Integrand size = 18, antiderivative size = 202 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=\frac {35 b^3 (11 A b-8 a B)}{64 a^5 (a+b x)^{3/2}}-\frac {A}{4 a x^4 (a+b x)^{3/2}}+\frac {11 A b-8 a B}{24 a^2 x^3 (a+b x)^{3/2}}-\frac {3 b (11 A b-8 a B)}{32 a^3 x^2 (a+b x)^{3/2}}+\frac {21 b^2 (11 A b-8 a B)}{64 a^4 x (a+b x)^{3/2}}+\frac {105 b^3 (11 A b-8 a B)}{64 a^6 \sqrt {a+b x}}-\frac {105 b^3 (11 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{13/2}} \]
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Time = 0.06 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 44, 53, 65, 214} \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=-\frac {105 b^3 (11 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{13/2}}+\frac {105 b^3 (11 A b-8 a B)}{64 a^6 \sqrt {a+b x}}+\frac {35 b^3 (11 A b-8 a B)}{64 a^5 (a+b x)^{3/2}}+\frac {21 b^2 (11 A b-8 a B)}{64 a^4 x (a+b x)^{3/2}}-\frac {3 b (11 A b-8 a B)}{32 a^3 x^2 (a+b x)^{3/2}}+\frac {11 A b-8 a B}{24 a^2 x^3 (a+b x)^{3/2}}-\frac {A}{4 a x^4 (a+b x)^{3/2}} \]
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A}{4 a x^4 (a+b x)^{3/2}}+\frac {\left (-\frac {11 A b}{2}+4 a B\right ) \int \frac {1}{x^4 (a+b x)^{5/2}} \, dx}{4 a} \\ & = -\frac {A}{4 a x^4 (a+b x)^{3/2}}+\frac {11 A b-8 a B}{24 a^2 x^3 (a+b x)^{3/2}}+\frac {(3 b (11 A b-8 a B)) \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx}{16 a^2} \\ & = -\frac {A}{4 a x^4 (a+b x)^{3/2}}+\frac {11 A b-8 a B}{24 a^2 x^3 (a+b x)^{3/2}}-\frac {3 b (11 A b-8 a B)}{32 a^3 x^2 (a+b x)^{3/2}}-\frac {\left (21 b^2 (11 A b-8 a B)\right ) \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx}{64 a^3} \\ & = -\frac {A}{4 a x^4 (a+b x)^{3/2}}+\frac {11 A b-8 a B}{24 a^2 x^3 (a+b x)^{3/2}}-\frac {3 b (11 A b-8 a B)}{32 a^3 x^2 (a+b x)^{3/2}}+\frac {21 b^2 (11 A b-8 a B)}{64 a^4 x (a+b x)^{3/2}}+\frac {\left (105 b^3 (11 A b-8 a B)\right ) \int \frac {1}{x (a+b x)^{5/2}} \, dx}{128 a^4} \\ & = \frac {35 b^3 (11 A b-8 a B)}{64 a^5 (a+b x)^{3/2}}-\frac {A}{4 a x^4 (a+b x)^{3/2}}+\frac {11 A b-8 a B}{24 a^2 x^3 (a+b x)^{3/2}}-\frac {3 b (11 A b-8 a B)}{32 a^3 x^2 (a+b x)^{3/2}}+\frac {21 b^2 (11 A b-8 a B)}{64 a^4 x (a+b x)^{3/2}}+\frac {\left (105 b^3 (11 A b-8 a B)\right ) \int \frac {1}{x (a+b x)^{3/2}} \, dx}{128 a^5} \\ & = \frac {35 b^3 (11 A b-8 a B)}{64 a^5 (a+b x)^{3/2}}-\frac {A}{4 a x^4 (a+b x)^{3/2}}+\frac {11 A b-8 a B}{24 a^2 x^3 (a+b x)^{3/2}}-\frac {3 b (11 A b-8 a B)}{32 a^3 x^2 (a+b x)^{3/2}}+\frac {21 b^2 (11 A b-8 a B)}{64 a^4 x (a+b x)^{3/2}}+\frac {105 b^3 (11 A b-8 a B)}{64 a^6 \sqrt {a+b x}}+\frac {\left (105 b^3 (11 A b-8 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{128 a^6} \\ & = \frac {35 b^3 (11 A b-8 a B)}{64 a^5 (a+b x)^{3/2}}-\frac {A}{4 a x^4 (a+b x)^{3/2}}+\frac {11 A b-8 a B}{24 a^2 x^3 (a+b x)^{3/2}}-\frac {3 b (11 A b-8 a B)}{32 a^3 x^2 (a+b x)^{3/2}}+\frac {21 b^2 (11 A b-8 a B)}{64 a^4 x (a+b x)^{3/2}}+\frac {105 b^3 (11 A b-8 a B)}{64 a^6 \sqrt {a+b x}}+\frac {\left (105 b^2 (11 A b-8 a B)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{64 a^6} \\ & = \frac {35 b^3 (11 A b-8 a B)}{64 a^5 (a+b x)^{3/2}}-\frac {A}{4 a x^4 (a+b x)^{3/2}}+\frac {11 A b-8 a B}{24 a^2 x^3 (a+b x)^{3/2}}-\frac {3 b (11 A b-8 a B)}{32 a^3 x^2 (a+b x)^{3/2}}+\frac {21 b^2 (11 A b-8 a B)}{64 a^4 x (a+b x)^{3/2}}+\frac {105 b^3 (11 A b-8 a B)}{64 a^6 \sqrt {a+b x}}-\frac {105 b^3 (11 A b-8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{13/2}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.75 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=\frac {\frac {\sqrt {a} \left (3465 A b^5 x^5+21 a^2 b^3 x^3 (33 A-160 B x)+420 a b^4 x^4 (11 A-6 B x)-16 a^5 (3 A+4 B x)+8 a^4 b x (11 A+18 B x)-18 a^3 b^2 x^2 (11 A+28 B x)\right )}{x^4 (a+b x)^{3/2}}+315 b^3 (-11 A b+8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{192 a^{13/2}} \]
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Time = 0.59 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(-\frac {33 \left (\frac {35 x^{4} \left (A b -\frac {8 B a}{11}\right ) b^{3} \left (b x +a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2}+\left (\frac {32 B x}{99}+\frac {8 A}{33}\right ) a^{\frac {11}{2}}+x b \left (-\frac {70 x^{3} \left (-\frac {6 B x}{11}+A \right ) b^{3} a^{\frac {3}{2}}}{3}-\frac {7 x^{2} \left (-\frac {160 B x}{33}+A \right ) b^{2} a^{\frac {5}{2}}}{2}+b x \left (\frac {28 B x}{11}+A \right ) a^{\frac {7}{2}}+\left (-\frac {8 B x}{11}-\frac {4 A}{9}\right ) a^{\frac {9}{2}}-\frac {35 A \sqrt {a}\, b^{4} x^{4}}{2}\right )\right )}{32 a^{\frac {13}{2}} \left (b x +a \right )^{\frac {3}{2}} x^{4}}\) | \(138\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (-1545 A \,b^{3} x^{3}+984 B a \,b^{2} x^{3}+518 a A \,b^{2} x^{2}-272 B \,a^{2} b \,x^{2}-184 a^{2} A b x +64 a^{3} B x +48 a^{3} A \right )}{192 a^{6} x^{4}}+\frac {b^{3} \left (-\frac {2 \left (-640 A b +512 B a \right )}{\sqrt {b x +a}}+\frac {256 a \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}-\frac {2 \left (1155 A b -840 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{128 a^{6}}\) | \(149\) |
derivativedivides | \(2 b^{3} \left (-\frac {\frac {\left (-\frac {515 A b}{128}+\frac {41 B a}{16}\right ) \left (b x +a \right )^{\frac {7}{2}}+\left (\frac {5153}{384} a b A -\frac {403}{48} a^{2} B \right ) \left (b x +a \right )^{\frac {5}{2}}+\left (-\frac {5855}{384} a^{2} b A +\frac {445}{48} a^{3} B \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {765}{128} A \,a^{3} b -\frac {55}{16} B \,a^{4}\right ) \sqrt {b x +a}}{b^{4} x^{4}}+\frac {105 \left (11 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}}{a^{6}}-\frac {-5 A b +4 B a}{a^{6} \sqrt {b x +a}}-\frac {-A b +B a}{3 a^{5} \left (b x +a \right )^{\frac {3}{2}}}\right )\) | \(169\) |
default | \(2 b^{3} \left (-\frac {\frac {\left (-\frac {515 A b}{128}+\frac {41 B a}{16}\right ) \left (b x +a \right )^{\frac {7}{2}}+\left (\frac {5153}{384} a b A -\frac {403}{48} a^{2} B \right ) \left (b x +a \right )^{\frac {5}{2}}+\left (-\frac {5855}{384} a^{2} b A +\frac {445}{48} a^{3} B \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {765}{128} A \,a^{3} b -\frac {55}{16} B \,a^{4}\right ) \sqrt {b x +a}}{b^{4} x^{4}}+\frac {105 \left (11 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}}{a^{6}}-\frac {-5 A b +4 B a}{a^{6} \sqrt {b x +a}}-\frac {-A b +B a}{3 a^{5} \left (b x +a \right )^{\frac {3}{2}}}\right )\) | \(169\) |
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Time = 0.24 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.45 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=\left [-\frac {315 \, {\left ({\left (8 \, B a b^{5} - 11 \, A b^{6}\right )} x^{6} + 2 \, {\left (8 \, B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + {\left (8 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (48 \, A a^{6} + 315 \, {\left (8 \, B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + 420 \, {\left (8 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4} + 63 \, {\left (8 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{3} - 18 \, {\left (8 \, B a^{5} b - 11 \, A a^{4} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{6} - 11 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{384 \, {\left (a^{7} b^{2} x^{6} + 2 \, a^{8} b x^{5} + a^{9} x^{4}\right )}}, -\frac {315 \, {\left ({\left (8 \, B a b^{5} - 11 \, A b^{6}\right )} x^{6} + 2 \, {\left (8 \, B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + {\left (8 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (48 \, A a^{6} + 315 \, {\left (8 \, B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + 420 \, {\left (8 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4} + 63 \, {\left (8 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{3} - 18 \, {\left (8 \, B a^{5} b - 11 \, A a^{4} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{6} - 11 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{192 \, {\left (a^{7} b^{2} x^{6} + 2 \, a^{8} b x^{5} + a^{9} x^{4}\right )}}\right ] \]
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Timed out. \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=-\frac {1}{384} \, b^{4} {\left (\frac {2 \, {\left (128 \, B a^{6} - 128 \, A a^{5} b + 315 \, {\left (8 \, B a - 11 \, A b\right )} {\left (b x + a\right )}^{5} - 1155 \, {\left (8 \, B a^{2} - 11 \, A a b\right )} {\left (b x + a\right )}^{4} + 1533 \, {\left (8 \, B a^{3} - 11 \, A a^{2} b\right )} {\left (b x + a\right )}^{3} - 837 \, {\left (8 \, B a^{4} - 11 \, A a^{3} b\right )} {\left (b x + a\right )}^{2} + 128 \, {\left (8 \, B a^{5} - 11 \, A a^{4} b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {11}{2}} a^{6} b - 4 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{7} b + 6 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{8} b - 4 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{9} b + {\left (b x + a\right )}^{\frac {3}{2}} a^{10} b} + \frac {315 \, {\left (8 \, B a - 11 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {13}{2}} b}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=-\frac {105 \, {\left (8 \, B a b^{3} - 11 \, A b^{4}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a^{6}} - \frac {2 \, {\left (12 \, {\left (b x + a\right )} B a b^{3} + B a^{2} b^{3} - 15 \, {\left (b x + a\right )} A b^{4} - A a b^{4}\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6}} - \frac {984 \, {\left (b x + a\right )}^{\frac {7}{2}} B a b^{3} - 3224 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{2} b^{3} + 3560 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{3} b^{3} - 1320 \, \sqrt {b x + a} B a^{4} b^{3} - 1545 \, {\left (b x + a\right )}^{\frac {7}{2}} A b^{4} + 5153 \, {\left (b x + a\right )}^{\frac {5}{2}} A a b^{4} - 5855 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{2} b^{4} + 2295 \, \sqrt {b x + a} A a^{3} b^{4}}{192 \, a^{6} b^{4} x^{4}} \]
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Time = 0.53 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=\frac {\frac {2\,\left (A\,b^4-B\,a\,b^3\right )}{3\,a}+\frac {2\,\left (11\,A\,b^4-8\,B\,a\,b^3\right )\,\left (a+b\,x\right )}{3\,a^2}-\frac {279\,\left (11\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^2}{64\,a^3}+\frac {511\,\left (11\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^3}{64\,a^4}-\frac {385\,\left (11\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^4}{64\,a^5}+\frac {105\,\left (11\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^5}{64\,a^6}}{{\left (a+b\,x\right )}^{11/2}-4\,a\,{\left (a+b\,x\right )}^{9/2}+a^4\,{\left (a+b\,x\right )}^{3/2}-4\,a^3\,{\left (a+b\,x\right )}^{5/2}+6\,a^2\,{\left (a+b\,x\right )}^{7/2}}-\frac {105\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (11\,A\,b-8\,B\,a\right )}{64\,a^{13/2}} \]
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