Integrand size = 13, antiderivative size = 87 \[ \int \frac {\sqrt {a+b x}}{x^4} \, dx=-\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2}+\frac {b^2 \sqrt {a+b x}}{8 a^2 x}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}} \]
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Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {43, 44, 65, 214} \[ \int \frac {\sqrt {a+b x}}{x^4} \, dx=-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}}+\frac {b^2 \sqrt {a+b x}}{8 a^2 x}-\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2} \]
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x}}{3 x^3}+\frac {1}{6} b \int \frac {1}{x^3 \sqrt {a+b x}} \, dx \\ & = -\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2}-\frac {b^2 \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{8 a} \\ & = -\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2}+\frac {b^2 \sqrt {a+b x}}{8 a^2 x}+\frac {b^3 \int \frac {1}{x \sqrt {a+b x}} \, dx}{16 a^2} \\ & = -\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2}+\frac {b^2 \sqrt {a+b x}}{8 a^2 x}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{8 a^2} \\ & = -\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2}+\frac {b^2 \sqrt {a+b x}}{8 a^2 x}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {a+b x}}{x^4} \, dx=-\frac {\sqrt {a+b x} \left (8 a^2+2 a b x-3 b^2 x^2\right )}{24 a^2 x^3}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-3 b^{2} x^{2}+2 a b x +8 a^{2}\right )}{24 x^{3} a^{2}}-\frac {b^{3} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {5}{2}}}\) | \(56\) |
pseudoelliptic | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{3} x^{3}-\sqrt {b x +a}\, \left (\sqrt {a}\, b^{2} x^{2}-\frac {2 a^{\frac {3}{2}} b x}{3}-\frac {8 a^{\frac {5}{2}}}{3}\right )}{8 a^{\frac {5}{2}} x^{3}}\) | \(61\) |
derivativedivides | \(2 b^{3} \left (-\frac {-\frac {\left (b x +a \right )^{\frac {5}{2}}}{16 a^{2}}+\frac {\left (b x +a \right )^{\frac {3}{2}}}{6 a}+\frac {\sqrt {b x +a}}{16}}{b^{3} x^{3}}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {5}{2}}}\right )\) | \(66\) |
default | \(2 b^{3} \left (-\frac {-\frac {\left (b x +a \right )^{\frac {5}{2}}}{16 a^{2}}+\frac {\left (b x +a \right )^{\frac {3}{2}}}{6 a}+\frac {\sqrt {b x +a}}{16}}{b^{3} x^{3}}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {5}{2}}}\right )\) | \(66\) |
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Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {a+b x}}{x^4} \, dx=\left [\frac {3 \, \sqrt {a} b^{3} x^{3} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3 \, a b^{2} x^{2} - 2 \, a^{2} b x - 8 \, a^{3}\right )} \sqrt {b x + a}}{48 \, a^{3} x^{3}}, \frac {3 \, \sqrt {-a} b^{3} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b^{2} x^{2} - 2 \, a^{2} b x - 8 \, a^{3}\right )} \sqrt {b x + a}}{24 \, a^{3} x^{3}}\right ] \]
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Time = 6.52 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {a+b x}}{x^4} \, dx=- \frac {a}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 \sqrt {b}}{12 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {b^{\frac {3}{2}}}{24 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {b^{\frac {5}{2}}}{8 a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {5}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {a+b x}}{x^4} \, dx=\frac {b^{3} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{16 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{3} - 8 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{3} - 3 \, \sqrt {b x + a} a^{2} b^{3}}{24 \, {\left ({\left (b x + a\right )}^{3} a^{2} - 3 \, {\left (b x + a\right )}^{2} a^{3} + 3 \, {\left (b x + a\right )} a^{4} - a^{5}\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {a+b x}}{x^4} \, dx=\frac {\frac {3 \, b^{4} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{4} - 8 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{4} - 3 \, \sqrt {b x + a} a^{2} b^{4}}{a^{2} b^{3} x^{3}}}{24 \, b} \]
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Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a+b x}}{x^4} \, dx=\frac {{\left (a+b\,x\right )}^{5/2}}{8\,a^2\,x^3}-\frac {{\left (a+b\,x\right )}^{3/2}}{3\,a\,x^3}-\frac {\sqrt {a+b\,x}}{8\,x^3}+\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{8\,a^{5/2}} \]
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