Integrand size = 18, antiderivative size = 177 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=-\frac {A \sqrt {a+b x}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x}}{40 a^2 x^4}-\frac {7 b (9 A b-10 a B) \sqrt {a+b x}}{240 a^3 x^3}+\frac {7 b^2 (9 A b-10 a B) \sqrt {a+b x}}{192 a^4 x^2}-\frac {7 b^3 (9 A b-10 a B) \sqrt {a+b x}}{128 a^5 x}+\frac {7 b^4 (9 A b-10 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{11/2}} \]
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Time = 0.05 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 44, 65, 214} \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\frac {7 b^4 (9 A b-10 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{11/2}}-\frac {7 b^3 \sqrt {a+b x} (9 A b-10 a B)}{128 a^5 x}+\frac {7 b^2 \sqrt {a+b x} (9 A b-10 a B)}{192 a^4 x^2}-\frac {7 b \sqrt {a+b x} (9 A b-10 a B)}{240 a^3 x^3}+\frac {\sqrt {a+b x} (9 A b-10 a B)}{40 a^2 x^4}-\frac {A \sqrt {a+b x}}{5 a x^5} \]
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Rule 44
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A \sqrt {a+b x}}{5 a x^5}+\frac {\left (-\frac {9 A b}{2}+5 a B\right ) \int \frac {1}{x^5 \sqrt {a+b x}} \, dx}{5 a} \\ & = -\frac {A \sqrt {a+b x}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x}}{40 a^2 x^4}+\frac {(7 b (9 A b-10 a B)) \int \frac {1}{x^4 \sqrt {a+b x}} \, dx}{80 a^2} \\ & = -\frac {A \sqrt {a+b x}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x}}{40 a^2 x^4}-\frac {7 b (9 A b-10 a B) \sqrt {a+b x}}{240 a^3 x^3}-\frac {\left (7 b^2 (9 A b-10 a B)\right ) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{96 a^3} \\ & = -\frac {A \sqrt {a+b x}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x}}{40 a^2 x^4}-\frac {7 b (9 A b-10 a B) \sqrt {a+b x}}{240 a^3 x^3}+\frac {7 b^2 (9 A b-10 a B) \sqrt {a+b x}}{192 a^4 x^2}+\frac {\left (7 b^3 (9 A b-10 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{128 a^4} \\ & = -\frac {A \sqrt {a+b x}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x}}{40 a^2 x^4}-\frac {7 b (9 A b-10 a B) \sqrt {a+b x}}{240 a^3 x^3}+\frac {7 b^2 (9 A b-10 a B) \sqrt {a+b x}}{192 a^4 x^2}-\frac {7 b^3 (9 A b-10 a B) \sqrt {a+b x}}{128 a^5 x}-\frac {\left (7 b^4 (9 A b-10 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{256 a^5} \\ & = -\frac {A \sqrt {a+b x}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x}}{40 a^2 x^4}-\frac {7 b (9 A b-10 a B) \sqrt {a+b x}}{240 a^3 x^3}+\frac {7 b^2 (9 A b-10 a B) \sqrt {a+b x}}{192 a^4 x^2}-\frac {7 b^3 (9 A b-10 a B) \sqrt {a+b x}}{128 a^5 x}-\frac {\left (7 b^3 (9 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^5} \\ & = -\frac {A \sqrt {a+b x}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x}}{40 a^2 x^4}-\frac {7 b (9 A b-10 a B) \sqrt {a+b x}}{240 a^3 x^3}+\frac {7 b^2 (9 A b-10 a B) \sqrt {a+b x}}{192 a^4 x^2}-\frac {7 b^3 (9 A b-10 a B) \sqrt {a+b x}}{128 a^5 x}+\frac {7 b^4 (9 A b-10 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{11/2}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \left (-945 A b^4 x^4+210 a b^3 x^3 (3 A+5 B x)-96 a^4 (4 A+5 B x)-28 a^2 b^2 x^2 (18 A+25 B x)+16 a^3 b x (27 A+35 B x)\right )}{1920 a^5 x^5}+\frac {7 b^4 (9 A b-10 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{11/2}} \]
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Time = 0.54 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.67
method | result | size |
pseudoelliptic | \(\frac {\frac {63 x^{5} \left (A b -\frac {10 B a}{9}\right ) b^{4} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128}+\frac {9 \sqrt {b x +a}\, \left (\frac {35 x^{3} \left (\frac {5 B x}{3}+A \right ) b^{3} a^{\frac {3}{2}}}{24}-\frac {7 x^{2} \left (\frac {25 B x}{18}+A \right ) b^{2} a^{\frac {5}{2}}}{6}+b x \left (\frac {35 B x}{27}+A \right ) a^{\frac {7}{2}}+\frac {2 \left (-5 B x -4 A \right ) a^{\frac {9}{2}}}{9}-\frac {35 A \sqrt {a}\, b^{4} x^{4}}{16}\right )}{40}}{a^{\frac {11}{2}} x^{5}}\) | \(118\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (945 A \,b^{4} x^{4}-1050 B a \,b^{3} x^{4}-630 A a \,b^{3} x^{3}+700 B \,a^{2} b^{2} x^{3}+504 A \,a^{2} b^{2} x^{2}-560 B \,a^{3} b \,x^{2}-432 A \,a^{3} b x +480 B \,a^{4} x +384 A \,a^{4}\right )}{1920 a^{5} x^{5}}+\frac {7 b^{4} \left (9 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {11}{2}}}\) | \(131\) |
derivativedivides | \(2 b^{4} \left (-\frac {\frac {7 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{5}}-\frac {49 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{4}}+\frac {7 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{3}}-\frac {79 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a^{2}}+\frac {\left (193 A b -186 B a \right ) \sqrt {b x +a}}{256 a}}{b^{5} x^{5}}+\frac {7 \left (9 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {11}{2}}}\right )\) | \(147\) |
default | \(2 b^{4} \left (-\frac {\frac {7 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{5}}-\frac {49 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{4}}+\frac {7 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{3}}-\frac {79 \left (9 A b -10 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a^{2}}+\frac {\left (193 A b -186 B a \right ) \sqrt {b x +a}}{256 a}}{b^{5} x^{5}}+\frac {7 \left (9 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {11}{2}}}\right )\) | \(147\) |
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Time = 0.25 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.74 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\left [-\frac {105 \, {\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} \sqrt {a} x^{5} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (384 \, A a^{5} - 105 \, {\left (10 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 70 \, {\left (10 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} - 56 \, {\left (10 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3840 \, a^{6} x^{5}}, \frac {105 \, {\left (10 \, B a b^{4} - 9 \, A b^{5}\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (384 \, A a^{5} - 105 \, {\left (10 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 70 \, {\left (10 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} - 56 \, {\left (10 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{1920 \, a^{6} x^{5}}\right ] \]
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Timed out. \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\frac {1}{3840} \, b^{5} {\left (\frac {2 \, {\left (105 \, {\left (10 \, B a - 9 \, A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 490 \, {\left (10 \, B a^{2} - 9 \, A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 896 \, {\left (10 \, B a^{3} - 9 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 790 \, {\left (10 \, B a^{4} - 9 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (186 \, B a^{5} - 193 \, A a^{4} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{5} a^{5} b - 5 \, {\left (b x + a\right )}^{4} a^{6} b + 10 \, {\left (b x + a\right )}^{3} a^{7} b - 10 \, {\left (b x + a\right )}^{2} a^{8} b + 5 \, {\left (b x + a\right )} a^{9} b - a^{10} b} + \frac {105 \, {\left (10 \, B a - 9 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {11}{2}} b}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\frac {\frac {105 \, {\left (10 \, B a b^{5} - 9 \, A b^{6}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{5}} + \frac {1050 \, {\left (b x + a\right )}^{\frac {9}{2}} B a b^{5} - 4900 \, {\left (b x + a\right )}^{\frac {7}{2}} B a^{2} b^{5} + 8960 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{3} b^{5} - 7900 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{4} b^{5} + 2790 \, \sqrt {b x + a} B a^{5} b^{5} - 945 \, {\left (b x + a\right )}^{\frac {9}{2}} A b^{6} + 4410 \, {\left (b x + a\right )}^{\frac {7}{2}} A a b^{6} - 8064 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{2} b^{6} + 7110 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{3} b^{6} - 2895 \, \sqrt {b x + a} A a^{4} b^{6}}{a^{5} b^{5} x^{5}}}{1920 \, b} \]
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Time = 0.12 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{x^6 \sqrt {a+b x}} \, dx=\frac {\frac {7\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{5/2}}{15\,a^3}-\frac {79\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{3/2}}{192\,a^2}-\frac {49\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{7/2}}{192\,a^4}+\frac {7\,\left (9\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{9/2}}{128\,a^5}+\frac {\left (193\,A\,b^5-186\,B\,a\,b^4\right )\,\sqrt {a+b\,x}}{128\,a}}{5\,a\,{\left (a+b\,x\right )}^4-5\,a^4\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^5-10\,a^2\,{\left (a+b\,x\right )}^3+10\,a^3\,{\left (a+b\,x\right )}^2+a^5}+\frac {7\,b^4\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (9\,A\,b-10\,B\,a\right )}{128\,a^{11/2}} \]
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