Integrand size = 20, antiderivative size = 216 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}+\frac {160 b^2 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^4 x^{7/2}}-\frac {64 b^3 (12 A b-13 a B) \sqrt {a+b x}}{3003 a^5 x^{5/2}}+\frac {256 b^4 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^6 x^{3/2}}-\frac {512 b^5 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^7 \sqrt {x}} \]
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Time = 0.06 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=-\frac {512 b^5 \sqrt {a+b x} (12 A b-13 a B)}{9009 a^7 \sqrt {x}}+\frac {256 b^4 \sqrt {a+b x} (12 A b-13 a B)}{9009 a^6 x^{3/2}}-\frac {64 b^3 \sqrt {a+b x} (12 A b-13 a B)}{3003 a^5 x^{5/2}}+\frac {160 b^2 \sqrt {a+b x} (12 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac {20 b \sqrt {a+b x} (12 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac {2 \sqrt {a+b x} (12 A b-13 a B)}{143 a^2 x^{11/2}}-\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {\left (2 \left (-6 A b+\frac {13 a B}{2}\right )\right ) \int \frac {1}{x^{13/2} \sqrt {a+b x}} \, dx}{13 a} \\ & = -\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}+\frac {(10 b (12 A b-13 a B)) \int \frac {1}{x^{11/2} \sqrt {a+b x}} \, dx}{143 a^2} \\ & = -\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}-\frac {\left (80 b^2 (12 A b-13 a B)\right ) \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx}{1287 a^3} \\ & = -\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}+\frac {160 b^2 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^4 x^{7/2}}+\frac {\left (160 b^3 (12 A b-13 a B)\right ) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{3003 a^4} \\ & = -\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}+\frac {160 b^2 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^4 x^{7/2}}-\frac {64 b^3 (12 A b-13 a B) \sqrt {a+b x}}{3003 a^5 x^{5/2}}-\frac {\left (128 b^4 (12 A b-13 a B)\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{3003 a^5} \\ & = -\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}+\frac {160 b^2 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^4 x^{7/2}}-\frac {64 b^3 (12 A b-13 a B) \sqrt {a+b x}}{3003 a^5 x^{5/2}}+\frac {256 b^4 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^6 x^{3/2}}+\frac {\left (256 b^5 (12 A b-13 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{9009 a^6} \\ & = -\frac {2 A \sqrt {a+b x}}{13 a x^{13/2}}+\frac {2 (12 A b-13 a B) \sqrt {a+b x}}{143 a^2 x^{11/2}}-\frac {20 b (12 A b-13 a B) \sqrt {a+b x}}{1287 a^3 x^{9/2}}+\frac {160 b^2 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^4 x^{7/2}}-\frac {64 b^3 (12 A b-13 a B) \sqrt {a+b x}}{3003 a^5 x^{5/2}}+\frac {256 b^4 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^6 x^{3/2}}-\frac {512 b^5 (12 A b-13 a B) \sqrt {a+b x}}{9009 a^7 \sqrt {x}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=-\frac {2 \sqrt {a+b x} \left (3072 A b^6 x^6-256 a b^5 x^5 (6 A+13 B x)+128 a^2 b^4 x^4 (9 A+13 B x)-96 a^3 b^3 x^3 (10 A+13 B x)+63 a^6 (11 A+13 B x)+40 a^4 b^2 x^2 (21 A+26 B x)-14 a^5 b x (54 A+65 B x)\right )}{9009 a^7 x^{13/2}} \]
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Time = 0.51 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.69
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (3072 A \,b^{6} x^{6}-3328 B a \,b^{5} x^{6}-1536 A a \,b^{5} x^{5}+1664 B \,a^{2} b^{4} x^{5}+1152 A \,a^{2} b^{4} x^{4}-1248 B \,a^{3} b^{3} x^{4}-960 A \,a^{3} b^{3} x^{3}+1040 B \,a^{4} b^{2} x^{3}+840 A \,a^{4} b^{2} x^{2}-910 B \,a^{5} b \,x^{2}-756 A \,a^{5} b x +819 B \,a^{6} x +693 A \,a^{6}\right )}{9009 x^{\frac {13}{2}} a^{7}}\) | \(149\) |
default | \(-\frac {2 \sqrt {b x +a}\, \left (3072 A \,b^{6} x^{6}-3328 B a \,b^{5} x^{6}-1536 A a \,b^{5} x^{5}+1664 B \,a^{2} b^{4} x^{5}+1152 A \,a^{2} b^{4} x^{4}-1248 B \,a^{3} b^{3} x^{4}-960 A \,a^{3} b^{3} x^{3}+1040 B \,a^{4} b^{2} x^{3}+840 A \,a^{4} b^{2} x^{2}-910 B \,a^{5} b \,x^{2}-756 A \,a^{5} b x +819 B \,a^{6} x +693 A \,a^{6}\right )}{9009 x^{\frac {13}{2}} a^{7}}\) | \(149\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (3072 A \,b^{6} x^{6}-3328 B a \,b^{5} x^{6}-1536 A a \,b^{5} x^{5}+1664 B \,a^{2} b^{4} x^{5}+1152 A \,a^{2} b^{4} x^{4}-1248 B \,a^{3} b^{3} x^{4}-960 A \,a^{3} b^{3} x^{3}+1040 B \,a^{4} b^{2} x^{3}+840 A \,a^{4} b^{2} x^{2}-910 B \,a^{5} b \,x^{2}-756 A \,a^{5} b x +819 B \,a^{6} x +693 A \,a^{6}\right )}{9009 x^{\frac {13}{2}} a^{7}}\) | \(149\) |
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Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=-\frac {2 \, {\left (693 \, A a^{6} - 256 \, {\left (13 \, B a b^{5} - 12 \, A b^{6}\right )} x^{6} + 128 \, {\left (13 \, B a^{2} b^{4} - 12 \, A a b^{5}\right )} x^{5} - 96 \, {\left (13 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{4} + 80 \, {\left (13 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} - 70 \, {\left (13 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x^{2} + 63 \, {\left (13 \, B a^{6} - 12 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{9009 \, a^{7} x^{\frac {13}{2}}} \]
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Timed out. \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=\frac {512 \, \sqrt {b x^{2} + a x} B b^{5}}{693 \, a^{6} x} - \frac {2048 \, \sqrt {b x^{2} + a x} A b^{6}}{3003 \, a^{7} x} - \frac {256 \, \sqrt {b x^{2} + a x} B b^{4}}{693 \, a^{5} x^{2}} + \frac {1024 \, \sqrt {b x^{2} + a x} A b^{5}}{3003 \, a^{6} x^{2}} + \frac {64 \, \sqrt {b x^{2} + a x} B b^{3}}{231 \, a^{4} x^{3}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{4}}{1001 \, a^{5} x^{3}} - \frac {160 \, \sqrt {b x^{2} + a x} B b^{2}}{693 \, a^{3} x^{4}} + \frac {640 \, \sqrt {b x^{2} + a x} A b^{3}}{3003 \, a^{4} x^{4}} + \frac {20 \, \sqrt {b x^{2} + a x} B b}{99 \, a^{2} x^{5}} - \frac {80 \, \sqrt {b x^{2} + a x} A b^{2}}{429 \, a^{3} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{11 \, a x^{6}} + \frac {24 \, \sqrt {b x^{2} + a x} A b}{143 \, a^{2} x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{13 \, a x^{7}} \]
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Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=\frac {2 \, {\left ({\left (2 \, {\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (13 \, B a b^{12} - 12 \, A b^{13}\right )} {\left (b x + a\right )}}{a^{7}} - \frac {13 \, {\left (13 \, B a^{2} b^{12} - 12 \, A a b^{13}\right )}}{a^{7}}\right )} + \frac {143 \, {\left (13 \, B a^{3} b^{12} - 12 \, A a^{2} b^{13}\right )}}{a^{7}}\right )} - \frac {429 \, {\left (13 \, B a^{4} b^{12} - 12 \, A a^{3} b^{13}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {3003 \, {\left (13 \, B a^{5} b^{12} - 12 \, A a^{4} b^{13}\right )}}{a^{7}}\right )} {\left (b x + a\right )} - \frac {3003 \, {\left (13 \, B a^{6} b^{12} - 12 \, A a^{5} b^{13}\right )}}{a^{7}}\right )} {\left (b x + a\right )} + \frac {9009 \, {\left (B a^{7} b^{12} - A a^{6} b^{13}\right )}}{a^{7}}\right )} \sqrt {b x + a} b}{9009 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {13}{2}} {\left | b \right |}} \]
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Time = 0.99 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.63 \[ \int \frac {A+B x}{x^{15/2} \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{13\,a}+\frac {x\,\left (1638\,B\,a^6-1512\,A\,a^5\,b\right )}{9009\,a^7}-\frac {160\,b^2\,x^3\,\left (12\,A\,b-13\,B\,a\right )}{9009\,a^4}+\frac {64\,b^3\,x^4\,\left (12\,A\,b-13\,B\,a\right )}{3003\,a^5}-\frac {256\,b^4\,x^5\,\left (12\,A\,b-13\,B\,a\right )}{9009\,a^6}+\frac {512\,b^5\,x^6\,\left (12\,A\,b-13\,B\,a\right )}{9009\,a^7}+\frac {20\,b\,x^2\,\left (12\,A\,b-13\,B\,a\right )}{1287\,a^3}\right )}{x^{13/2}} \]
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