Integrand size = 26, antiderivative size = 79 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^{11}} \, dx=-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1125, 660, 45} \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^{11}} \, dx=-\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )} \]
[In]
[Out]
Rule 45
Rule 660
Rule 1125
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{x^6} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \frac {a b+b^2 x}{x^6} \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \left (\frac {a b}{x^6}+\frac {b^2}{x^5}\right ) \, dx,x,x^2\right )}{2 \left (a b+b^2 x^2\right )} \\ & = -\frac {a \sqrt {a^2+2 a b x^2+b^2 x^4}}{10 x^{10} \left (a+b x^2\right )}-\frac {b \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 x^8 \left (a+b x^2\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^{11}} \, dx=-\frac {\sqrt {\left (a+b x^2\right )^2} \left (4 a+5 b x^2\right )}{40 x^{10} \left (a+b x^2\right )} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.30
method | result | size |
pseudoelliptic | \(-\frac {\left (5 b \,x^{2}+4 a \right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{40 x^{10}}\) | \(24\) |
risch | \(\frac {\left (-\frac {b \,x^{2}}{8}-\frac {a}{10}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{x^{10} \left (b \,x^{2}+a \right )}\) | \(35\) |
gosper | \(-\frac {\left (5 b \,x^{2}+4 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{40 x^{10} \left (b \,x^{2}+a \right )}\) | \(36\) |
default | \(-\frac {\left (5 b \,x^{2}+4 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{40 x^{10} \left (b \,x^{2}+a \right )}\) | \(36\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^{11}} \, dx=-\frac {5 \, b x^{2} + 4 \, a}{40 \, x^{10}} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^{11}} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^{11}} \, dx=-\frac {5 \, b x^{2} + 4 \, a}{40 \, x^{10}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^{11}} \, dx=-\frac {5 \, b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 4 \, a \mathrm {sgn}\left (b x^{2} + a\right )}{40 \, x^{10}} \]
[In]
[Out]
Time = 13.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{x^{11}} \, dx=-\frac {\left (5\,b\,x^2+4\,a\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{40\,x^{10}\,\left (b\,x^2+a\right )} \]
[In]
[Out]