Integrand size = 29, antiderivative size = 240 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {231 \sqrt {b} (13 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}} \]
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Time = 0.08 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {27, 79, 44, 53, 65, 211} \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {231 \sqrt {b} (13 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}-\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5} \]
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Rule 27
Rule 44
Rule 53
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x}{x^{5/2} (a+b x)^6} \, dx \\ & = \frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}-\frac {\left (-\frac {13 A b}{2}+\frac {3 a B}{2}\right ) \int \frac {1}{x^{5/2} (a+b x)^5} \, dx}{5 a b} \\ & = \frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {(11 (13 A b-3 a B)) \int \frac {1}{x^{5/2} (a+b x)^4} \, dx}{80 a^2 b} \\ & = \frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {(33 (13 A b-3 a B)) \int \frac {1}{x^{5/2} (a+b x)^3} \, dx}{160 a^3 b} \\ & = \frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {(231 (13 A b-3 a B)) \int \frac {1}{x^{5/2} (a+b x)^2} \, dx}{640 a^4 b} \\ & = \frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {(231 (13 A b-3 a B)) \int \frac {1}{x^{5/2} (a+b x)} \, dx}{256 a^5 b} \\ & = -\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}-\frac {(231 (13 A b-3 a B)) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{256 a^6} \\ & = -\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {(231 b (13 A b-3 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 a^7} \\ & = -\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {(231 b (13 A b-3 a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 a^7} \\ & = -\frac {77 (13 A b-3 a B)}{128 a^6 b x^{3/2}}+\frac {231 (13 A b-3 a B)}{128 a^7 \sqrt {x}}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}+\frac {13 A b-3 a B}{40 a^2 b x^{3/2} (a+b x)^4}+\frac {11 (13 A b-3 a B)}{240 a^3 b x^{3/2} (a+b x)^3}+\frac {33 (13 A b-3 a B)}{320 a^4 b x^{3/2} (a+b x)^2}+\frac {231 (13 A b-3 a B)}{640 a^5 b x^{3/2} (a+b x)}+\frac {231 \sqrt {b} (13 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.72 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-\frac {\sqrt {a} \left (-45045 A b^6 x^6+1280 a^6 (A+3 B x)+1155 a b^5 x^5 (-182 A+9 B x)+462 a^2 b^4 x^4 (-832 A+105 B x)+66 a^3 b^3 x^3 (-5135 A+1344 B x)+55 a^4 b^2 x^2 (-2509 A+1422 B x)+5 a^5 b x (-3328 A+6369 B x)\right )}{x^{3/2} (a+b x)^5}+3465 \sqrt {b} (13 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{1920 a^{15/2}} \]
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Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {2 \left (-18 A b x +3 a B x +a A \right )}{3 a^{7} x^{\frac {3}{2}}}+\frac {b \left (\frac {2 \left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {b^{3} a \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{192}+2 \left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+2 \left (\frac {12131}{384} A \,a^{3} b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+2 \left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 \sqrt {b a}}\right )}{a^{7}}\) | \(165\) |
derivativedivides | \(\frac {2 b \left (\frac {\left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {b^{3} a \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+\left (\frac {12131}{384} A \,a^{3} b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+\left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{256 \sqrt {b a}}\right )}{a^{7}}-\frac {2 A}{3 a^{6} x^{\frac {3}{2}}}-\frac {2 \left (-6 A b +B a \right )}{a^{7} \sqrt {x}}\) | \(168\) |
default | \(\frac {2 b \left (\frac {\left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {b^{3} a \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+\left (\frac {12131}{384} A \,a^{3} b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+\left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{256 \sqrt {b a}}\right )}{a^{7}}-\frac {2 A}{3 a^{6} x^{\frac {3}{2}}}-\frac {2 \left (-6 A b +B a \right )}{a^{7} \sqrt {x}}\) | \(168\) |
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Time = 0.44 (sec) , antiderivative size = 734, normalized size of antiderivative = 3.06 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {3465 \, {\left ({\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{7} + 5 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{6} + 10 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{5} + 10 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + 5 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{3} + {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{7} b^{5} x^{7} + 5 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{5} + 10 \, a^{10} b^{2} x^{4} + 5 \, a^{11} b x^{3} + a^{12} x^{2}\right )}}, \frac {3465 \, {\left ({\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{7} + 5 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{6} + 10 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{5} + 10 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{4} + 5 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{3} + {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{7} b^{5} x^{7} + 5 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{5} + 10 \, a^{10} b^{2} x^{4} + 5 \, a^{11} b x^{3} + a^{12} x^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.32 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x}{1920 \, {\left (a^{7} b^{5} x^{\frac {13}{2}} + 5 \, a^{8} b^{4} x^{\frac {11}{2}} + 10 \, a^{9} b^{3} x^{\frac {9}{2}} + 10 \, a^{10} b^{2} x^{\frac {7}{2}} + 5 \, a^{11} b x^{\frac {5}{2}} + a^{12} x^{\frac {3}{2}}\right )}} - \frac {231 \, {\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{7}} \]
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Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.75 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {231 \, {\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{7}} - \frac {2 \, {\left (3 \, B a x - 18 \, A b x + A a\right )}}{3 \, a^{7} x^{\frac {3}{2}}} - \frac {6555 \, B a b^{5} x^{\frac {9}{2}} - 22005 \, A b^{6} x^{\frac {9}{2}} + 29310 \, B a^{2} b^{4} x^{\frac {7}{2}} - 96290 \, A a b^{5} x^{\frac {7}{2}} + 50304 \, B a^{3} b^{3} x^{\frac {5}{2}} - 160384 \, A a^{2} b^{4} x^{\frac {5}{2}} + 39810 \, B a^{4} b^{2} x^{\frac {3}{2}} - 121310 \, A a^{3} b^{3} x^{\frac {3}{2}} + 12645 \, B a^{5} b \sqrt {x} - 35595 \, A a^{4} b^{2} \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{7}} \]
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Time = 10.24 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {2\,x\,\left (13\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {2\,A}{3\,a}+\frac {869\,b^2\,x^3\,\left (13\,A\,b-3\,B\,a\right )}{64\,a^4}+\frac {77\,b^3\,x^4\,\left (13\,A\,b-3\,B\,a\right )}{5\,a^5}+\frac {539\,b^4\,x^5\,\left (13\,A\,b-3\,B\,a\right )}{64\,a^6}+\frac {231\,b^5\,x^6\,\left (13\,A\,b-3\,B\,a\right )}{128\,a^7}+\frac {2123\,b\,x^2\,\left (13\,A\,b-3\,B\,a\right )}{384\,a^3}}{a^5\,x^{3/2}+b^5\,x^{13/2}+5\,a^4\,b\,x^{5/2}+5\,a\,b^4\,x^{11/2}+10\,a^3\,b^2\,x^{7/2}+10\,a^2\,b^3\,x^{9/2}}+\frac {231\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (13\,A\,b-3\,B\,a\right )}{128\,a^{15/2}} \]
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