac {33}{2}} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {200 A a^{7} b^{\frac {35}{2}} x \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {196 A a^{6} b^{\frac {37}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {56 A a^{5} b^{\frac {39}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {70 A a^{4} b^{\frac {41}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {560 A a^{3} b^{\frac {43}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {1120 A a^{2} b^{\frac {45}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {896 A a b^{\frac {47}{2}} x^{7} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {256 A b^{\frac {49}{2}} x^{8} \sqrt {\frac {a}{b x} + 1}}{315 a^{9} b^{16} x^{4} + 1260 a^{8} b^{17} x^{5} + 1890 a^{7} b^{18} x^{6} + 1260 a^{6} b^{19} x^{7} + 315 a^{5} b^{20} x^{8}} - \frac {10 B a^{6} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {18 B a^{5} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} - \frac {10 B a^{4} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {10 B a^{3} b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {60 B a^{2} b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {80 B a b^{\frac {29}{2}} x^{5} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} + \frac {32 B b^{\frac {31}{2}} x^{6} \sqrt {\frac {a}{b x} + 1}}{35 a^{7} b^{9} x^{3} + 105 a^{6} b^{10} x^{4} + 105 a^{5} b^{11} x^{5} + 35 a^{4} b^{12} x^{6}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x}{x^{11/2} \sqrt {a+b x}} \, dx=\frac {32 \, \sqrt {b x^{2} + a x} B b^{3}}{35 \, a^{4} x} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{4}}{315 \, a^{5} x} - \frac {16 \, \sqrt {b x^{2} + a x} B b^{2}}{35 \, a^{3} x^{2}} + \frac {128 \, \sqrt {b x^{2} + a x} A b^{3}}{315 \, a^{4} x^{2}} + \frac {12 \, \sqrt {b x^{2} + a x} B b}{35 \, a^{2} x^{3}} - \frac {32 \, \sqrt {b x^{2} + a x} A b^{2}}{105 \, a^{3} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{7 \, a x^{4}} + \frac {16 \, \sqrt {b x^{2} + a x} A b}{63 \, a^{2} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{9 \, a x^{5}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x}{x^{11/2} \sqrt {a+b x}} \, dx=\frac {2 \, {\left ({\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (9 \, B a b^{8} - 8 \, A b^{9}\right )} {\left (b x + a\right )}}{a^{5}} - \frac {9 \, {\left (9 \, B a^{2} b^{8} - 8 \, A a b^{9}\right )}}{a^{5}}\right )} + \frac {63 \, {\left (9 \, B a^{3} b^{8} - 8 \, A a^{2} b^{9}\right )}}{a^{5}}\right )} - \frac {105 \, {\left (9 \, B a^{4} b^{8} - 8 \, A a^{3} b^{9}\right )}}{a^{5}}\right )} {\left (b x + a\right )} + \frac {315 \, {\left (B a^{5} b^{8} - A a^{4} b^{9}\right )}}{a^{5}}\right )} \sqrt {b x + a} b}{315 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {9}{2}} {\left | b \right |}} \]
[In]
[Out]
Time = 0.96 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.66 \[ \int \frac {A+B x}{x^{11/2} \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{9\,a}+\frac {x\,\left (90\,B\,a^4-80\,A\,a^3\,b\right )}{315\,a^5}+\frac {x^4\,\left (256\,A\,b^4-288\,B\,a\,b^3\right )}{315\,a^5}-\frac {16\,b^2\,x^3\,\left (8\,A\,b-9\,B\,a\right )}{315\,a^4}+\frac {4\,b\,x^2\,\left (8\,A\,b-9\,B\,a\right )}{105\,a^3}\right )}{x^{9/2}} \]
[In]
[Out]