Integrand size = 20, antiderivative size = 183 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac {2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}-\frac {16 b (10 A b-13 a B) (a+b x)^{3/2}}{1287 a^3 x^{9/2}}+\frac {32 b^2 (10 A b-13 a B) (a+b x)^{3/2}}{3003 a^4 x^{7/2}}-\frac {128 b^3 (10 A b-13 a B) (a+b x)^{3/2}}{15015 a^5 x^{5/2}}+\frac {256 b^4 (10 A b-13 a B) (a+b x)^{3/2}}{45045 a^6 x^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 183, 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} \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=\frac {256 b^4 (a+b x)^{3/2} (10 A b-13 a B)}{45045 a^6 x^{3/2}}-\frac {128 b^3 (a+b x)^{3/2} (10 A b-13 a B)}{15015 a^5 x^{5/2}}+\frac {32 b^2 (a+b x)^{3/2} (10 A b-13 a B)}{3003 a^4 x^{7/2}}-\frac {16 b (a+b x)^{3/2} (10 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac {2 (a+b x)^{3/2} (10 A b-13 a B)}{143 a^2 x^{11/2}}-\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac {\left (2 \left (-5 A b+\frac {13 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x}}{x^{13/2}} \, dx}{13 a} \\ & = -\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac {2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}+\frac {(8 b (10 A b-13 a B)) \int \frac {\sqrt {a+b x}}{x^{11/2}} \, dx}{143 a^2} \\ & = -\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac {2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}-\frac {16 b (10 A b-13 a B) (a+b x)^{3/2}}{1287 a^3 x^{9/2}}-\frac {\left (16 b^2 (10 A b-13 a B)\right ) \int \frac {\sqrt {a+b x}}{x^{9/2}} \, dx}{429 a^3} \\ & = -\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac {2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}-\frac {16 b (10 A b-13 a B) (a+b x)^{3/2}}{1287 a^3 x^{9/2}}+\frac {32 b^2 (10 A b-13 a B) (a+b x)^{3/2}}{3003 a^4 x^{7/2}}+\frac {\left (64 b^3 (10 A b-13 a B)\right ) \int \frac {\sqrt {a+b x}}{x^{7/2}} \, dx}{3003 a^4} \\ & = -\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac {2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}-\frac {16 b (10 A b-13 a B) (a+b x)^{3/2}}{1287 a^3 x^{9/2}}+\frac {32 b^2 (10 A b-13 a B) (a+b x)^{3/2}}{3003 a^4 x^{7/2}}-\frac {128 b^3 (10 A b-13 a B) (a+b x)^{3/2}}{15015 a^5 x^{5/2}}-\frac {\left (128 b^4 (10 A b-13 a B)\right ) \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx}{15015 a^5} \\ & = -\frac {2 A (a+b x)^{3/2}}{13 a x^{13/2}}+\frac {2 (10 A b-13 a B) (a+b x)^{3/2}}{143 a^2 x^{11/2}}-\frac {16 b (10 A b-13 a B) (a+b x)^{3/2}}{1287 a^3 x^{9/2}}+\frac {32 b^2 (10 A b-13 a B) (a+b x)^{3/2}}{3003 a^4 x^{7/2}}-\frac {128 b^3 (10 A b-13 a B) (a+b x)^{3/2}}{15015 a^5 x^{5/2}}+\frac {256 b^4 (10 A b-13 a B) (a+b x)^{3/2}}{45045 a^6 x^{3/2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {2 (a+b x)^{3/2} \left (-1280 A b^5 x^5+315 a^5 (11 A+13 B x)+128 a b^4 x^4 (15 A+13 B x)-96 a^2 b^3 x^3 (25 A+26 B x)+80 a^3 b^2 x^2 (35 A+39 B x)-70 a^4 b x (45 A+52 B x)\right )}{45045 a^6 x^{13/2}} \]
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Time = 0.51 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-1280 A \,b^{5} x^{5}+1664 B a \,b^{4} x^{5}+1920 a A \,b^{4} x^{4}-2496 B \,a^{2} b^{3} x^{4}-2400 a^{2} A \,b^{3} x^{3}+3120 B \,a^{3} b^{2} x^{3}+2800 a^{3} A \,b^{2} x^{2}-3640 B \,a^{4} b \,x^{2}-3150 a^{4} A b x +4095 a^{5} B x +3465 a^{5} A \right )}{45045 x^{\frac {13}{2}} a^{6}}\) | \(125\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-1280 A \,b^{5} x^{5}+1664 B a \,b^{4} x^{5}+1920 a A \,b^{4} x^{4}-2496 B \,a^{2} b^{3} x^{4}-2400 a^{2} A \,b^{3} x^{3}+3120 B \,a^{3} b^{2} x^{3}+2800 a^{3} A \,b^{2} x^{2}-3640 B \,a^{4} b \,x^{2}-3150 a^{4} A b x +4095 a^{5} B x +3465 a^{5} A \right )}{45045 x^{\frac {13}{2}} a^{6}}\) | \(125\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-1280 A \,b^{6} x^{6}+1664 B a \,b^{5} x^{6}+640 A a \,b^{5} x^{5}-832 B \,a^{2} b^{4} x^{5}-480 A \,a^{2} b^{4} x^{4}+624 B \,a^{3} b^{3} x^{4}+400 A \,a^{3} b^{3} x^{3}-520 B \,a^{4} b^{2} x^{3}-350 A \,a^{4} b^{2} x^{2}+455 B \,a^{5} b \,x^{2}+315 A \,a^{5} b x +4095 B \,a^{6} x +3465 A \,a^{6}\right )}{45045 x^{\frac {13}{2}} a^{6}}\) | \(149\) |
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Time = 0.22 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {2 \, {\left (3465 \, A a^{6} + 128 \, {\left (13 \, B a b^{5} - 10 \, A b^{6}\right )} x^{6} - 64 \, {\left (13 \, B a^{2} b^{4} - 10 \, A a b^{5}\right )} x^{5} + 48 \, {\left (13 \, B a^{3} b^{3} - 10 \, A a^{2} b^{4}\right )} x^{4} - 40 \, {\left (13 \, B a^{4} b^{2} - 10 \, A a^{3} b^{3}\right )} x^{3} + 35 \, {\left (13 \, B a^{5} b - 10 \, A a^{4} b^{2}\right )} x^{2} + 315 \, {\left (13 \, B a^{6} + A a^{5} b\right )} x\right )} \sqrt {b x + a}}{45045 \, a^{6} x^{\frac {13}{2}}} \]
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Timed out. \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {256 \, \sqrt {b x^{2} + a x} B b^{5}}{3465 \, a^{5} x} + \frac {512 \, \sqrt {b x^{2} + a x} A b^{6}}{9009 \, a^{6} x} + \frac {128 \, \sqrt {b x^{2} + a x} B b^{4}}{3465 \, a^{4} x^{2}} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{5}}{9009 \, a^{5} x^{2}} - \frac {32 \, \sqrt {b x^{2} + a x} B b^{3}}{1155 \, a^{3} x^{3}} + \frac {64 \, \sqrt {b x^{2} + a x} A b^{4}}{3003 \, a^{4} x^{3}} + \frac {16 \, \sqrt {b x^{2} + a x} B b^{2}}{693 \, a^{2} x^{4}} - \frac {160 \, \sqrt {b x^{2} + a x} A b^{3}}{9009 \, a^{3} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} B b}{99 \, a x^{5}} + \frac {20 \, \sqrt {b x^{2} + a x} A b^{2}}{1287 \, a^{2} x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{11 \, x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A b}{143 \, a x^{6}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{13 \, x^{7}} \]
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Time = 0.31 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (13 \, B a b^{12} - 10 \, A b^{13}\right )} {\left (b x + a\right )}}{a^{6}} - \frac {13 \, {\left (13 \, B a^{2} b^{12} - 10 \, A a b^{13}\right )}}{a^{6}}\right )} + \frac {143 \, {\left (13 \, B a^{3} b^{12} - 10 \, A a^{2} b^{13}\right )}}{a^{6}}\right )} - \frac {429 \, {\left (13 \, B a^{4} b^{12} - 10 \, A a^{3} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )} + \frac {3003 \, {\left (13 \, B a^{5} b^{12} - 10 \, A a^{4} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )} - \frac {15015 \, {\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{6}}\right )} {\left (b x + a\right )}^{\frac {3}{2}} b}{45045 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {13}{2}} {\left | b \right |}} \]
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Time = 0.88 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{15/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{13}+\frac {2\,x\,\left (A\,b+13\,B\,a\right )}{143\,a}+\frac {16\,b^2\,x^3\,\left (10\,A\,b-13\,B\,a\right )}{9009\,a^3}-\frac {32\,b^3\,x^4\,\left (10\,A\,b-13\,B\,a\right )}{15015\,a^4}+\frac {128\,b^4\,x^5\,\left (10\,A\,b-13\,B\,a\right )}{45045\,a^5}-\frac {256\,b^5\,x^6\,\left (10\,A\,b-13\,B\,a\right )}{45045\,a^6}-\frac {2\,b\,x^2\,\left (10\,A\,b-13\,B\,a\right )}{1287\,a^2}\right )}{x^{13/2}} \]
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