Integrand size = 15, antiderivative size = 95 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=-\frac {\sqrt {a+b x^3}}{9 x^9}-\frac {b \sqrt {a+b x^3}}{36 a x^6}+\frac {b^2 \sqrt {a+b x^3}}{24 a^2 x^3}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{5/2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 43, 44, 65, 214} \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{5/2}}+\frac {b^2 \sqrt {a+b x^3}}{24 a^2 x^3}-\frac {\sqrt {a+b x^3}}{9 x^9}-\frac {b \sqrt {a+b x^3}}{36 a x^6} \]
[In]
[Out]
Rule 43
Rule 44
Rule 65
Rule 214
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^4} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {a+b x^3}}{9 x^9}+\frac {1}{18} b \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt {a+b x^3}}{9 x^9}-\frac {b \sqrt {a+b x^3}}{36 a x^6}-\frac {b^2 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^3\right )}{24 a} \\ & = -\frac {\sqrt {a+b x^3}}{9 x^9}-\frac {b \sqrt {a+b x^3}}{36 a x^6}+\frac {b^2 \sqrt {a+b x^3}}{24 a^2 x^3}+\frac {b^3 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{48 a^2} \\ & = -\frac {\sqrt {a+b x^3}}{9 x^9}-\frac {b \sqrt {a+b x^3}}{36 a x^6}+\frac {b^2 \sqrt {a+b x^3}}{24 a^2 x^3}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{24 a^2} \\ & = -\frac {\sqrt {a+b x^3}}{9 x^9}-\frac {b \sqrt {a+b x^3}}{36 a x^6}+\frac {b^2 \sqrt {a+b x^3}}{24 a^2 x^3}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{5/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\frac {\sqrt {a+b x^3} \left (-8 a^2-2 a b x^3+3 b^2 x^6\right )}{72 a^2 x^9}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{5/2}} \]
[In]
[Out]
Time = 3.78 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.65
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (-3 b^{2} x^{6}+2 a b \,x^{3}+8 a^{2}\right )}{72 x^{9} a^{2}}-\frac {b^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{24 a^{\frac {5}{2}}}\) | \(62\) |
default | \(-\frac {b^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{24 a^{\frac {5}{2}}}-\frac {\sqrt {b \,x^{3}+a}}{9 x^{9}}-\frac {b \sqrt {b \,x^{3}+a}}{36 a \,x^{6}}+\frac {b^{2} \sqrt {b \,x^{3}+a}}{24 a^{2} x^{3}}\) | \(76\) |
elliptic | \(-\frac {b^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{24 a^{\frac {5}{2}}}-\frac {\sqrt {b \,x^{3}+a}}{9 x^{9}}-\frac {b \sqrt {b \,x^{3}+a}}{36 a \,x^{6}}+\frac {b^{2} \sqrt {b \,x^{3}+a}}{24 a^{2} x^{3}}\) | \(76\) |
pseudoelliptic | \(\frac {-3 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right ) b^{3} x^{9}+3 b^{2} x^{6} \sqrt {b \,x^{3}+a}\, \sqrt {a}-2 a^{\frac {3}{2}} b \,x^{3} \sqrt {b \,x^{3}+a}-8 a^{\frac {5}{2}} \sqrt {b \,x^{3}+a}}{72 a^{\frac {5}{2}} x^{9}}\) | \(84\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\left [\frac {3 \, \sqrt {a} b^{3} x^{9} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (3 \, a b^{2} x^{6} - 2 \, a^{2} b x^{3} - 8 \, a^{3}\right )} \sqrt {b x^{3} + a}}{144 \, a^{3} x^{9}}, \frac {3 \, \sqrt {-a} b^{3} x^{9} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b^{2} x^{6} - 2 \, a^{2} b x^{3} - 8 \, a^{3}\right )} \sqrt {b x^{3} + a}}{72 \, a^{3} x^{9}}\right ] \]
[In]
[Out]
Time = 4.80 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=- \frac {a}{9 \sqrt {b} x^{\frac {21}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {5 \sqrt {b}}{36 x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {b^{\frac {3}{2}}}{72 a x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {b^{\frac {5}{2}}}{24 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{24 a^{\frac {5}{2}}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\frac {b^{3} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{48 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} b^{3} - 8 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a b^{3} - 3 \, \sqrt {b x^{3} + a} a^{2} b^{3}}{72 \, {\left ({\left (b x^{3} + a\right )}^{3} a^{2} - 3 \, {\left (b x^{3} + a\right )}^{2} a^{3} + 3 \, {\left (b x^{3} + a\right )} a^{4} - a^{5}\right )}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\frac {\frac {3 \, b^{4} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} b^{4} - 8 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a b^{4} - 3 \, \sqrt {b x^{3} + a} a^{2} b^{4}}{a^{2} b^{3} x^{9}}}{72 \, b} \]
[In]
[Out]
Time = 5.93 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\frac {b^3\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{48\,a^{5/2}}-\frac {\sqrt {b\,x^3+a}}{9\,x^9}-\frac {b\,\sqrt {b\,x^3+a}}{36\,a\,x^6}+\frac {b^2\,\sqrt {b\,x^3+a}}{24\,a^2\,x^3} \]
[In]
[Out]