Integrand size = 24, antiderivative size = 59 \[ \int \frac {\left (-b+a x^3\right ) \sqrt {b+a x^3}}{x^7} \, dx=\frac {\left (2 b-3 a x^3\right ) \sqrt {b+a x^3}}{12 x^6}-\frac {5 a^2 \text {arctanh}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )}{12 \sqrt {b}} \]
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Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 79, 43, 65, 214} \[ \int \frac {\left (-b+a x^3\right ) \sqrt {b+a x^3}}{x^7} \, dx=-\frac {5 a^2 \text {arctanh}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )}{12 \sqrt {b}}-\frac {5 a \sqrt {a x^3+b}}{12 x^3}+\frac {\left (a x^3+b\right )^{3/2}}{6 x^6} \]
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Rule 43
Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(-b+a x) \sqrt {b+a x}}{x^3} \, dx,x,x^3\right ) \\ & = \frac {\left (b+a x^3\right )^{3/2}}{6 x^6}+\frac {1}{12} (5 a) \text {Subst}\left (\int \frac {\sqrt {b+a x}}{x^2} \, dx,x,x^3\right ) \\ & = -\frac {5 a \sqrt {b+a x^3}}{12 x^3}+\frac {\left (b+a x^3\right )^{3/2}}{6 x^6}+\frac {1}{24} \left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {b+a x}} \, dx,x,x^3\right ) \\ & = -\frac {5 a \sqrt {b+a x^3}}{12 x^3}+\frac {\left (b+a x^3\right )^{3/2}}{6 x^6}+\frac {1}{12} (5 a) \text {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b+a x^3}\right ) \\ & = -\frac {5 a \sqrt {b+a x^3}}{12 x^3}+\frac {\left (b+a x^3\right )^{3/2}}{6 x^6}-\frac {5 a^2 \text {arctanh}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )}{12 \sqrt {b}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-b+a x^3\right ) \sqrt {b+a x^3}}{x^7} \, dx=\frac {\left (2 b-3 a x^3\right ) \sqrt {b+a x^3}}{12 x^6}-\frac {5 a^2 \text {arctanh}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )}{12 \sqrt {b}} \]
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Time = 0.94 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {\sqrt {a \,x^{3}+b}\, \left (3 a \,x^{3}-2 b \right )}{12 x^{6}}-\frac {5 a^{2} \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{12 \sqrt {b}}\) | \(48\) |
elliptic | \(\frac {b \sqrt {a \,x^{3}+b}}{6 x^{6}}-\frac {a \sqrt {a \,x^{3}+b}}{4 x^{3}}-\frac {5 a^{2} \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{12 \sqrt {b}}\) | \(54\) |
pseudoelliptic | \(\frac {-5 \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right ) a^{2} x^{6}-3 a \,x^{3} \sqrt {a \,x^{3}+b}\, \sqrt {b}+2 b^{\frac {3}{2}} \sqrt {a \,x^{3}+b}}{12 x^{6} \sqrt {b}}\) | \(64\) |
default | \(a \left (-\frac {\sqrt {a \,x^{3}+b}}{3 x^{3}}-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3 \sqrt {b}}\right )-b \left (-\frac {\sqrt {a \,x^{3}+b}}{6 x^{6}}-\frac {a \sqrt {a \,x^{3}+b}}{12 b \,x^{3}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{12 b^{\frac {3}{2}}}\right )\) | \(97\) |
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Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.34 \[ \int \frac {\left (-b+a x^3\right ) \sqrt {b+a x^3}}{x^7} \, dx=\left [\frac {5 \, a^{2} \sqrt {b} x^{6} \log \left (\frac {a x^{3} - 2 \, \sqrt {a x^{3} + b} \sqrt {b} + 2 \, b}{x^{3}}\right ) - 2 \, {\left (3 \, a b x^{3} - 2 \, b^{2}\right )} \sqrt {a x^{3} + b}}{24 \, b x^{6}}, \frac {5 \, a^{2} \sqrt {-b} x^{6} \arctan \left (\frac {\sqrt {a x^{3} + b} \sqrt {-b}}{b}\right ) - {\left (3 \, a b x^{3} - 2 \, b^{2}\right )} \sqrt {a x^{3} + b}}{12 \, b x^{6}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (51) = 102\).
Time = 28.78 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.17 \[ \int \frac {\left (-b+a x^3\right ) \sqrt {b+a x^3}}{x^7} \, dx=- \frac {a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}}{3 x^{\frac {3}{2}}} + \frac {a^{\frac {3}{2}}}{12 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} + \frac {\sqrt {a} b}{4 x^{\frac {9}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} - \frac {5 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{12 \sqrt {b}} + \frac {b^{2}}{6 \sqrt {a} x^{\frac {15}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (47) = 94\).
Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.68 \[ \int \frac {\left (-b+a x^3\right ) \sqrt {b+a x^3}}{x^7} \, dx=\frac {1}{6} \, {\left (\frac {a \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right )}{\sqrt {b}} - \frac {2 \, \sqrt {a x^{3} + b}}{x^{3}}\right )} a + \frac {1}{24} \, {\left (\frac {a^{2} \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (a x^{3} + b\right )}^{\frac {3}{2}} a^{2} + \sqrt {a x^{3} + b} a^{2} b\right )}}{{\left (a x^{3} + b\right )}^{2} b - 2 \, {\left (a x^{3} + b\right )} b^{2} + b^{3}}\right )} b \]
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Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.19 \[ \int \frac {\left (-b+a x^3\right ) \sqrt {b+a x^3}}{x^7} \, dx=\frac {\frac {5 \, a^{3} \arctan \left (\frac {\sqrt {a x^{3} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {3 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} a^{3} - 5 \, \sqrt {a x^{3} + b} a^{3} b}{a^{2} x^{6}}}{12 \, a} \]
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Time = 5.89 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.25 \[ \int \frac {\left (-b+a x^3\right ) \sqrt {b+a x^3}}{x^7} \, dx=\frac {b\,\sqrt {a\,x^3+b}}{6\,x^6}-\frac {a\,\sqrt {a\,x^3+b}}{4\,x^3}+\frac {5\,a^2\,\ln \left (\frac {{\left (\sqrt {a\,x^3+b}-\sqrt {b}\right )}^3\,\left (\sqrt {a\,x^3+b}+\sqrt {b}\right )}{x^6}\right )}{24\,\sqrt {b}} \]
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