Integrand size = 13, antiderivative size = 90 \[ \int \frac {1}{x^4 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x}}{3 a x^3}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}+\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}} \]
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Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {44, 65, 214} \[ \int \frac {1}{x^4 \sqrt {a+b x}} \, dx=\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {\sqrt {a+b x}}{3 a x^3} \]
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Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b x}}{3 a x^3}-\frac {(5 b) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{6 a} \\ & = -\frac {\sqrt {a+b x}}{3 a x^3}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}+\frac {\left (5 b^2\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{8 a^2} \\ & = -\frac {\sqrt {a+b x}}{3 a x^3}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}-\frac {\left (5 b^3\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{16 a^3} \\ & = -\frac {\sqrt {a+b x}}{3 a x^3}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{8 a^3} \\ & = -\frac {\sqrt {a+b x}}{3 a x^3}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}+\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^4 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x} \left (8 a^2-10 a b x+15 b^2 x^2\right )}{24 a^3 x^3}+\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}} \]
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Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (15 b^{2} x^{2}-10 a b x +8 a^{2}\right )}{24 a^{3} x^{3}}+\frac {5 b^{3} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {7}{2}}}\) | \(56\) |
pseudoelliptic | \(\frac {\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{3} x^{3}}{8}-\frac {5 \sqrt {b x +a}\, \left (\sqrt {a}\, b^{2} x^{2}-\frac {2 a^{\frac {3}{2}} b x}{3}+\frac {8 a^{\frac {5}{2}}}{15}\right )}{8}}{a^{\frac {7}{2}} x^{3}}\) | \(61\) |
derivativedivides | \(2 b^{3} \left (-\frac {\sqrt {b x +a}}{6 a \,b^{3} x^{3}}+\frac {\frac {5 \sqrt {b x +a}}{24 a \,b^{2} x^{2}}+\frac {5 \left (-\frac {3 \sqrt {b x +a}}{8 a b x}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{6 a}}{a}\right )\) | \(90\) |
default | \(2 b^{3} \left (-\frac {\sqrt {b x +a}}{6 a \,b^{3} x^{3}}+\frac {\frac {5 \sqrt {b x +a}}{24 a \,b^{2} x^{2}}+\frac {5 \left (-\frac {3 \sqrt {b x +a}}{8 a b x}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{6 a}}{a}\right )\) | \(90\) |
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Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.61 \[ \int \frac {1}{x^4 \sqrt {a+b x}} \, dx=\left [\frac {15 \, \sqrt {a} b^{3} x^{3} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a}}{48 \, a^{4} x^{3}}, -\frac {15 \, \sqrt {-a} b^{3} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a}}{24 \, a^{4} x^{3}}\right ] \]
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Time = 10.86 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^4 \sqrt {a+b x}} \, dx=- \frac {1}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {\sqrt {b}}{12 a x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 b^{\frac {3}{2}}}{24 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 b^{\frac {5}{2}}}{8 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {5 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {7}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x^4 \sqrt {a+b x}} \, dx=-\frac {5 \, b^{3} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{16 \, a^{\frac {7}{2}}} - \frac {15 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{3} - 40 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{3} + 33 \, \sqrt {b x + a} a^{2} b^{3}}{24 \, {\left ({\left (b x + a\right )}^{3} a^{3} - 3 \, {\left (b x + a\right )}^{2} a^{4} + 3 \, {\left (b x + a\right )} a^{5} - a^{6}\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^4 \sqrt {a+b x}} \, dx=-\frac {\frac {15 \, b^{4} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {15 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{4} - 40 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{4} + 33 \, \sqrt {b x + a} a^{2} b^{4}}{a^{3} b^{3} x^{3}}}{24 \, b} \]
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Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^4 \sqrt {a+b x}} \, dx=\frac {5\,{\left (a+b\,x\right )}^{3/2}}{3\,a^2\,x^3}-\frac {11\,\sqrt {a+b\,x}}{8\,a\,x^3}-\frac {5\,{\left (a+b\,x\right )}^{5/2}}{8\,a^3\,x^3}-\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{8\,a^{7/2}} \]
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