Integrand size = 18, antiderivative size = 177 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=\frac {(7 A b-10 a B) \sqrt {a+b x}}{40 a x^4}+\frac {b (7 A b-10 a B) \sqrt {a+b x}}{240 a^2 x^3}-\frac {b^2 (7 A b-10 a B) \sqrt {a+b x}}{192 a^3 x^2}+\frac {b^3 (7 A b-10 a B) \sqrt {a+b x}}{128 a^4 x}-\frac {A (a+b x)^{3/2}}{5 a x^5}-\frac {b^4 (7 A b-10 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{9/2}} \]
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Time = 0.06 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 43, 44, 65, 214} \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=-\frac {b^4 (7 A b-10 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{9/2}}+\frac {b^3 \sqrt {a+b x} (7 A b-10 a B)}{128 a^4 x}-\frac {b^2 \sqrt {a+b x} (7 A b-10 a B)}{192 a^3 x^2}+\frac {b \sqrt {a+b x} (7 A b-10 a B)}{240 a^2 x^3}+\frac {\sqrt {a+b x} (7 A b-10 a B)}{40 a x^4}-\frac {A (a+b x)^{3/2}}{5 a x^5} \]
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Rule 43
Rule 44
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A (a+b x)^{3/2}}{5 a x^5}+\frac {\left (-\frac {7 A b}{2}+5 a B\right ) \int \frac {\sqrt {a+b x}}{x^5} \, dx}{5 a} \\ & = \frac {(7 A b-10 a B) \sqrt {a+b x}}{40 a x^4}-\frac {A (a+b x)^{3/2}}{5 a x^5}-\frac {(b (7 A b-10 a B)) \int \frac {1}{x^4 \sqrt {a+b x}} \, dx}{80 a} \\ & = \frac {(7 A b-10 a B) \sqrt {a+b x}}{40 a x^4}+\frac {b (7 A b-10 a B) \sqrt {a+b x}}{240 a^2 x^3}-\frac {A (a+b x)^{3/2}}{5 a x^5}+\frac {\left (b^2 (7 A b-10 a B)\right ) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{96 a^2} \\ & = \frac {(7 A b-10 a B) \sqrt {a+b x}}{40 a x^4}+\frac {b (7 A b-10 a B) \sqrt {a+b x}}{240 a^2 x^3}-\frac {b^2 (7 A b-10 a B) \sqrt {a+b x}}{192 a^3 x^2}-\frac {A (a+b x)^{3/2}}{5 a x^5}-\frac {\left (b^3 (7 A b-10 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{128 a^3} \\ & = \frac {(7 A b-10 a B) \sqrt {a+b x}}{40 a x^4}+\frac {b (7 A b-10 a B) \sqrt {a+b x}}{240 a^2 x^3}-\frac {b^2 (7 A b-10 a B) \sqrt {a+b x}}{192 a^3 x^2}+\frac {b^3 (7 A b-10 a B) \sqrt {a+b x}}{128 a^4 x}-\frac {A (a+b x)^{3/2}}{5 a x^5}+\frac {\left (b^4 (7 A b-10 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{256 a^4} \\ & = \frac {(7 A b-10 a B) \sqrt {a+b x}}{40 a x^4}+\frac {b (7 A b-10 a B) \sqrt {a+b x}}{240 a^2 x^3}-\frac {b^2 (7 A b-10 a B) \sqrt {a+b x}}{192 a^3 x^2}+\frac {b^3 (7 A b-10 a B) \sqrt {a+b x}}{128 a^4 x}-\frac {A (a+b x)^{3/2}}{5 a x^5}+\frac {\left (b^3 (7 A b-10 a B)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{128 a^4} \\ & = \frac {(7 A b-10 a B) \sqrt {a+b x}}{40 a x^4}+\frac {b (7 A b-10 a B) \sqrt {a+b x}}{240 a^2 x^3}-\frac {b^2 (7 A b-10 a B) \sqrt {a+b x}}{192 a^3 x^2}+\frac {b^3 (7 A b-10 a B) \sqrt {a+b x}}{128 a^4 x}-\frac {A (a+b x)^{3/2}}{5 a x^5}-\frac {b^4 (7 A b-10 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{9/2}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=\frac {\frac {\sqrt {a} \sqrt {a+b x} \left (105 A b^4 x^4-16 a^3 b x (3 A+5 B x)-96 a^4 (4 A+5 B x)-10 a b^3 x^3 (7 A+15 B x)+4 a^2 b^2 x^2 (14 A+25 B x)\right )}{x^5}-15 b^4 (7 A b-10 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1920 a^{9/2}} \]
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Time = 0.53 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.67
method | result | size |
pseudoelliptic | \(\frac {-\frac {7 x^{5} b^{4} \left (A b -\frac {10 B a}{7}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128}+\frac {7 \left (-\frac {5 x^{3} \left (\frac {15 B x}{7}+A \right ) b^{3} a^{\frac {3}{2}}}{4}+b^{2} x^{2} \left (\frac {25 B x}{14}+A \right ) a^{\frac {5}{2}}-\frac {6 x \left (\frac {5 B x}{3}+A \right ) b \,a^{\frac {7}{2}}}{7}+\frac {12 \left (-5 B x -4 A \right ) a^{\frac {9}{2}}}{7}+\frac {15 A \sqrt {a}\, b^{4} x^{4}}{8}\right ) \sqrt {b x +a}}{240}}{a^{\frac {9}{2}} x^{5}}\) | \(118\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (-105 A \,b^{4} x^{4}+150 B a \,b^{3} x^{4}+70 A a \,b^{3} x^{3}-100 B \,a^{2} b^{2} x^{3}-56 A \,a^{2} b^{2} x^{2}+80 B \,a^{3} b \,x^{2}+48 A \,a^{3} b x +480 B \,a^{4} x +384 A \,a^{4}\right )}{1920 x^{5} a^{4}}-\frac {b^{4} \left (7 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {9}{2}}}\) | \(131\) |
derivativedivides | \(2 b^{4} \left (-\frac {-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{4}}+\frac {7 \left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{3}}-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{2}}+\frac {\left (79 A b -58 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a}+\left (\frac {7 A b}{256}-\frac {5 B a}{128}\right ) \sqrt {b x +a}}{b^{5} x^{5}}-\frac {\left (7 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {9}{2}}}\right )\) | \(143\) |
default | \(2 b^{4} \left (-\frac {-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{4}}+\frac {7 \left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{3}}-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{2}}+\frac {\left (79 A b -58 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a}+\left (\frac {7 A b}{256}-\frac {5 B a}{128}\right ) \sqrt {b x +a}}{b^{5} x^{5}}-\frac {\left (7 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {9}{2}}}\right )\) | \(143\) |
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Time = 0.23 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=\left [-\frac {15 \, {\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} \sqrt {a} x^{5} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (384 \, A a^{5} + 15 \, {\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{4} - 10 \, {\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{3} + 8 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3840 \, a^{5} x^{5}}, -\frac {15 \, {\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (384 \, A a^{5} + 15 \, {\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{4} - 10 \, {\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{3} + 8 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt {b x + a}}{1920 \, a^{5} x^{5}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (165) = 330\).
Time = 177.45 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=- \frac {A a}{5 \sqrt {b} x^{\frac {11}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {9 A \sqrt {b}}{40 x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {A b^{\frac {3}{2}}}{240 a x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {7 A b^{\frac {5}{2}}}{960 a^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {7 A b^{\frac {7}{2}}}{384 a^{3} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {7 A b^{\frac {9}{2}}}{128 a^{4} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {7 A b^{5} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{128 a^{\frac {9}{2}}} - \frac {B a}{4 \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {7 B \sqrt {b}}{24 x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {B b^{\frac {3}{2}}}{96 a x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 B b^{\frac {5}{2}}}{192 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 B b^{\frac {7}{2}}}{64 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {5 B b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{64 a^{\frac {7}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=-\frac {1}{3840} \, b^{5} {\left (\frac {2 \, {\left (15 \, {\left (10 \, B a - 7 \, A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 70 \, {\left (10 \, B a^{2} - 7 \, A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 128 \, {\left (10 \, B a^{3} - 7 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (58 \, B a^{4} - 79 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 15 \, {\left (10 \, B a^{5} - 7 \, A a^{4} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{5} a^{4} b - 5 \, {\left (b x + a\right )}^{4} a^{5} b + 10 \, {\left (b x + a\right )}^{3} a^{6} b - 10 \, {\left (b x + a\right )}^{2} a^{7} b + 5 \, {\left (b x + a\right )} a^{8} b - a^{9} b} + \frac {15 \, {\left (10 \, B a - 7 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}} b}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=-\frac {\frac {15 \, {\left (10 \, B a b^{5} - 7 \, A b^{6}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {150 \, {\left (b x + a\right )}^{\frac {9}{2}} B a b^{5} - 700 \, {\left (b x + a\right )}^{\frac {7}{2}} B a^{2} b^{5} + 1280 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{3} b^{5} - 580 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{4} b^{5} - 150 \, \sqrt {b x + a} B a^{5} b^{5} - 105 \, {\left (b x + a\right )}^{\frac {9}{2}} A b^{6} + 490 \, {\left (b x + a\right )}^{\frac {7}{2}} A a b^{6} - 896 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{2} b^{6} + 790 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{3} b^{6} + 105 \, \sqrt {b x + a} A a^{4} b^{6}}{a^{4} b^{5} x^{5}}}{1920 \, b} \]
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Time = 0.51 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=\frac {\left (\frac {7\,A\,b^5}{128}-\frac {5\,B\,a\,b^4}{64}\right )\,\sqrt {a+b\,x}-\frac {\left (7\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{5/2}}{15\,a^2}+\frac {7\,\left (7\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{7/2}}{192\,a^3}-\frac {\left (7\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{9/2}}{128\,a^4}+\frac {\left (79\,A\,b^5-58\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{3/2}}{192\,a}}{5\,a\,{\left (a+b\,x\right )}^4-5\,a^4\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^5-10\,a^2\,{\left (a+b\,x\right )}^3+10\,a^3\,{\left (a+b\,x\right )}^2+a^5}-\frac {b^4\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (7\,A\,b-10\,B\,a\right )}{128\,a^{9/2}} \]
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