Integrand size = 29, antiderivative size = 240 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {231 (3 A b-13 a B) \sqrt {x}}{128 b^7}-\frac {77 (3 A b-13 a B) x^{3/2}}{128 a b^6}+\frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}-\frac {231 \sqrt {a} (3 A b-13 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 b^{15/2}} \]
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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, 43, 52, 65, 211} \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {231 \sqrt {a} (3 A b-13 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 b^{15/2}}+\frac {231 \sqrt {x} (3 A b-13 a B)}{128 b^7}-\frac {77 x^{3/2} (3 A b-13 a B)}{128 a b^6}+\frac {231 x^{5/2} (3 A b-13 a B)}{640 a b^5 (a+b x)}+\frac {33 x^{7/2} (3 A b-13 a B)}{320 a b^4 (a+b x)^2}+\frac {11 x^{9/2} (3 A b-13 a B)}{240 a b^3 (a+b x)^3}+\frac {x^{11/2} (3 A b-13 a B)}{40 a b^2 (a+b x)^4}+\frac {x^{13/2} (A b-a B)}{5 a b (a+b x)^5} \]
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Rule 27
Rule 43
Rule 52
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{11/2} (A+B x)}{(a+b x)^6} \, dx \\ & = \frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}-\frac {\left (\frac {3 A b}{2}-\frac {13 a B}{2}\right ) \int \frac {x^{11/2}}{(a+b x)^5} \, dx}{5 a b} \\ & = \frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}-\frac {(11 (3 A b-13 a B)) \int \frac {x^{9/2}}{(a+b x)^4} \, dx}{80 a b^2} \\ & = \frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}-\frac {(33 (3 A b-13 a B)) \int \frac {x^{7/2}}{(a+b x)^3} \, dx}{160 a b^3} \\ & = \frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}-\frac {(231 (3 A b-13 a B)) \int \frac {x^{5/2}}{(a+b x)^2} \, dx}{640 a b^4} \\ & = \frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}-\frac {(231 (3 A b-13 a B)) \int \frac {x^{3/2}}{a+b x} \, dx}{256 a b^5} \\ & = -\frac {77 (3 A b-13 a B) x^{3/2}}{128 a b^6}+\frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}+\frac {(231 (3 A b-13 a B)) \int \frac {\sqrt {x}}{a+b x} \, dx}{256 b^6} \\ & = \frac {231 (3 A b-13 a B) \sqrt {x}}{128 b^7}-\frac {77 (3 A b-13 a B) x^{3/2}}{128 a b^6}+\frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}-\frac {(231 a (3 A b-13 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 b^7} \\ & = \frac {231 (3 A b-13 a B) \sqrt {x}}{128 b^7}-\frac {77 (3 A b-13 a B) x^{3/2}}{128 a b^6}+\frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}-\frac {(231 a (3 A b-13 a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 b^7} \\ & = \frac {231 (3 A b-13 a B) \sqrt {x}}{128 b^7}-\frac {77 (3 A b-13 a B) x^{3/2}}{128 a b^6}+\frac {(A b-a B) x^{13/2}}{5 a b (a+b x)^5}+\frac {(3 A b-13 a B) x^{11/2}}{40 a b^2 (a+b x)^4}+\frac {11 (3 A b-13 a B) x^{9/2}}{240 a b^3 (a+b x)^3}+\frac {33 (3 A b-13 a B) x^{7/2}}{320 a b^4 (a+b x)^2}+\frac {231 (3 A b-13 a B) x^{5/2}}{640 a b^5 (a+b x)}-\frac {231 \sqrt {a} (3 A b-13 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 b^{15/2}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.70 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {\sqrt {b} \sqrt {x} \left (-45045 a^6 B+66 a^3 b^3 x^2 (1344 A-5135 B x)+5 a b^5 x^4 (6369 A-3328 B x)+55 a^2 b^4 x^3 (1422 A-2509 B x)+462 a^4 b^2 x (105 A-832 B x)+1155 a^5 b (9 A-182 B x)+1280 b^6 x^5 (3 A+B x)\right )}{(a+b x)^5}+3465 \sqrt {a} (-3 A b+13 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{1920 b^{15/2}} \]
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Time = 2.14 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {2 \left (B b x +3 A b -18 B a \right ) \sqrt {x}}{3 b^{7}}-\frac {a \left (\frac {2 \left (-\frac {843}{256} A \,b^{5}+\frac {2373}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+2 \left (-\frac {1327}{128} A a \,b^{4}+\frac {12131}{384} B \,a^{2} b^{3}\right ) x^{\frac {7}{2}}+2 \left (-\frac {131}{10} A \,a^{2} b^{3}+\frac {1253}{30} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}-\frac {a^{3} b \left (2931 A b -9629 B a \right ) x^{\frac {3}{2}}}{192}+2 \left (-\frac {437}{256} A \,a^{4} b +\frac {1467}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (3 A b -13 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 \sqrt {b a}}\right )}{b^{7}}\) | \(165\) |
derivativedivides | \(\frac {\frac {2 B b \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}-12 a B \sqrt {x}}{b^{7}}-\frac {2 a \left (\frac {\left (-\frac {843}{256} A \,b^{5}+\frac {2373}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\left (-\frac {1327}{128} A a \,b^{4}+\frac {12131}{384} B \,a^{2} b^{3}\right ) x^{\frac {7}{2}}+\left (-\frac {131}{10} A \,a^{2} b^{3}+\frac {1253}{30} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}-\frac {a^{3} b \left (2931 A b -9629 B a \right ) x^{\frac {3}{2}}}{384}+\left (-\frac {437}{256} A \,a^{4} b +\frac {1467}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (3 A b -13 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{256 \sqrt {b a}}\right )}{b^{7}}\) | \(169\) |
default | \(\frac {\frac {2 B b \,x^{\frac {3}{2}}}{3}+2 A b \sqrt {x}-12 a B \sqrt {x}}{b^{7}}-\frac {2 a \left (\frac {\left (-\frac {843}{256} A \,b^{5}+\frac {2373}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\left (-\frac {1327}{128} A a \,b^{4}+\frac {12131}{384} B \,a^{2} b^{3}\right ) x^{\frac {7}{2}}+\left (-\frac {131}{10} A \,a^{2} b^{3}+\frac {1253}{30} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}-\frac {a^{3} b \left (2931 A b -9629 B a \right ) x^{\frac {3}{2}}}{384}+\left (-\frac {437}{256} A \,a^{4} b +\frac {1467}{256} a^{5} B \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (3 A b -13 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{256 \sqrt {b a}}\right )}{b^{7}}\) | \(169\) |
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Time = 0.35 (sec) , antiderivative size = 703, normalized size of antiderivative = 2.93 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {3465 \, {\left (13 \, B a^{6} - 3 \, A a^{5} b + {\left (13 \, B a b^{5} - 3 \, A b^{6}\right )} x^{5} + 5 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x^{4} + 10 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (1280 \, B b^{6} x^{6} - 45045 \, B a^{6} + 10395 \, A a^{5} b - 1280 \, {\left (13 \, B a b^{5} - 3 \, A b^{6}\right )} x^{5} - 10615 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x^{4} - 26070 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} - 29568 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} - 16170 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{3840 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}}, \frac {3465 \, {\left (13 \, B a^{6} - 3 \, A a^{5} b + {\left (13 \, B a b^{5} - 3 \, A b^{6}\right )} x^{5} + 5 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x^{4} + 10 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (1280 \, B b^{6} x^{6} - 45045 \, B a^{6} + 10395 \, A a^{5} b - 1280 \, {\left (13 \, B a b^{5} - 3 \, A b^{6}\right )} x^{5} - 10615 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x^{4} - 26070 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} - 29568 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} - 16170 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {x}}{1920 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.96 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {45 \, {\left (791 \, B a^{2} b^{4} - 281 \, A a b^{5}\right )} x^{\frac {9}{2}} + 10 \, {\left (12131 \, B a^{3} b^{3} - 3981 \, A a^{2} b^{4}\right )} x^{\frac {7}{2}} + 128 \, {\left (1253 \, B a^{4} b^{2} - 393 \, A a^{3} b^{3}\right )} x^{\frac {5}{2}} + 10 \, {\left (9629 \, B a^{5} b - 2931 \, A a^{4} b^{2}\right )} x^{\frac {3}{2}} + 15 \, {\left (1467 \, B a^{6} - 437 \, A a^{5} b\right )} \sqrt {x}}{1920 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} + \frac {231 \, {\left (13 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} b^{7}} + \frac {2 \, {\left (B b x^{\frac {3}{2}} - 3 \, {\left (6 \, B a - A b\right )} \sqrt {x}\right )}}{3 \, b^{7}} \]
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Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.80 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {231 \, {\left (13 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} b^{7}} - \frac {35595 \, B a^{2} b^{4} x^{\frac {9}{2}} - 12645 \, A a b^{5} x^{\frac {9}{2}} + 121310 \, B a^{3} b^{3} x^{\frac {7}{2}} - 39810 \, A a^{2} b^{4} x^{\frac {7}{2}} + 160384 \, B a^{4} b^{2} x^{\frac {5}{2}} - 50304 \, A a^{3} b^{3} x^{\frac {5}{2}} + 96290 \, B a^{5} b x^{\frac {3}{2}} - 29310 \, A a^{4} b^{2} x^{\frac {3}{2}} + 22005 \, B a^{6} \sqrt {x} - 6555 \, A a^{5} b \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} b^{7}} + \frac {2 \, {\left (B b^{12} x^{\frac {3}{2}} - 18 \, B a b^{11} \sqrt {x} + 3 \, A b^{12} \sqrt {x}\right )}}{3 \, b^{18}} \]
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Time = 10.16 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02 \[ \int \frac {x^{11/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\sqrt {x}\,\left (\frac {2\,A}{b^6}-\frac {12\,B\,a}{b^7}\right )+\frac {x^{3/2}\,\left (\frac {977\,A\,a^4\,b^2}{64}-\frac {9629\,B\,a^5\,b}{192}\right )-x^{9/2}\,\left (\frac {2373\,B\,a^2\,b^4}{128}-\frac {843\,A\,a\,b^5}{128}\right )-\sqrt {x}\,\left (\frac {1467\,B\,a^6}{128}-\frac {437\,A\,a^5\,b}{128}\right )+x^{5/2}\,\left (\frac {131\,A\,a^3\,b^3}{5}-\frac {1253\,B\,a^4\,b^2}{15}\right )+x^{7/2}\,\left (\frac {1327\,A\,a^2\,b^4}{64}-\frac {12131\,B\,a^3\,b^3}{192}\right )}{a^5\,b^7+5\,a^4\,b^8\,x+10\,a^3\,b^9\,x^2+10\,a^2\,b^{10}\,x^3+5\,a\,b^{11}\,x^4+b^{12}\,x^5}+\frac {2\,B\,x^{3/2}}{3\,b^6}+\frac {231\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\sqrt {x}\,\left (3\,A\,b-13\,B\,a\right )}{13\,B\,a^2-3\,A\,a\,b}\right )\,\left (3\,A\,b-13\,B\,a\right )}{128\,b^{15/2}} \]
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