Integrand size = 29, antiderivative size = 213 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {63 (A b-11 a B) \sqrt {x}}{128 a b^6}+\frac {(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac {(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac {3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac {21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}+\frac {21 (A b-11 a B) x^{3/2}}{128 a b^5 (a+b x)}+\frac {63 (A b-11 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 \sqrt {a} b^{13/2}} \]
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Time = 0.07 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {27, 79, 43, 52, 65, 211} \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {63 (A b-11 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 \sqrt {a} b^{13/2}}-\frac {63 \sqrt {x} (A b-11 a B)}{128 a b^6}+\frac {21 x^{3/2} (A b-11 a B)}{128 a b^5 (a+b x)}+\frac {21 x^{5/2} (A b-11 a B)}{320 a b^4 (a+b x)^2}+\frac {3 x^{7/2} (A b-11 a B)}{80 a b^3 (a+b x)^3}+\frac {x^{9/2} (A b-11 a B)}{40 a b^2 (a+b x)^4}+\frac {x^{11/2} (A b-a B)}{5 a b (a+b x)^5} \]
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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^{9/2} (A+B x)}{(a+b x)^6} \, dx \\ & = \frac {(A b-a B) x^{11/2}}{5 a b (a+b x)^5}-\frac {(A b-11 a B) \int \frac {x^{9/2}}{(a+b x)^5} \, dx}{10 a b} \\ & = \frac {(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac {(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}-\frac {(9 (A b-11 a B)) \int \frac {x^{7/2}}{(a+b x)^4} \, dx}{80 a b^2} \\ & = \frac {(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac {(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac {3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}-\frac {(21 (A b-11 a B)) \int \frac {x^{5/2}}{(a+b x)^3} \, dx}{160 a b^3} \\ & = \frac {(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac {(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac {3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac {21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}-\frac {(21 (A b-11 a B)) \int \frac {x^{3/2}}{(a+b x)^2} \, dx}{128 a b^4} \\ & = \frac {(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac {(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac {3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac {21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}+\frac {21 (A b-11 a B) x^{3/2}}{128 a b^5 (a+b x)}-\frac {(63 (A b-11 a B)) \int \frac {\sqrt {x}}{a+b x} \, dx}{256 a b^5} \\ & = -\frac {63 (A b-11 a B) \sqrt {x}}{128 a b^6}+\frac {(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac {(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac {3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac {21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}+\frac {21 (A b-11 a B) x^{3/2}}{128 a b^5 (a+b x)}+\frac {(63 (A b-11 a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 b^6} \\ & = -\frac {63 (A b-11 a B) \sqrt {x}}{128 a b^6}+\frac {(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac {(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac {3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac {21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}+\frac {21 (A b-11 a B) x^{3/2}}{128 a b^5 (a+b x)}+\frac {(63 (A b-11 a B)) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 b^6} \\ & = -\frac {63 (A b-11 a B) \sqrt {x}}{128 a b^6}+\frac {(A b-a B) x^{11/2}}{5 a b (a+b x)^5}+\frac {(A b-11 a B) x^{9/2}}{40 a b^2 (a+b x)^4}+\frac {3 (A b-11 a B) x^{7/2}}{80 a b^3 (a+b x)^3}+\frac {21 (A b-11 a B) x^{5/2}}{320 a b^4 (a+b x)^2}+\frac {21 (A b-11 a B) x^{3/2}}{128 a b^5 (a+b x)}+\frac {63 (A b-11 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 \sqrt {a} b^{13/2}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.70 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {\sqrt {b} \sqrt {x} \left (3465 a^5 B-105 a^4 b (3 A-154 B x)+5 b^5 x^4 (-193 A+256 B x)+42 a^3 b^2 x (-35 A+704 B x)+5 a b^4 x^3 (-474 A+2123 B x)+6 a^2 b^3 x^2 (-448 A+4345 B x)\right )}{(a+b x)^5}+\frac {315 (A b-11 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a}}}{640 b^{13/2}} \]
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Time = 2.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {2 B \sqrt {x}}{b^{6}}+\frac {\frac {2 \left (\left (-\frac {193}{256} A \,b^{5}+\frac {843}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\left (\frac {1327}{128} B \,a^{2} b^{3}-\frac {237}{128} A a \,b^{4}\right ) x^{\frac {7}{2}}+\left (\frac {131}{10} B \,a^{3} b^{2}-\frac {21}{10} A \,a^{2} b^{3}\right ) x^{\frac {5}{2}}-\frac {a^{3} b \left (147 A b -977 B a \right ) x^{\frac {3}{2}}}{128}+\left (\frac {437}{256} a^{5} B -\frac {63}{256} A \,a^{4} b \right ) \sqrt {x}\right )}{\left (b x +a \right )^{5}}+\frac {63 \left (A b -11 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 \sqrt {b a}}}{b^{6}}\) | \(150\) |
default | \(\frac {2 B \sqrt {x}}{b^{6}}+\frac {\frac {2 \left (\left (-\frac {193}{256} A \,b^{5}+\frac {843}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\left (\frac {1327}{128} B \,a^{2} b^{3}-\frac {237}{128} A a \,b^{4}\right ) x^{\frac {7}{2}}+\left (\frac {131}{10} B \,a^{3} b^{2}-\frac {21}{10} A \,a^{2} b^{3}\right ) x^{\frac {5}{2}}-\frac {a^{3} b \left (147 A b -977 B a \right ) x^{\frac {3}{2}}}{128}+\left (\frac {437}{256} a^{5} B -\frac {63}{256} A \,a^{4} b \right ) \sqrt {x}\right )}{\left (b x +a \right )^{5}}+\frac {63 \left (A b -11 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 \sqrt {b a}}}{b^{6}}\) | \(150\) |
risch | \(\frac {2 B \sqrt {x}}{b^{6}}+\frac {\frac {2 \left (-\frac {193}{256} A \,b^{5}+\frac {843}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+2 \left (\frac {1327}{128} B \,a^{2} b^{3}-\frac {237}{128} A a \,b^{4}\right ) x^{\frac {7}{2}}+2 \left (\frac {131}{10} B \,a^{3} b^{2}-\frac {21}{10} A \,a^{2} b^{3}\right ) x^{\frac {5}{2}}-\frac {a^{3} b \left (147 A b -977 B a \right ) x^{\frac {3}{2}}}{64}+2 \left (\frac {437}{256} a^{5} B -\frac {63}{256} A \,a^{4} b \right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {63 \left (A b -11 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 \sqrt {b a}}}{b^{6}}\) | \(150\) |
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Time = 0.41 (sec) , antiderivative size = 673, normalized size of antiderivative = 3.16 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [\frac {315 \, {\left (11 \, B a^{6} - A a^{5} b + {\left (11 \, B a b^{5} - A b^{6}\right )} x^{5} + 5 \, {\left (11 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 10 \, {\left (11 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (11 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (11 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (1280 \, B a b^{6} x^{5} + 3465 \, B a^{6} b - 315 \, A a^{5} b^{2} + 965 \, {\left (11 \, B a^{2} b^{5} - A a b^{6}\right )} x^{4} + 2370 \, {\left (11 \, B a^{3} b^{4} - A a^{2} b^{5}\right )} x^{3} + 2688 \, {\left (11 \, B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{2} + 1470 \, {\left (11 \, B a^{5} b^{2} - A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{1280 \, {\left (a b^{12} x^{5} + 5 \, a^{2} b^{11} x^{4} + 10 \, a^{3} b^{10} x^{3} + 10 \, a^{4} b^{9} x^{2} + 5 \, a^{5} b^{8} x + a^{6} b^{7}\right )}}, \frac {315 \, {\left (11 \, B a^{6} - A a^{5} b + {\left (11 \, B a b^{5} - A b^{6}\right )} x^{5} + 5 \, {\left (11 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 10 \, {\left (11 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (11 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (11 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (1280 \, B a b^{6} x^{5} + 3465 \, B a^{6} b - 315 \, A a^{5} b^{2} + 965 \, {\left (11 \, B a^{2} b^{5} - A a b^{6}\right )} x^{4} + 2370 \, {\left (11 \, B a^{3} b^{4} - A a^{2} b^{5}\right )} x^{3} + 2688 \, {\left (11 \, B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{2} + 1470 \, {\left (11 \, B a^{5} b^{2} - A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{640 \, {\left (a b^{12} x^{5} + 5 \, a^{2} b^{11} x^{4} + 10 \, a^{3} b^{10} x^{3} + 10 \, a^{4} b^{9} x^{2} + 5 \, a^{5} b^{8} x + a^{6} b^{7}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.97 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {5 \, {\left (843 \, B a b^{4} - 193 \, A b^{5}\right )} x^{\frac {9}{2}} + 10 \, {\left (1327 \, B a^{2} b^{3} - 237 \, A a b^{4}\right )} x^{\frac {7}{2}} + 128 \, {\left (131 \, B a^{3} b^{2} - 21 \, A a^{2} b^{3}\right )} x^{\frac {5}{2}} + 10 \, {\left (977 \, B a^{4} b - 147 \, A a^{3} b^{2}\right )} x^{\frac {3}{2}} + 5 \, {\left (437 \, B a^{5} - 63 \, A a^{4} b\right )} \sqrt {x}}{640 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} + \frac {2 \, B \sqrt {x}}{b^{6}} - \frac {63 \, {\left (11 \, B a - A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} b^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.75 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {2 \, B \sqrt {x}}{b^{6}} - \frac {63 \, {\left (11 \, B a - A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} b^{6}} + \frac {4215 \, B a b^{4} x^{\frac {9}{2}} - 965 \, A b^{5} x^{\frac {9}{2}} + 13270 \, B a^{2} b^{3} x^{\frac {7}{2}} - 2370 \, A a b^{4} x^{\frac {7}{2}} + 16768 \, B a^{3} b^{2} x^{\frac {5}{2}} - 2688 \, A a^{2} b^{3} x^{\frac {5}{2}} + 9770 \, B a^{4} b x^{\frac {3}{2}} - 1470 \, A a^{3} b^{2} x^{\frac {3}{2}} + 2185 \, B a^{5} \sqrt {x} - 315 \, A a^{4} b \sqrt {x}}{640 \, {\left (b x + a\right )}^{5} b^{6}} \]
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Time = 0.19 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.94 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {2\,B\,\sqrt {x}}{b^6}-\frac {x^{3/2}\,\left (\frac {147\,A\,a^3\,b^2}{64}-\frac {977\,B\,a^4\,b}{64}\right )-x^{7/2}\,\left (\frac {1327\,B\,a^2\,b^3}{64}-\frac {237\,A\,a\,b^4}{64}\right )-\sqrt {x}\,\left (\frac {437\,B\,a^5}{128}-\frac {63\,A\,a^4\,b}{128}\right )+x^{9/2}\,\left (\frac {193\,A\,b^5}{128}-\frac {843\,B\,a\,b^4}{128}\right )+x^{5/2}\,\left (\frac {21\,A\,a^2\,b^3}{5}-\frac {131\,B\,a^3\,b^2}{5}\right )}{a^5\,b^6+5\,a^4\,b^7\,x+10\,a^3\,b^8\,x^2+10\,a^2\,b^9\,x^3+5\,a\,b^{10}\,x^4+b^{11}\,x^5}+\frac {63\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (A\,b-11\,B\,a\right )}{128\,\sqrt {a}\,b^{13/2}} \]
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