Integrand size = 18, antiderivative size = 146 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^5} \, dx=\frac {(5 A b-8 a B) \sqrt {a+b x}}{24 a x^3}+\frac {b (5 A b-8 a B) \sqrt {a+b x}}{96 a^2 x^2}-\frac {b^2 (5 A b-8 a B) \sqrt {a+b x}}{64 a^3 x}-\frac {A (a+b x)^{3/2}}{4 a x^4}+\frac {b^3 (5 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{7/2}} \]
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Time = 0.05 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 43, 44, 65, 214} \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^5} \, dx=\frac {b^3 (5 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{7/2}}-\frac {b^2 \sqrt {a+b x} (5 A b-8 a B)}{64 a^3 x}+\frac {b \sqrt {a+b x} (5 A b-8 a B)}{96 a^2 x^2}+\frac {\sqrt {a+b x} (5 A b-8 a B)}{24 a x^3}-\frac {A (a+b x)^{3/2}}{4 a x^4} \]
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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}}{4 a x^4}+\frac {\left (-\frac {5 A b}{2}+4 a B\right ) \int \frac {\sqrt {a+b x}}{x^4} \, dx}{4 a} \\ & = \frac {(5 A b-8 a B) \sqrt {a+b x}}{24 a x^3}-\frac {A (a+b x)^{3/2}}{4 a x^4}-\frac {(b (5 A b-8 a B)) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{48 a} \\ & = \frac {(5 A b-8 a B) \sqrt {a+b x}}{24 a x^3}+\frac {b (5 A b-8 a B) \sqrt {a+b x}}{96 a^2 x^2}-\frac {A (a+b x)^{3/2}}{4 a x^4}+\frac {\left (b^2 (5 A b-8 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{64 a^2} \\ & = \frac {(5 A b-8 a B) \sqrt {a+b x}}{24 a x^3}+\frac {b (5 A b-8 a B) \sqrt {a+b x}}{96 a^2 x^2}-\frac {b^2 (5 A b-8 a B) \sqrt {a+b x}}{64 a^3 x}-\frac {A (a+b x)^{3/2}}{4 a x^4}-\frac {\left (b^3 (5 A b-8 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{128 a^3} \\ & = \frac {(5 A b-8 a B) \sqrt {a+b x}}{24 a x^3}+\frac {b (5 A b-8 a B) \sqrt {a+b x}}{96 a^2 x^2}-\frac {b^2 (5 A b-8 a B) \sqrt {a+b x}}{64 a^3 x}-\frac {A (a+b x)^{3/2}}{4 a x^4}-\frac {\left (b^2 (5 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^3} \\ & = \frac {(5 A b-8 a B) \sqrt {a+b x}}{24 a x^3}+\frac {b (5 A b-8 a B) \sqrt {a+b x}}{96 a^2 x^2}-\frac {b^2 (5 A b-8 a B) \sqrt {a+b x}}{64 a^3 x}-\frac {A (a+b x)^{3/2}}{4 a x^4}+\frac {b^3 (5 A b-8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{7/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^5} \, dx=-\frac {\sqrt {a+b x} \left (15 A b^3 x^3+8 a^2 b x (A+2 B x)+16 a^3 (3 A+4 B x)-2 a b^2 x^2 (5 A+12 B x)\right )}{192 a^3 x^4}+\frac {b^3 (5 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{7/2}} \]
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Time = 0.53 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(-\frac {\left (-\frac {15}{8} A \,b^{4}+3 B a \,b^{3}\right ) x^{4} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\sqrt {b x +a}\, \left (-\frac {5 x^{2} b^{2} \left (\frac {12 B x}{5}+A \right ) a^{\frac {3}{2}}}{4}+b x \left (2 B x +A \right ) a^{\frac {5}{2}}+\left (8 B x +6 A \right ) a^{\frac {7}{2}}+\frac {15 A \sqrt {a}\, b^{3} x^{3}}{8}\right )}{24 a^{\frac {7}{2}} x^{4}}\) | \(102\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (15 A \,b^{3} x^{3}-24 B a \,b^{2} x^{3}-10 a A \,b^{2} x^{2}+16 B \,a^{2} b \,x^{2}+8 a^{2} A b x +64 a^{3} B x +48 a^{3} A \right )}{192 x^{4} a^{3}}+\frac {b^{3} \left (5 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{64 a^{\frac {7}{2}}}\) | \(107\) |
derivativedivides | \(2 b^{3} \left (-\frac {\frac {\left (5 A b -8 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{128 a^{3}}-\frac {11 \left (5 A b -8 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{384 a^{2}}+\frac {\left (73 A b -40 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a}+\left (\frac {5 A b}{128}-\frac {B a}{16}\right ) \sqrt {b x +a}}{b^{4} x^{4}}+\frac {\left (5 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {7}{2}}}\right )\) | \(122\) |
default | \(2 b^{3} \left (-\frac {\frac {\left (5 A b -8 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{128 a^{3}}-\frac {11 \left (5 A b -8 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{384 a^{2}}+\frac {\left (73 A b -40 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a}+\left (\frac {5 A b}{128}-\frac {B a}{16}\right ) \sqrt {b x +a}}{b^{4} x^{4}}+\frac {\left (5 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {7}{2}}}\right )\) | \(122\) |
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Time = 0.23 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^5} \, dx=\left [-\frac {3 \, {\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} \sqrt {a} x^{4} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (48 \, A a^{4} - 3 \, {\left (8 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} + 2 \, {\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt {b x + a}}{384 \, a^{4} x^{4}}, \frac {3 \, {\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (48 \, A a^{4} - 3 \, {\left (8 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} + 2 \, {\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt {b x + a}}{192 \, a^{4} x^{4}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (134) = 268\).
Time = 80.36 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.03 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^5} \, dx=- \frac {A a}{4 \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {7 A \sqrt {b}}{24 x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {A b^{\frac {3}{2}}}{96 a x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 A b^{\frac {5}{2}}}{192 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 A b^{\frac {7}{2}}}{64 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {5 A b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{64 a^{\frac {7}{2}}} - \frac {B a}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 B \sqrt {b}}{12 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {B b^{\frac {3}{2}}}{24 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {B b^{\frac {5}{2}}}{8 a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {B b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {5}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^5} \, dx=\frac {1}{384} \, b^{4} {\left (\frac {2 \, {\left (3 \, {\left (8 \, B a - 5 \, A b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 11 \, {\left (8 \, B a^{2} - 5 \, A a b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + {\left (40 \, B a^{3} - 73 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 3 \, {\left (8 \, B a^{4} - 5 \, A a^{3} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{4} a^{3} b - 4 \, {\left (b x + a\right )}^{3} a^{4} b + 6 \, {\left (b x + a\right )}^{2} a^{5} b - 4 \, {\left (b x + a\right )} a^{6} b + a^{7} b} + \frac {3 \, {\left (8 \, B a - 5 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}} b}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^5} \, dx=\frac {\frac {3 \, {\left (8 \, B a b^{4} - 5 \, A b^{5}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {24 \, {\left (b x + a\right )}^{\frac {7}{2}} B a b^{4} - 88 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{2} b^{4} + 40 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{3} b^{4} + 24 \, \sqrt {b x + a} B a^{4} b^{4} - 15 \, {\left (b x + a\right )}^{\frac {7}{2}} A b^{5} + 55 \, {\left (b x + a\right )}^{\frac {5}{2}} A a b^{5} - 73 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{2} b^{5} - 15 \, \sqrt {b x + a} A a^{3} b^{5}}{a^{3} b^{4} x^{4}}}{192 \, b} \]
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Time = 0.57 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^5} \, dx=\frac {b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (5\,A\,b-8\,B\,a\right )}{64\,a^{7/2}}-\frac {\left (\frac {5\,A\,b^4}{64}-\frac {B\,a\,b^3}{8}\right )\,\sqrt {a+b\,x}-\frac {11\,\left (5\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^{5/2}}{192\,a^2}+\frac {\left (5\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^{7/2}}{64\,a^3}+\frac {\left (73\,A\,b^4-40\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^{3/2}}{192\,a}}{{\left (a+b\,x\right )}^4-4\,a^3\,\left (a+b\,x\right )-4\,a\,{\left (a+b\,x\right )}^3+6\,a^2\,{\left (a+b\,x\right )}^2+a^4} \]
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