Integrand size = 26, antiderivative size = 77 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5} \, dx=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 14} \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5} \, dx=-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )} \]
[In]
[Out]
Rule 14
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^5} \, dx}{a b+b^2 x^3} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a b}{x^5}+\frac {b^2}{x^2}\right ) \, dx}{a b+b^2 x^3} \\ & = -\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5} \, dx=-\frac {\sqrt {\left (a+b x^3\right )^2} \left (a+4 b x^3\right )}{4 x^4 \left (a+b x^3\right )} \]
[In]
[Out]
Time = 4.52 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.44
method | result | size |
gosper | \(-\frac {\left (4 b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{4 \left (b \,x^{3}+a \right ) x^{4}}\) | \(34\) |
default | \(-\frac {\left (4 b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{4 \left (b \,x^{3}+a \right ) x^{4}}\) | \(34\) |
risch | \(\frac {\left (-b \,x^{3}-\frac {a}{4}\right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}}{x^{4} \left (b \,x^{3}+a \right )}\) | \(35\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5} \, dx=-\frac {4 \, b x^{3} + a}{4 \, x^{4}} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5} \, dx=-\frac {4 \, b x^{3} + a}{4 \, x^{4}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5} \, dx=-\frac {4 \, b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a \mathrm {sgn}\left (b x^{3} + a\right )}{4 \, x^{4}} \]
[In]
[Out]
Time = 8.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5} \, dx=-\frac {\left (4\,b\,x^3+a\right )\,\sqrt {{\left (b\,x^3+a\right )}^2}}{4\,x^4\,\left (b\,x^3+a\right )} \]
[In]
[Out]