Integrand size = 29, antiderivative size = 195 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {(A b-a B) x^{7/2}}{5 a b (a+b x)^5}-\frac {(3 A b+7 a B) x^{5/2}}{40 a b^2 (a+b x)^4}-\frac {(3 A b+7 a B) x^{3/2}}{48 a b^3 (a+b x)^3}-\frac {(3 A b+7 a B) \sqrt {x}}{64 a b^4 (a+b x)^2}+\frac {(3 A b+7 a B) \sqrt {x}}{128 a^2 b^4 (a+b x)}+\frac {(3 A b+7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{5/2} b^{9/2}} \]
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Time = 0.06 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {27, 79, 43, 44, 65, 211} \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {(7 a B+3 A b) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{5/2} b^{9/2}}+\frac {\sqrt {x} (7 a B+3 A b)}{128 a^2 b^4 (a+b x)}-\frac {\sqrt {x} (7 a B+3 A b)}{64 a b^4 (a+b x)^2}-\frac {x^{3/2} (7 a B+3 A b)}{48 a b^3 (a+b x)^3}-\frac {x^{5/2} (7 a B+3 A b)}{40 a b^2 (a+b x)^4}+\frac {x^{7/2} (A b-a B)}{5 a b (a+b x)^5} \]
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Rule 27
Rule 43
Rule 44
Rule 65
Rule 79
Rule 211
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{5/2} (A+B x)}{(a+b x)^6} \, dx \\ & = \frac {(A b-a B) x^{7/2}}{5 a b (a+b x)^5}+\frac {(3 A b+7 a B) \int \frac {x^{5/2}}{(a+b x)^5} \, dx}{10 a b} \\ & = \frac {(A b-a B) x^{7/2}}{5 a b (a+b x)^5}-\frac {(3 A b+7 a B) x^{5/2}}{40 a b^2 (a+b x)^4}+\frac {(3 A b+7 a B) \int \frac {x^{3/2}}{(a+b x)^4} \, dx}{16 a b^2} \\ & = \frac {(A b-a B) x^{7/2}}{5 a b (a+b x)^5}-\frac {(3 A b+7 a B) x^{5/2}}{40 a b^2 (a+b x)^4}-\frac {(3 A b+7 a B) x^{3/2}}{48 a b^3 (a+b x)^3}+\frac {(3 A b+7 a B) \int \frac {\sqrt {x}}{(a+b x)^3} \, dx}{32 a b^3} \\ & = \frac {(A b-a B) x^{7/2}}{5 a b (a+b x)^5}-\frac {(3 A b+7 a B) x^{5/2}}{40 a b^2 (a+b x)^4}-\frac {(3 A b+7 a B) x^{3/2}}{48 a b^3 (a+b x)^3}-\frac {(3 A b+7 a B) \sqrt {x}}{64 a b^4 (a+b x)^2}+\frac {(3 A b+7 a B) \int \frac {1}{\sqrt {x} (a+b x)^2} \, dx}{128 a b^4} \\ & = \frac {(A b-a B) x^{7/2}}{5 a b (a+b x)^5}-\frac {(3 A b+7 a B) x^{5/2}}{40 a b^2 (a+b x)^4}-\frac {(3 A b+7 a B) x^{3/2}}{48 a b^3 (a+b x)^3}-\frac {(3 A b+7 a B) \sqrt {x}}{64 a b^4 (a+b x)^2}+\frac {(3 A b+7 a B) \sqrt {x}}{128 a^2 b^4 (a+b x)}+\frac {(3 A b+7 a B) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{256 a^2 b^4} \\ & = \frac {(A b-a B) x^{7/2}}{5 a b (a+b x)^5}-\frac {(3 A b+7 a B) x^{5/2}}{40 a b^2 (a+b x)^4}-\frac {(3 A b+7 a B) x^{3/2}}{48 a b^3 (a+b x)^3}-\frac {(3 A b+7 a B) \sqrt {x}}{64 a b^4 (a+b x)^2}+\frac {(3 A b+7 a B) \sqrt {x}}{128 a^2 b^4 (a+b x)}+\frac {(3 A b+7 a B) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{128 a^2 b^4} \\ & = \frac {(A b-a B) x^{7/2}}{5 a b (a+b x)^5}-\frac {(3 A b+7 a B) x^{5/2}}{40 a b^2 (a+b x)^4}-\frac {(3 A b+7 a B) x^{3/2}}{48 a b^3 (a+b x)^3}-\frac {(3 A b+7 a B) \sqrt {x}}{64 a b^4 (a+b x)^2}+\frac {(3 A b+7 a B) \sqrt {x}}{128 a^2 b^4 (a+b x)}+\frac {(3 A b+7 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{5/2} b^{9/2}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.74 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {x} \left (-105 a^5 B+45 A b^5 x^4+105 a b^4 x^3 (2 A+B x)-14 a^3 b^2 x (15 A+64 B x)-5 a^4 b (9 A+98 B x)-2 a^2 b^3 x^2 (192 A+395 B x)\right )}{1920 a^2 b^4 (a+b x)^5}+\frac {(3 A b+7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{5/2} b^{9/2}} \]
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Time = 1.98 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {\frac {\left (3 A b +7 B a \right ) x^{\frac {9}{2}}}{128 a^{2}}+\frac {\left (21 A b -79 B a \right ) x^{\frac {7}{2}}}{192 b a}-\frac {\left (3 A b +7 B a \right ) x^{\frac {5}{2}}}{15 b^{2}}-\frac {7 a \left (3 A b +7 B a \right ) x^{\frac {3}{2}}}{192 b^{3}}-\frac {\left (3 A b +7 B a \right ) a^{2} \sqrt {x}}{128 b^{4}}}{\left (b x +a \right )^{5}}+\frac {\left (3 A b +7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{2} b^{4} \sqrt {b a}}\) | \(137\) |
default | \(\frac {\frac {\left (3 A b +7 B a \right ) x^{\frac {9}{2}}}{128 a^{2}}+\frac {\left (21 A b -79 B a \right ) x^{\frac {7}{2}}}{192 b a}-\frac {\left (3 A b +7 B a \right ) x^{\frac {5}{2}}}{15 b^{2}}-\frac {7 a \left (3 A b +7 B a \right ) x^{\frac {3}{2}}}{192 b^{3}}-\frac {\left (3 A b +7 B a \right ) a^{2} \sqrt {x}}{128 b^{4}}}{\left (b x +a \right )^{5}}+\frac {\left (3 A b +7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {b a}}\right )}{128 a^{2} b^{4} \sqrt {b a}}\) | \(137\) |
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Time = 0.42 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.37 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {15 \, {\left (7 \, B a^{6} + 3 \, A a^{5} b + {\left (7 \, B a b^{5} + 3 \, A b^{6}\right )} x^{5} + 5 \, {\left (7 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} x^{4} + 10 \, {\left (7 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (7 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (7 \, B a^{5} b + 3 \, 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 (105 \, B a^{6} b + 45 \, A a^{5} b^{2} - 15 \, {\left (7 \, B a^{2} b^{5} + 3 \, A a b^{6}\right )} x^{4} + 10 \, {\left (79 \, B a^{3} b^{4} - 21 \, A a^{2} b^{5}\right )} x^{3} + 128 \, {\left (7 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4}\right )} x^{2} + 70 \, {\left (7 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{3} b^{10} x^{5} + 5 \, a^{4} b^{9} x^{4} + 10 \, a^{5} b^{8} x^{3} + 10 \, a^{6} b^{7} x^{2} + 5 \, a^{7} b^{6} x + a^{8} b^{5}\right )}}, -\frac {15 \, {\left (7 \, B a^{6} + 3 \, A a^{5} b + {\left (7 \, B a b^{5} + 3 \, A b^{6}\right )} x^{5} + 5 \, {\left (7 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} x^{4} + 10 \, {\left (7 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (7 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (7 \, B a^{5} b + 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (105 \, B a^{6} b + 45 \, A a^{5} b^{2} - 15 \, {\left (7 \, B a^{2} b^{5} + 3 \, A a b^{6}\right )} x^{4} + 10 \, {\left (79 \, B a^{3} b^{4} - 21 \, A a^{2} b^{5}\right )} x^{3} + 128 \, {\left (7 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4}\right )} x^{2} + 70 \, {\left (7 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{3} b^{10} x^{5} + 5 \, a^{4} b^{9} x^{4} + 10 \, a^{5} b^{8} x^{3} + 10 \, a^{6} b^{7} x^{2} + 5 \, a^{7} b^{6} x + a^{8} b^{5}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.05 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {15 \, {\left (7 \, B a b^{4} + 3 \, A b^{5}\right )} x^{\frac {9}{2}} - 10 \, {\left (79 \, B a^{2} b^{3} - 21 \, A a b^{4}\right )} x^{\frac {7}{2}} - 128 \, {\left (7 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{\frac {5}{2}} - 70 \, {\left (7 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{\frac {3}{2}} - 15 \, {\left (7 \, B a^{5} + 3 \, A a^{4} b\right )} \sqrt {x}}{1920 \, {\left (a^{2} b^{9} x^{5} + 5 \, a^{3} b^{8} x^{4} + 10 \, a^{4} b^{7} x^{3} + 10 \, a^{5} b^{6} x^{2} + 5 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} + \frac {{\left (7 \, B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{2} b^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.80 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {{\left (7 \, B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{2} b^{4}} + \frac {105 \, B a b^{4} x^{\frac {9}{2}} + 45 \, A b^{5} x^{\frac {9}{2}} - 790 \, B a^{2} b^{3} x^{\frac {7}{2}} + 210 \, A a b^{4} x^{\frac {7}{2}} - 896 \, B a^{3} b^{2} x^{\frac {5}{2}} - 384 \, A a^{2} b^{3} x^{\frac {5}{2}} - 490 \, B a^{4} b x^{\frac {3}{2}} - 210 \, A a^{3} b^{2} x^{\frac {3}{2}} - 105 \, B a^{5} \sqrt {x} - 45 \, A a^{4} b \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{2} b^{4}} \]
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Time = 10.11 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.90 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (3\,A\,b+7\,B\,a\right )}{128\,a^{5/2}\,b^{9/2}}-\frac {\frac {x^{5/2}\,\left (3\,A\,b+7\,B\,a\right )}{15\,b^2}-\frac {x^{9/2}\,\left (3\,A\,b+7\,B\,a\right )}{128\,a^2}+\frac {a^2\,\sqrt {x}\,\left (3\,A\,b+7\,B\,a\right )}{128\,b^4}-\frac {x^{7/2}\,\left (21\,A\,b-79\,B\,a\right )}{192\,a\,b}+\frac {7\,a\,x^{3/2}\,\left (3\,A\,b+7\,B\,a\right )}{192\,b^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \]
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